2022
Moreno, Andrés J.; Earp, Henrique N. Sá
The Weitzenbock formula for the Fueter–Dirac operator Journal Article
Em: Communications in Analysis and Geometry, vol. 30, iss. 1, pp. 153 – 205, 2022.
Resumo | Links | BibTeX | Tags:
@article{nokey,
title = {The Weitzenbock formula for the Fueter\textendashDirac operator},
author = {Andr\'{e}s J. Moreno and Henrique N. S\'{a} Earp},
url = {https://www.intlpress.com/site/pub/pages/journals/items/cag/content/vols/0030/0001/a003/},
doi = {http://dx.doi.org/10.4310/CAG.2022.v30.n1.a3},
year = {2022},
date = {2022-07-22},
journal = {Communications in Analysis and Geometry},
volume = {30},
issue = {1},
pages = {153 \textendash 205},
abstract = {We find a Weitzenb\"{o}ck formula for the Fueter\textendashDirac operator which controls infinitesimal deformations of an associative submanifold in a \textendashmanifold with a -structure. We establish a vanishing theorem to conclude rigidity under some positivity assumptions on curvature, which are particularly mild in the nearly parallel case. As applications, we find a different proof of rigidity for one of Lotay’s associatives in the round -sphere from those given by Kawai [14, 15]. We also provide simpler proofs of previous results by Gayet for the Bryant-Salamon metric [11]. Finally, we obtain an original example of a rigid associative in a compact manifold with locally conformal calibrated -structure obtained by Fern\'{a}ndez\textendashFino\textendashRaffero [9].},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We find a Weitzenböck formula for the Fueter–Dirac operator which controls infinitesimal deformations of an associative submanifold in a –manifold with a -structure. We establish a vanishing theorem to conclude rigidity under some positivity assumptions on curvature, which are particularly mild in the nearly parallel case. As applications, we find a different proof of rigidity for one of Lotay’s associatives in the round -sphere from those given by Kawai [14, 15]. We also provide simpler proofs of previous results by Gayet for the Bryant-Salamon metric [11]. Finally, we obtain an original example of a rigid associative in a compact manifold with locally conformal calibrated -structure obtained by Fernández–Fino–Raffero [9].
Loubeau, Eric; Moreno, Andrés J.; Earp, Henrique N. Sá; Saavedra, Julieth
Harmonic Sp(2)-Invariant G2-Structures on the 7-Sphere Journal Article
Em: The Journal of Geometric Analysis, vol. 32, iss. 240, 2022.
Resumo | Links | BibTeX | Tags: 7-Sphere, G2-Structures
@article{nokey,
title = {Harmonic Sp(2)-Invariant G2-Structures on the 7-Sphere},
author = {Eric Loubeau and Andr\'{e}s J. Moreno and Henrique N. S\'{a} Earp and Julieth Saavedra },
url = {http://dx.doi.org/10.1007/s12220-022-00953-9},
doi = {http://dx.doi.org/10.1007/s12220-022-00953-9},
year = {2022},
date = {2022-07-18},
journal = {The Journal of Geometric Analysis},
volume = {32},
issue = {240},
abstract = {We describe the 10-dimensional space of Sp(2)-invariant G2-structures on the homogeneous 7-sphere S7=Sp(2)/Sp(1) as Ω3+(S7)Sp(2)≃R+×Gl+(3,R). In those terms, we formulate a general Ansatz for G2-structures, which realises representatives in each of the 7 possible isometric classes of homogeneous G2-structures. Moreover, the well-known nearly parallel round and squashed metrics occur naturally as opposite poles in an S3-family, the equator of which is a new S2-family of coclosed G2-structures satisfying the harmonicity condition divT=0. We show general existence of harmonic representatives of G2-structures in each isometric class through explicit solutions of the associated flow and describe the qualitative behaviour of the flow. We study the stability of the Dirichlet gradient flow near these critical points, showing explicit examples of degenerate and nondegenerate local maxima and minima, at various regimes of the general Ansatz. Finally, for metrics outside of the Ansatz, we identify families of harmonic G2-structures, prove long-time existence of the flow and study the stability properties of some well-chosen examples.},
keywords = {7-Sphere, G2-Structures},
pubstate = {published},
tppubtype = {article}
}
We describe the 10-dimensional space of Sp(2)-invariant G2-structures on the homogeneous 7-sphere S7=Sp(2)/Sp(1) as Ω3+(S7)Sp(2)≃R+×Gl+(3,R). In those terms, we formulate a general Ansatz for G2-structures, which realises representatives in each of the 7 possible isometric classes of homogeneous G2-structures. Moreover, the well-known nearly parallel round and squashed metrics occur naturally as opposite poles in an S3-family, the equator of which is a new S2-family of coclosed G2-structures satisfying the harmonicity condition divT=0. We show general existence of harmonic representatives of G2-structures in each isometric class through explicit solutions of the associated flow and describe the qualitative behaviour of the flow. We study the stability of the Dirichlet gradient flow near these critical points, showing explicit examples of degenerate and nondegenerate local maxima and minima, at various regimes of the general Ansatz. Finally, for metrics outside of the Ansatz, we identify families of harmonic G2-structures, prove long-time existence of the flow and study the stability properties of some well-chosen examples.
Lotay, Jason D.; Earp, Henrique N. Sá; Saavedra, Julieth
Flows of G2-structures on contact Calabi–Yau 7-manifolds Journal Article
Em: Annals of Global Analysis and Geometry, vol. 62, pp. 367–389, 2022.
Resumo | Links | BibTeX | Tags: Calabi–Yau, G2-Structures
@article{nokey,
title = {Flows of G2-structures on contact Calabi\textendashYau 7-manifolds},
author = {Jason D. Lotay and Henrique N. S\'{a} Earp and Julieth Saavedra },
url = {https://link.springer.com/article/10.1007/s10455-022-09854-0},
doi = {http://dx.doi.org/10.1007/s10455-022-09854-0},
year = {2022},
date = {2022-06-21},
journal = {Annals of Global Analysis and Geometry},
volume = {62},
pages = {367\textendash389},
abstract = {We study the Laplacian flow and coflow on contact Calabi\textendashYau 7-manifolds. We show that the natural initial condition leads to an ancient solution of the Laplacian flow with a finite time Type I singularity which is not a soliton, whereas it produces an immortal (though neither eternal nor self-similar) solution of the Laplacian coflow which has an infinite time singularity of Type IIb, unless the transverse Calabi\textendashYau geometry is flat. The flows in each case collapse (under normalised volume) to a lower-dimensional limit, which is either R, for the Laplacian flow, or standard C3, for the Laplacian coflow. We also study the Hitchin flow in this setting, which we show coincides with the Laplacian coflow, up to reparametrisation of time, and defines an (incomplete) Calabi\textendashYau structure on the spacetime track of the flow.},
keywords = {Calabi\textendashYau, G2-Structures},
pubstate = {published},
tppubtype = {article}
}
We study the Laplacian flow and coflow on contact Calabi–Yau 7-manifolds. We show that the natural initial condition leads to an ancient solution of the Laplacian flow with a finite time Type I singularity which is not a soliton, whereas it produces an immortal (though neither eternal nor self-similar) solution of the Laplacian coflow which has an infinite time singularity of Type IIb, unless the transverse Calabi–Yau geometry is flat. The flows in each case collapse (under normalised volume) to a lower-dimensional limit, which is either R, for the Laplacian flow, or standard C3, for the Laplacian coflow. We also study the Hitchin flow in this setting, which we show coincides with the Laplacian coflow, up to reparametrisation of time, and defines an (incomplete) Calabi–Yau structure on the spacetime track of the flow.
Almeida, Charles; Jardim, Marcos; S.Tikhomirov, Alexander
Irreducible components of the moduli space of rank 2 sheaves of odd determinant on projective space Journal Article
Em: Advances in Mathematics, vol. 402, 2022.
Resumo | Links | BibTeX | Tags:
@article{nokey,
title = {Irreducible components of the moduli space of rank 2 sheaves of odd determinant on projective space},
author = {Charles Almeida and Marcos Jardim and Alexander S.Tikhomirov},
url = {https://www.sciencedirect.com/science/article/abs/pii/S0001870822001797?via%3Dihub},
doi = {https://doi.org/10.1016/j.aim.2022.108363},
year = {2022},
date = {2022-03-29},
journal = {Advances in Mathematics},
volume = {402},
abstract = {We describe new irreducible components of the moduli space of rank 2 semistable torsion free sheaves on the three-dimensional projective space whose generic point corresponds to non-locally free sheaves whose singular locus is either 0-dimensional or consists of a line plus disjoint points. In particular, we prove that the moduli spaces of semistable sheaves with Chern classes and always contain at least one rational irreducible component. As an application, we prove that the number of such components grows as the second Chern class grows, and compute the exact number of irreducible components of the moduli spaces of rank 2 semistable torsion free sheaves with Chern classes for non negative values for m; all components turn out to be rational. Furthermore, we also prove that these moduli spaces are connected, showing that some of sheaves here considered are smoothable.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We describe new irreducible components of the moduli space of rank 2 semistable torsion free sheaves on the three-dimensional projective space whose generic point corresponds to non-locally free sheaves whose singular locus is either 0-dimensional or consists of a line plus disjoint points. In particular, we prove that the moduli spaces of semistable sheaves with Chern classes and always contain at least one rational irreducible component. As an application, we prove that the number of such components grows as the second Chern class grows, and compute the exact number of irreducible components of the moduli spaces of rank 2 semistable torsion free sheaves with Chern classes for non negative values for m; all components turn out to be rational. Furthermore, we also prove that these moduli spaces are connected, showing that some of sheaves here considered are smoothable.
Grajales, Brian; Grama, Lino
Invariant Einstein metrics on real flag manifolds with two or three isotropy summands Journal Article
Em: Journal of Geometry and Physics, vol. 176, 2022.
Resumo | Links | BibTeX | Tags:
@article{nokey,
title = {Invariant Einstein metrics on real flag manifolds with two or three isotropy summands},
author = {Brian Grajales and Lino Grama},
url = {https://www.sciencedirect.com/science/article/pii/S0393044022000444?via%3Dihub},
doi = {https://doi.org/10.1016/j.geomphys.2022.104494},
year = {2022},
date = {2022-03-22},
journal = {Journal of Geometry and Physics},
volume = {176},
abstract = {We study the existence of invariant Einstein metrics on real flag manifolds associated to simple and non-compact split real forms of complex classical Lie algebras whose isotropy representation decomposes into two or three irreducible subrepresentations. In this situation, one can have equivalent submodules, leading to the existence of non-diagonal homogeneous Riemannian metrics. In particular, we prove the existence of non-diagonal Einstein metrics on real flag manifolds.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We study the existence of invariant Einstein metrics on real flag manifolds associated to simple and non-compact split real forms of complex classical Lie algebras whose isotropy representation decomposes into two or three irreducible subrepresentations. In this situation, one can have equivalent submodules, leading to the existence of non-diagonal homogeneous Riemannian metrics. In particular, we prove the existence of non-diagonal Einstein metrics on real flag manifolds.
Corrêa, Maurício; Jardim, Marcos; Muniz, Alan
Moduli of distributions via singular schemes Journal Article
Em: Mathematische Zeitschrift volume, vol. 301, pp. 2709–2731, 2022.
Resumo | Links | BibTeX | Tags: Hilbert scheme, Syzygy
@article{nokey,
title = {Moduli of distributions via singular schemes},
author = {Maur\'{i}cio Corr\^{e}a and Marcos Jardim and Alan Muniz },
url = {https://link.springer.com/article/10.1007/s00209-022-03001-y},
doi = {https://doi.org/10.1007/s00209-022-03001-y},
year = {2022},
date = {2022-03-05},
journal = {Mathematische Zeitschrift volume},
volume = {301},
pages = {2709\textendash2731},
abstract = {Let X be a smooth projective variety. We show that the map that sends a codimension one distribution on X to its singular scheme is a morphism from the moduli space of distributions into a Hilbert scheme. We describe its fibers and, when X=Pn, compute them via syzygies. As an application, we describe the moduli spaces of degree 1 distributions on P3. We also give the minimal graded free resolution for the ideal of the singular scheme of a generic distribution on P3.},
keywords = {Hilbert scheme, Syzygy},
pubstate = {published},
tppubtype = {article}
}
Let X be a smooth projective variety. We show that the map that sends a codimension one distribution on X to its singular scheme is a morphism from the moduli space of distributions into a Hilbert scheme. We describe its fibers and, when X=Pn, compute them via syzygies. As an application, we describe the moduli spaces of degree 1 distributions on P3. We also give the minimal graded free resolution for the ideal of the singular scheme of a generic distribution on P3.
Franco, Emilio; Jardim, Marcos
Mirror symmetry for Nahm branes Journal Article
Em: Épijournal de Géométrie Algébrique, vol. 6, 2022.
Resumo | Links | BibTeX | Tags:
@article{nokey,
title = {Mirror symmetry for Nahm branes},
author = {Emilio Franco and Marcos Jardim},
url = {https://epiga.episciences.org/9150},
doi = {https://doi.org/10.46298/epiga.2022.6604},
year = {2022},
date = {2022-03-01},
journal = {\'{E}pijournal de G\'{e}om\'{e}trie Alg\'{e}brique},
volume = {6},
abstract = {The Dirac--Higgs bundle is a hyperholomorphic bundle over the moduli space of stable Higgs bundles of coprime rank and degree. We provide an algebraic generalization to the case of trivial degree and the rank higher than 1. This allow us to generalize to this case the Nahm transform defined by Frejlich and the second named author, which, out of a stable Higgs bundle, produces a vector bundle with connection over the moduli space of rank 1 Higgs bundles. By performing the higher rank Nahm transform we obtain a hyperholomorphic bundle with connection over the moduli space of stable Higgs bundles of rank n and degree 0, twisted by the gerbe of liftings of the projective universal bundle. Such hyperholomorphic vector bundles over the moduli space of stable Higgs bundles can be seen, in the physicist's language, as BBB-branes twisted by the above mentioned gerbe. We refer to these objects as Nahm branes. Finally, we study the behaviour of Nahm branes under Fourier--Mukai transform over the smooth locus of the Hitchin fibration, checking that the resulting objects are supported on a Lagrangian multisection of the Hitchin fibration, so they describe partial data of BAA-branes.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
The Dirac--Higgs bundle is a hyperholomorphic bundle over the moduli space of stable Higgs bundles of coprime rank and degree. We provide an algebraic generalization to the case of trivial degree and the rank higher than 1. This allow us to generalize to this case the Nahm transform defined by Frejlich and the second named author, which, out of a stable Higgs bundle, produces a vector bundle with connection over the moduli space of rank 1 Higgs bundles. By performing the higher rank Nahm transform we obtain a hyperholomorphic bundle with connection over the moduli space of stable Higgs bundles of rank n and degree 0, twisted by the gerbe of liftings of the projective universal bundle. Such hyperholomorphic vector bundles over the moduli space of stable Higgs bundles can be seen, in the physicist's language, as BBB-branes twisted by the above mentioned gerbe. We refer to these objects as Nahm branes. Finally, we study the behaviour of Nahm branes under Fourier--Mukai transform over the smooth locus of the Hitchin fibration, checking that the resulting objects are supported on a Lagrangian multisection of the Hitchin fibration, so they describe partial data of BAA-branes.
Correa, Eder M.
Homogeneous Contact Manifolds and Resolutions of Calabi–Yau Cones Journal Article
Em: Communications in Mathematical Physics, vol. 367, pp. 1095–1151, 2022.
Resumo | Links | BibTeX | Tags: Calabi–Yau
@article{nokey,
title = {Homogeneous Contact Manifolds and Resolutions of Calabi\textendashYau Cones},
author = {Eder M. Correa},
url = {https://link.springer.com/article/10.1007/s00220-019-03337-3},
doi = {https://doi.org/10.1007/s00220-019-03337-3},
year = {2022},
date = {2022-02-09},
journal = {Communications in Mathematical Physics},
volume = {367},
pages = {1095\textendash1151},
abstract = {In the present work we provide a constructive method to describe contact structures on compact homogeneous contact manifolds. The main feature of our approach is to describe the Cartan\textendashEhresmann connection (gauge field) for principal U(1)-bundles over complex flag manifolds by using elements of representation theory of simple Lie algebras. This description allows us to compute explicitly the expression of the contact form for any Boothby\textendashWang fibration over complex flag manifolds (Boothby and Wang in Ann Math 68:721\textendash734, 1958) as well as their underlying Sasaki structures. By following Conlon and Hein (Duke Math J 162:2855\textendash2902, 2013), Van Coevering (Math Ann, 2009. https://doi.org/10.1007/s00208-009-0446-1) and Goto (J Math Soc Jpn 64:1005\textendash1052, 2012), as an application of our results we use the Cartan\textendashRemmert reduction (Grauert in Math Ann 146:331\textendash368, 1962) and the Calabi Ansatz technique (Calabi in Ann Sci \'{E}cole Norm Sup (4) 12:269\textendash294, 1979) to provide many explicit examples of crepant resolutions of Calabi\textendashYau cones with certain homogeneous Sasaki\textendashEinstein manifolds realized as links of isolated singularities. These concrete examples illustrate the existence part of the conjecture introduced in Martelli and Sparks (Phys Rev D 79(6):065009, 2009).},
keywords = {Calabi\textendashYau},
pubstate = {published},
tppubtype = {article}
}
In the present work we provide a constructive method to describe contact structures on compact homogeneous contact manifolds. The main feature of our approach is to describe the Cartan–Ehresmann connection (gauge field) for principal U(1)-bundles over complex flag manifolds by using elements of representation theory of simple Lie algebras. This description allows us to compute explicitly the expression of the contact form for any Boothby–Wang fibration over complex flag manifolds (Boothby and Wang in Ann Math 68:721–734, 1958) as well as their underlying Sasaki structures. By following Conlon and Hein (Duke Math J 162:2855–2902, 2013), Van Coevering (Math Ann, 2009. https://doi.org/10.1007/s00208-009-0446-1) and Goto (J Math Soc Jpn 64:1005–1052, 2012), as an application of our results we use the Cartan–Remmert reduction (Grauert in Math Ann 146:331–368, 1962) and the Calabi Ansatz technique (Calabi in Ann Sci École Norm Sup (4) 12:269–294, 1979) to provide many explicit examples of crepant resolutions of Calabi–Yau cones with certain homogeneous Sasaki–Einstein manifolds realized as links of isolated singularities. These concrete examples illustrate the existence part of the conjecture introduced in Martelli and Sparks (Phys Rev D 79(6):065009, 2009).
2021
Grama, Lino; Oliveira, Ailton R.
Scalar Curvatures of Invariant Almost Hermitian Structures on Generalized Flag Manifolds Journal Article
Em: SIGMA, vol. 17, 2021.
Resumo | Links | BibTeX | Tags:
@article{nokey,
title = {Scalar Curvatures of Invariant Almost Hermitian Structures on Generalized Flag Manifolds},
author = {Lino Grama and Ailton R. Oliveira},
url = {https://www.emis.de/journals/SIGMA/2021/109/},
doi = {https://doi.org/10.3842/SIGMA.2021.109},
year = {2021},
date = {2021-12-21},
journal = {SIGMA},
volume = {17},
abstract = {In this paper we study invariant almost Hermitian geometry on generalized flag manifolds. We will focus on providing examples of K\"{a}hler like scalar curvature metric, that is, almost Hermitian structures (g,J) satisfying s=2sC, where s is Riemannian scalar curvature and sC is the Chern scalar curvature.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
In this paper we study invariant almost Hermitian geometry on generalized flag manifolds. We will focus on providing examples of Kähler like scalar curvature metric, that is, almost Hermitian structures (g,J) satisfying s=2sC, where s is Riemannian scalar curvature and sC is the Chern scalar curvature.
Miyamoto, Henrique K.; Costa, Sueli I. R.; Earp, Henrique N. Sá
Constructive Spherical Codes by Hopf Foliations Journal Article
Em: IEEE Transactions on Information Theory, vol. 67, iss. 12, pp. 7925 - 7939, 2021.
Resumo | Links | BibTeX | Tags:
@article{nokey,
title = {Constructive Spherical Codes by Hopf Foliations},
author = {Henrique K. Miyamoto and Sueli I. R. Costa and Henrique N. S\'{a} Earp},
url = {https://ieeexplore.ieee.org/document/9541191},
doi = {10.1109/TIT.2021.3114094},
year = {2021},
date = {2021-09-20},
journal = {IEEE Transactions on Information Theory},
volume = {67},
issue = {12},
pages = {7925 - 7939},
abstract = {We present a new systematic approach to constructing spherical codes in dimensions 2k , based on Hopf foliations. Using the fact that a sphere S2n−1 is foliated by manifolds Sn−1cosη×Sn−1sinη , η∈[0,π/2] , we distribute points in dimension 2k via a recursive algorithm from a basic construction in R4 . Our procedure outperforms some current constructive methods in several small-distance regimes and constitutes a compromise between achieving a large number of codewords for a minimum given distance and effective constructiveness with low encoding computational cost. Bounds for the asymptotic density are derived and compared with other constructions. The encoding process has storage complexity O(n) and time complexity O(nlogn) . We also propose a sub-optimal decoding procedure, which does not require storing the codebook and has time complexity O(nlogn) .},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We present a new systematic approach to constructing spherical codes in dimensions 2k , based on Hopf foliations. Using the fact that a sphere S2n−1 is foliated by manifolds Sn−1cosη×Sn−1sinη , η∈[0,π/2] , we distribute points in dimension 2k via a recursive algorithm from a basic construction in R4 . Our procedure outperforms some current constructive methods in several small-distance regimes and constitutes a compromise between achieving a large number of codewords for a minimum given distance and effective constructiveness with low encoding computational cost. Bounds for the asymptotic density are derived and compared with other constructions. The encoding process has storage complexity O(n) and time complexity O(nlogn) . We also propose a sub-optimal decoding procedure, which does not require storing the codebook and has time complexity O(nlogn) .
CALVO-ANDRADE, OMEGAR; CORRÊA, MAURÍCIO; JARDIM, MARCOS
Codimension one distributions and stable rank 2 reflexive sheaves on threefolds Journal Article
Em: An. Acad. Bras. Ciênc., vol. 93, iss. 3, 2021.
Resumo | Links | BibTeX | Tags:
@article{nokey,
title = {Codimension one distributions and stable rank 2 reflexive sheaves on threefolds},
author = {OMEGAR CALVO-ANDRADE and MAUR\'{I}CIO CORR\^{E}A and MARCOS JARDIM},
url = {https://www.scielo.br/j/aabc/a/g9ZNjMFqKBkYCVQbDjzJH5x/?lang=en},
doi = {https://doi.org/10.1590/0001-3765202120190909},
year = {2021},
date = {2021-08-30},
journal = {An. Acad. Bras. Ci\^{e}nc.},
volume = {93},
issue = {3},
abstract = {We show that codimension one distributions with at most isolated singularities on certain smooth projective threefolds with Picard rank one have stable tangent sheaves. The ideas in the proof of this fact are then applied to the characterization of certain irreducible components of the moduli space of stable rank 2 reflexive sheaves on $p3$, and to the construction of stable rank 2 reflexive sheaves with prescribed Chern classes on general threefolds. We also prove that if $sG$ is a subfoliation of a codimension one distribution $sF$ with isolated singularities, then $sing(sG)$ is a curve. As a consequence, we give a criterion to decide whether $sG$ is globally given as the intersection of $sF$ with another codimension one distribution. Turning our attention to codimension one distributions with non isolated singularities, we determine the number of connected components of the pure 1-dimensional component of the singular scheme.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We show that codimension one distributions with at most isolated singularities on certain smooth projective threefolds with Picard rank one have stable tangent sheaves. The ideas in the proof of this fact are then applied to the characterization of certain irreducible components of the moduli space of stable rank 2 reflexive sheaves on $p3$, and to the construction of stable rank 2 reflexive sheaves with prescribed Chern classes on general threefolds. We also prove that if $sG$ is a subfoliation of a codimension one distribution $sF$ with isolated singularities, then $sing(sG)$ is a curve. As a consequence, we give a criterion to decide whether $sG$ is globally given as the intersection of $sF$ with another codimension one distribution. Turning our attention to codimension one distributions with non isolated singularities, we determine the number of connected components of the pure 1-dimensional component of the singular scheme.
Galeano, Hugo; Jardim, Marcos; Muniz, Alan
Codimension one distributions of degree 2 on the three-dimensional projective space Journal Article
Em: Journal of Pure and Applied Algebra, vol. 226, iss. 2, 2021.
Resumo | Links | BibTeX | Tags:
@article{nokey,
title = {Codimension one distributions of degree 2 on the three-dimensional projective space},
author = {Hugo Galeano and Marcos Jardim and Alan Muniz},
url = {https://www.sciencedirect.com/science/article/abs/pii/S002240492100181X?via%3Dihub},
doi = {https://doi.org/10.1016/j.jpaa.2021.106840},
year = {2021},
date = {2021-07-19},
journal = {Journal of Pure and Applied Algebra},
volume = {226},
issue = {2},
abstract = {We establish a full classification of degree 2 codimension one distributions on according to invariants of their tangent sheaves.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We establish a full classification of degree 2 codimension one distributions on according to invariants of their tangent sheaves.
del Barco, Viviana; Moroianu, Andrei
Conformal Killing forms on 2-step nilpotent Riemannian Lie groups Journal Article
Em: Forum Mathematicum, vol. 33, iss. 5, 2021.
Resumo | Links | BibTeX | Tags: Conformal Killing forms, Riemannian Lie groups
@article{nokey,
title = {Conformal Killing forms on 2-step nilpotent Riemannian Lie groups},
author = {Viviana del Barco and Andrei Moroianu},
url = {https://www.degruyter.com/document/doi/10.1515/forum-2021-0026/html},
doi = {https://doi.org/10.1515/forum-2021-0026},
year = {2021},
date = {2021-07-18},
journal = {Forum Mathematicum},
volume = {33},
issue = {5},
abstract = {We study left-invariant conformal Killing 2- or 3-forms on simply connected 2-step nilpotent Riemannian Lie groups. We show that if the center of the group is of dimension greater than or equal to 4, then every such form is automatically coclosed (i.e. it is a Killing form). In addition, we prove that the only Riemannian 2-step nilpotent Lie groups with center of dimension at most 3 and admitting left-invariant non-coclosed conformal Killing 2- and 3-forms are the following: The Heisenberg Lie groups and their trivial 1-dimensional extensions, endowed with any left-invariant metric, and the simply connected Lie group corresponding to the free 2-step nilpotent Lie algebra on 3 generators, with a particular 1-parameter family of metrics. The explicit description of the space of conformal Killing 2- and 3-forms is provided in each case.},
keywords = {Conformal Killing forms, Riemannian Lie groups},
pubstate = {published},
tppubtype = {article}
}
We study left-invariant conformal Killing 2- or 3-forms on simply connected 2-step nilpotent Riemannian Lie groups. We show that if the center of the group is of dimension greater than or equal to 4, then every such form is automatically coclosed (i.e. it is a Killing form). In addition, we prove that the only Riemannian 2-step nilpotent Lie groups with center of dimension at most 3 and admitting left-invariant non-coclosed conformal Killing 2- and 3-forms are the following: The Heisenberg Lie groups and their trivial 1-dimensional extensions, endowed with any left-invariant metric, and the simply connected Lie group corresponding to the free 2-step nilpotent Lie algebra on 3 generators, with a particular 1-parameter family of metrics. The explicit description of the space of conformal Killing 2- and 3-forms is provided in each case.
do Prado, Rafaela F.; Grama, Lino
On the stability of harmonic maps under the homogeneous Ricci flow Journal Article
Em: Complex Manifolds, vol. 5, pp. 122–132, 2021.
Resumo | Links | BibTeX | Tags: Ricci flow
@article{nokey,
title = {On the stability of harmonic maps under the homogeneous Ricci flow},
author = {Rafaela F. do Prado and Lino Grama},
url = {https://www.degruyter.com/document/doi/10.1515/coma-2018-0007/html},
doi = {https://doi.org/10.1515/coma-2018-0007},
year = {2021},
date = {2021-06-21},
journal = {Complex Manifolds},
volume = {5},
pages = {122\textendash132},
abstract = {In this work we study properties of stability and non-stability of harmonic maps under the homo-geneous Ricci ow. We provide examples where the stability (non-stability) is preserved under the Ricci owand an example where the Ricci ow does not preserve the stability of an harmonic map.},
keywords = {Ricci flow},
pubstate = {published},
tppubtype = {article}
}
In this work we study properties of stability and non-stability of harmonic maps under the homo-geneous Ricci ow. We provide examples where the stability (non-stability) is preserved under the Ricci owand an example where the Ricci ow does not preserve the stability of an harmonic map.
Correa, Eder M.; Grama, Lino
Lax formalism for Gelfand-Tsetlin integrable systems Journal Article
Em: Bulletin des Sciences Mathématiques, vol. 170, 2021.
Resumo | Links | BibTeX | Tags: Gelfand-Tsetlin
@article{nokey,
title = {Lax formalism for Gelfand-Tsetlin integrable systems},
author = {Eder M. Correa and Lino Grama},
url = {https://www.sciencedirect.com/science/article/abs/pii/S0007449721000555?via%3Dihub},
doi = {https://doi.org/10.1016/j.bulsci.2021.102999},
year = {2021},
date = {2021-05-25},
journal = {Bulletin des Sciences Math\'{e}matiques},
volume = {170},
abstract = {In the present work, we study Hamiltonian systems on (co)adjoint orbits and propose a Lax pair formalism for Gelfand-Tsetlin integrable systems defined on (co)adjoint orbits of the compact Lie groups and . In the particular setting of (co)adjoint orbits of , by means of the associated Lax matrix we construct a family of algebraic curves which encodes the Gelfand-Tsetlin integrable systems as branch points. This family of algebraic curves enables us to explore some new insights into the relationship between the topology of singular Gelfand-Tsetlin fibers, singular algebraic curves and vanishing cycles. Further, we provide a new description for Guillemin and Sternberg's action coordinates in terms of hyperelliptic integrals.},
keywords = {Gelfand-Tsetlin},
pubstate = {published},
tppubtype = {article}
}
In the present work, we study Hamiltonian systems on (co)adjoint orbits and propose a Lax pair formalism for Gelfand-Tsetlin integrable systems defined on (co)adjoint orbits of the compact Lie groups and . In the particular setting of (co)adjoint orbits of , by means of the associated Lax matrix we construct a family of algebraic curves which encodes the Gelfand-Tsetlin integrable systems as branch points. This family of algebraic curves enables us to explore some new insights into the relationship between the topology of singular Gelfand-Tsetlin fibers, singular algebraic curves and vanishing cycles. Further, we provide a new description for Guillemin and Sternberg's action coordinates in terms of hyperelliptic integrals.
Menet, Grégoire; Nordström, Johannes; Earp, Henrique N. Sá
Construction of G2-instantons via twisted connected sums Journal Article
Em: Mathematical Research Letters, vol. 28, iss. 2, pp. 471 – 509, 2021.
Resumo | Links | BibTeX | Tags:
@article{nokey,
title = {Construction of G2-instantons via twisted connected sums},
author = {Gr\'{e}goire Menet and Johannes Nordstr\"{o}m and Henrique N. S\'{a} Earp},
url = {https://www.intlpress.com/site/pub/pages/journals/items/mrl/content/vols/0028/0002/a006/},
doi = {http://dx.doi.org/10.4310/MRL.2021.v28.n2.a6},
year = {2021},
date = {2021-05-13},
journal = {Mathematical Research Letters},
volume = {28},
issue = {2},
pages = {471 \textendash 509},
abstract = {We propose a method to construct G2\textendashinstantons over a compact twisted connected sum G2\textendashmanifold, applying a gluing result of S\'{a} Earp and Walpuski to instantons over a pair of 7-manifolds with a tubular end. In our example, the moduli spaces of the ingredient instantons are non-trivial, and their images in the moduli space over the asymptotic cross-section K3 surface intersect transversely. Such a pair of asymptotically stable holomorphic bundles is obtained using a twisted version of the Hartshorne\textendashSerre construction, which can be adapted to produce other examples. Moreover, their deformation theory and asymptotic behaviour are explicitly understood, results which may be of independent interest.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We propose a method to construct G2–instantons over a compact twisted connected sum G2–manifold, applying a gluing result of Sá Earp and Walpuski to instantons over a pair of 7-manifolds with a tubular end. In our example, the moduli spaces of the ingredient instantons are non-trivial, and their images in the moduli space over the asymptotic cross-section K3 surface intersect transversely. Such a pair of asymptotically stable holomorphic bundles is obtained using a twisted version of the Hartshorne–Serre construction, which can be adapted to produce other examples. Moreover, their deformation theory and asymptotic behaviour are explicitly understood, results which may be of independent interest.
Flach, Rodrigo A. von; Jardim, Marcos; Lanza, Valeriano
Obstruction theory for moduli spaces of framed flags of sheaves on the projective plane Journal Article
Em: Journal of Geometry and Physics, vol. 166, 2021.
Resumo | Links | BibTeX | Tags:
@article{nokey,
title = {Obstruction theory for moduli spaces of framed flags of sheaves on the projective plane},
author = {Rodrigo A.von Flach and Marcos Jardim and Valeriano Lanza},
url = {https://www.sciencedirect.com/science/article/pii/S0393044021001017?via%3Dihub},
doi = {https://doi.org/10.1016/j.geomphys.2021.104255},
year = {2021},
date = {2021-04-26},
journal = {Journal of Geometry and Physics},
volume = {166},
abstract = {In a previous paper, the first two named authors established an isomorphism between the moduli space of framed flags of sheaves on the projective plane and the moduli space of stable representations of a certain quiver. In the present note, we substitute one of the claims made, namely [5, Theorem 17], for a weaker claim regarding the existence of unobstructed points in the quiver moduli space. We also extend some of the results of the cited paper, concerning the maximal stability chamber within which the isomorphism mentioned holds, and the existence of a perfect obstruction theory for the quiver moduli space.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
In a previous paper, the first two named authors established an isomorphism between the moduli space of framed flags of sheaves on the projective plane and the moduli space of stable representations of a certain quiver. In the present note, we substitute one of the claims made, namely [5, Theorem 17], for a weaker claim regarding the existence of unobstructed points in the quiver moduli space. We also extend some of the results of the cited paper, concerning the maximal stability chamber within which the isomorphism mentioned holds, and the existence of a perfect obstruction theory for the quiver moduli space.
del Barco, Viviana; Moroianu, Andrei; Raffero, Alberto
Purely coclosed G2-structures on 2-step nilpotent Lie groups Journal Article
Em: Revista Matemática Complutense, vol. 35, pp. 323–359, 2021.
Resumo | Links | BibTeX | Tags: 2-Step nilpotent Lie algebra, G2-Strominger system, Metric Lie algebra, Purely coclosed G2-structure
@article{nokey,
title = {Purely coclosed G2-structures on 2-step nilpotent Lie groups},
author = {Viviana del Barco and Andrei Moroianu and Alberto Raffero},
url = {https://link.springer.com/article/10.1007/s13163-021-00392-0},
doi = {https://doi.org/10.1007/s13163-021-00392-0},
year = {2021},
date = {2021-04-02},
journal = {Revista Matem\'{a}tica Complutense},
volume = {35},
pages = {323\textendash359},
abstract = {We consider left-invariant (purely) coclosed G2-structures on 7-dimensional 2-step nilpotent Lie groups. According to the dimension of the commutator subgroup, we obtain various criteria characterizing the Riemannian metrics induced by left-invariant purely coclosed G2-structures. Then, we use them to determine the isomorphism classes of 2-step nilpotent Lie algebras admitting such type of structures. As an intermediate step, we show that every metric on a 2-step nilpotent Lie algebra admitting coclosed G2-structures is induced by one of them. Finally, we use our results to give the explicit description of the metrics induced by purely coclosed G2-structures on 2-step nilpotent Lie algebras with derived algebra of dimension at most two, up to automorphism.},
keywords = {2-Step nilpotent Lie algebra, G2-Strominger system, Metric Lie algebra, Purely coclosed G2-structure},
pubstate = {published},
tppubtype = {article}
}
We consider left-invariant (purely) coclosed G2-structures on 7-dimensional 2-step nilpotent Lie groups. According to the dimension of the commutator subgroup, we obtain various criteria characterizing the Riemannian metrics induced by left-invariant purely coclosed G2-structures. Then, we use them to determine the isomorphism classes of 2-step nilpotent Lie algebras admitting such type of structures. As an intermediate step, we show that every metric on a 2-step nilpotent Lie algebra admitting coclosed G2-structures is induced by one of them. Finally, we use our results to give the explicit description of the metrics induced by purely coclosed G2-structures on 2-step nilpotent Lie algebras with derived algebra of dimension at most two, up to automorphism.
Gneri, P. O.; Jardim, Marcos; Silva, D. D.
Derived categories of functors and quiver sheaves Journal Article
Em: Journal of Algebra and Its Applications, vol. 21, iss. 7, 2021.
Resumo | Links | BibTeX | Tags: derived category of functors, Quiver sheaves
@article{nokey,
title = {Derived categories of functors and quiver sheaves},
author = {P. O. Gneri and Marcos Jardim and D. D. Silva},
url = {https://www.worldscientific.com/doi/10.1142/S0219498822501274},
doi = {https://doi.org/10.1142/S0219498822501274},
year = {2021},
date = {2021-03-19},
journal = {Journal of Algebra and Its Applications},
volume = {21},
issue = {7},
abstract = {Let C be small category and A an arbitrary category. Consider the category C(A) whose objects are functors from C to A and whose morphisms are natural transformations. Let B be another category, and again, consider the category C(B). Now, given a functor F:A→B we construct the induced functor FC:C(A)→C(B). Assuming A and B to be abelian categories, it follows that the categories C(A) and C(B) are also abelian. We have two main goals: first, to find a relationship between the derived category D(C(A)) and the category C(D(A)); second relate the functors R(FC) and (RF)C:C(D(A))→C(D(B)). We apply the general results obtained to the special case of quiver sheaves.},
keywords = {derived category of functors, Quiver sheaves},
pubstate = {published},
tppubtype = {article}
}
Let C be small category and A an arbitrary category. Consider the category C(A) whose objects are functors from C to A and whose morphisms are natural transformations. Let B be another category, and again, consider the category C(B). Now, given a functor F:A→B we construct the induced functor FC:C(A)→C(B). Assuming A and B to be abelian categories, it follows that the categories C(A) and C(B) are also abelian. We have two main goals: first, to find a relationship between the derived category D(C(A)) and the category C(D(A)); second relate the functors R(FC) and (RF)C:C(D(A))→C(D(B)). We apply the general results obtained to the special case of quiver sheaves.
Grajales, Brian; Grama, Lino; Negreiros, Caio J. C.
Geodesic orbit spaces in real flag manifolds Journal Article
Em: Communications in Analysis and Geometry, vol. 28, iss. 8, pp. 1933 – 2003, 2021.
Resumo | Links | BibTeX | Tags:
@article{nokey,
title = {Geodesic orbit spaces in real flag manifolds},
author = {Brian Grajales and Lino Grama and Caio J. C. Negreiros},
url = {https://www.intlpress.com/site/pub/pages/journals/items/cag/content/vols/0028/0008/a007/},
doi = {DOI: https://dx.doi.org/10.4310/CAG.2020.v28.n8.a7},
year = {2021},
date = {2021-01-08},
journal = {Communications in Analysis and Geometry},
volume = {28},
issue = {8},
pages = {1933 \textendash 2003},
abstract = {We describe the invariant metrics on real flag manifolds and classify those with the following property: every geodesic is the orbit of a one-parameter subgroup. Such a metric is called g.o. (geodesic orbit). In contrast to the complex case, on real flag manifolds the isotropy representation can have equivalent submodules, which makes invariant metrics depend on more parameters and allows us to find more cases in which non-trivial g.o. metrics exist.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We describe the invariant metrics on real flag manifolds and classify those with the following property: every geodesic is the orbit of a one-parameter subgroup. Such a metric is called g.o. (geodesic orbit). In contrast to the complex case, on real flag manifolds the isotropy representation can have equivalent submodules, which makes invariant metrics depend on more parameters and allows us to find more cases in which non-trivial g.o. metrics exist.
2020
Fonseca, Tiago J.
On coefficients of Poincaré series and single-valued periods of modular forms Journal Article
Em: Research in the Mathematical Sciences volume, vol. 7, 2020.
Resumo | Links | BibTeX | Tags: Fourier coefficients, Poincaré series
@article{nokey,
title = {On coefficients of Poincar\'{e} series and single-valued periods of modular forms},
author = {Tiago J. Fonseca},
url = {https://link.springer.com/article/10.1007/s40687-020-00232-5},
doi = {https://doi.org/10.1007/s40687-020-00232-5},
year = {2020},
date = {2020-11-05},
journal = {Research in the Mathematical Sciences volume},
volume = {7},
abstract = {We prove that the field generated by the Fourier coefficients of weakly holomorphic Poincar\'{e} series of a given level Γ0(N) and integral weight k≥2 coincides with the field generated by the single-valued periods of a certain motive attached to Γ0(N). This clarifies the arithmetic nature of such Fourier coefficients and generalises previous formulas of Brown and Acres\textendashBroadhurst giving explicit series expansions for the single-valued periods of some modular forms. Our proof is based on Bringmann\textendashOno’s construction of harmonic lifts of Poincar\'{e} series.},
keywords = {Fourier coefficients, Poincar\'{e} series},
pubstate = {published},
tppubtype = {article}
}
We prove that the field generated by the Fourier coefficients of weakly holomorphic Poincaré series of a given level Γ0(N) and integral weight k≥2 coincides with the field generated by the single-valued periods of a certain motive attached to Γ0(N). This clarifies the arithmetic nature of such Fourier coefficients and generalises previous formulas of Brown and Acres–Broadhurst giving explicit series expansions for the single-valued periods of some modular forms. Our proof is based on Bringmann–Ono’s construction of harmonic lifts of Poincaré series.
Jardim, Marcos; Silva, D. D.
Instanton sheaves and representations of quivers Journal Article
Em: Proceedings of the Edinburgh Mathematical Society, vol. 63, 2020.
Resumo | Links | BibTeX | Tags: instanton sheaves, moduli spaces, representations of quivers, wall-crossing
@article{nokey,
title = {Instanton sheaves and representations of quivers},
author = {Marcos Jardim and D. D. Silva},
url = {https://www.cambridge.org/core/journals/proceedings-of-the-edinburgh-mathematical-society/article/abs/instanton-sheaves-and-representations-of-quivers/972F6C19519C40CCDC264A87BD40194D},
doi = {10.1017/S0013091520000292},
year = {2020},
date = {2020-09-04},
journal = {Proceedings of the Edinburgh Mathematical Society},
volume = {63},
abstract = {We study the moduli space of rank 2 instanton sheaves on ℙ3 in terms of representations of a quiver consisting of three vertices and four arrows between two pairs of vertices. Aiming at an alternative compactification for the moduli space of instanton sheaves, we show that for each rank 2 instanton sheaf, there is a stability parameter θ for which the corresponding quiver representation is θ-stable (in the sense of King), and that the space of stability parameters has a non-trivial wall-and-chamber decomposition. Looking more closely at instantons of low charge, we prove that there are stability parameters with respect to which every representation corresponding to a rank 2 instanton sheaf of charge 2 is stable and provide a complete description of the wall-and-chamber decomposition for representation corresponding to a rank 2 instanton sheaf of charge 1.},
keywords = {instanton sheaves, moduli spaces, representations of quivers, wall-crossing},
pubstate = {published},
tppubtype = {article}
}
We study the moduli space of rank 2 instanton sheaves on ℙ3 in terms of representations of a quiver consisting of three vertices and four arrows between two pairs of vertices. Aiming at an alternative compactification for the moduli space of instanton sheaves, we show that for each rank 2 instanton sheaf, there is a stability parameter θ for which the corresponding quiver representation is θ-stable (in the sense of King), and that the space of stability parameters has a non-trivial wall-and-chamber decomposition. Looking more closely at instantons of low charge, we prove that there are stability parameters with respect to which every representation corresponding to a rank 2 instanton sheaf of charge 2 is stable and provide a complete description of the wall-and-chamber decomposition for representation corresponding to a rank 2 instanton sheaf of charge 1.
Grama, Lino; Lima, Kennerson N. S.
Bifurcation and local rigidity of homogeneous solutions to the Yamabe problem on maximal flag manifolds Journal Article
Em: Differential Geometry and its Applications, vol. 73, 2020.
Resumo | Links | BibTeX | Tags:
@article{nokey,
title = {Bifurcation and local rigidity of homogeneous solutions to the Yamabe problem on maximal flag manifolds},
author = {Lino Grama and Kennerson N. S. Lima},
url = {https://www.sciencedirect.com/science/article/abs/pii/S0926224520300930?via%3Dihub},
doi = {https://doi.org/10.1016/j.difgeo.2020.101684},
year = {2020},
date = {2020-09-02},
journal = {Differential Geometry and its Applications},
volume = {73},
abstract = {We construct 1-parameter families of well known solutions to the Yamabe problem defined on maximal flag manifolds to determine bifurcation instants for these families looking at changes of the Morse index of these metrics when the parameter varies on the interval . A bifurcation point for such families is an accumulation point of others solutions to the Yamabe problem and a local rigidity point is a isolated solution of this problem, i.e., is not a bifurcation point.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We construct 1-parameter families of well known solutions to the Yamabe problem defined on maximal flag manifolds to determine bifurcation instants for these families looking at changes of the Morse index of these metrics when the parameter varies on the interval . A bifurcation point for such families is an accumulation point of others solutions to the Yamabe problem and a local rigidity point is a isolated solution of this problem, i.e., is not a bifurcation point.
del Barco, Viviana; Moroianu, Andrei
Higher degree Killing forms on 2-step nilmanifolds Journal Article
Em: Journal of Algebra, vol. 563, pp. 251-273, 2020.
Resumo | Links | BibTeX | Tags: Killing forms
@article{nokey,
title = {Higher degree Killing forms on 2-step nilmanifolds},
author = {Viviana del Barco and Andrei Moroianu},
url = {https://www.sciencedirect.com/science/article/abs/pii/S0021869320303768?via%3Dihub},
doi = {https://doi.org/10.1016/j.jalgebra.2020.07.020},
year = {2020},
date = {2020-08-13},
journal = {Journal of Algebra},
volume = {563},
pages = {251-273},
abstract = {We study left-invariant Killing forms of arbitrary degree on simply connected 2-step nilpotent Lie groups endowed with left-invariant Riemannian metrics, and classify them when the center of the group is at most two-dimensional.},
keywords = {Killing forms},
pubstate = {published},
tppubtype = {article}
}
We study left-invariant Killing forms of arbitrary degree on simply connected 2-step nilpotent Lie groups endowed with left-invariant Riemannian metrics, and classify them when the center of the group is at most two-dimensional.
Conti, Diego; del Barco, Viviana; Rossi, Federico A.
Diagram involutions and homogeneous Ricci-flat metrics Journal Article
Em: manuscripta mathematica, vol. 165, pp. 381–413, 2020.
Resumo | Links | BibTeX | Tags:
@article{nokey,
title = {Diagram involutions and homogeneous Ricci-flat metrics},
author = {Diego Conti and Viviana del Barco and Federico A. Rossi },
url = {https://link.springer.com/article/10.1007/s00229-020-01225-y},
doi = {https://doi.org/10.1007/s00229-020-01225-y},
year = {2020},
date = {2020-07-08},
journal = {manuscripta mathematica},
volume = {165},
pages = {381\textendash413},
abstract = {We introduce a combinatorial method to construct indefinite Ricci-flat metrics on nice nilpotent Lie groups. We prove that every nilpotent Lie group of dimension ≤6, every nice nilpotent Lie group of dimension ≤7 and every two-step nilpotent Lie group attached to a graph admits such a metric. We construct infinite families of Ricci-flat nilmanifolds associated to parabolic nilradicals in the simple Lie groups SL(n), SO(p,q), Sp(n,R). Most of these metrics are shown not to be flat.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We introduce a combinatorial method to construct indefinite Ricci-flat metrics on nice nilpotent Lie groups. We prove that every nilpotent Lie group of dimension ≤6, every nice nilpotent Lie group of dimension ≤7 and every two-step nilpotent Lie group attached to a graph admits such a metric. We construct infinite families of Ricci-flat nilmanifolds associated to parabolic nilradicals in the simple Lie groups SL(n), SO(p,q), Sp(n,R). Most of these metrics are shown not to be flat.
Grama, Lino; Seco, Lucas
Second Homotopy Group and Invariant Geometry of Flag Manifolds Journal Article
Em: Results in Mathematics, vol. 75, 2020.
Resumo | Links | BibTeX | Tags:
@article{nokey,
title = {Second Homotopy Group and Invariant Geometry of Flag Manifolds},
author = {Lino Grama and Lucas Seco },
url = {https://link.springer.com/article/10.1007/s00025-020-01213-4},
doi = {https://doi.org/10.1007/s00025-020-01213-4},
year = {2020},
date = {2020-06-03},
journal = {Results in Mathematics},
volume = {75},
abstract = {We use the Hopf fibration to explicitly compute generators of the second homotopy group of the flag manifolds of a compact Lie group. We show that these 2-spheres have nice geometrical properties such as being totally geodesic surfaces with respect to any invariant metric on the flag manifold, generalizing a result in Burstall and Rawnsley (Springer Lect. Notes Math. 2(84):1424, 1990). This illustrates how “rubber-band” topology can, in the presence of symmetry, single out very rigid objects. We characterize when these 2-spheres in the same homotopy class have the same geometry for all invariant metrics. This is done by exploring the action of Weyl group of the flag manifold, generalizing results of Patr\~{a}o and San Martin (Indag. Math. 26:547\textendash579, 2015) and de Siebenthal (Math. Helvetici 44(1):1\textendash3, 1969). This illustrates how some aspects of “continuum” invariant geometry can, in the presence of symmetry, be reduced to the study of discrete objects. We remark that the topology singling out very rigid objects and the study of a continuum object being reduced to discrete ones is a characteristic of situations with a lot of symmetry and, thus, are recurring themes in Lie theory.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We use the Hopf fibration to explicitly compute generators of the second homotopy group of the flag manifolds of a compact Lie group. We show that these 2-spheres have nice geometrical properties such as being totally geodesic surfaces with respect to any invariant metric on the flag manifold, generalizing a result in Burstall and Rawnsley (Springer Lect. Notes Math. 2(84):1424, 1990). This illustrates how “rubber-band” topology can, in the presence of symmetry, single out very rigid objects. We characterize when these 2-spheres in the same homotopy class have the same geometry for all invariant metrics. This is done by exploring the action of Weyl group of the flag manifold, generalizing results of Patrão and San Martin (Indag. Math. 26:547–579, 2015) and de Siebenthal (Math. Helvetici 44(1):1–3, 1969). This illustrates how some aspects of “continuum” invariant geometry can, in the presence of symmetry, be reduced to the study of discrete objects. We remark that the topology singling out very rigid objects and the study of a continuum object being reduced to discrete ones is a characteristic of situations with a lot of symmetry and, thus, are recurring themes in Lie theory.
Calvo-Andrade, Omegar; Díaz, Lázaro O. Rodríguez; Earp, Henrique N. Sá
Gauge theory and G2-geometry on Calabi–Yau links Journal Article
Em: Revista Matemática Iberoamericana, vol. 36, iss. 6, pp. 1753–1778, 2020.
Resumo | Links | BibTeX | Tags: Gauge Theory
@article{nokey,
title = {Gauge theory and G2-geometry on Calabi\textendashYau links},
author = {Omegar Calvo-Andrade and L\'{a}zaro O. Rodr\'{i}guez D\'{i}az and Henrique N. S\'{a} Earp},
doi = {http://dx.doi.org/10.4171/rmi/1182},
year = {2020},
date = {2020-02-21},
journal = {Revista Matem\'{a}tica Iberoamericana},
volume = {36},
issue = {6},
pages = {1753\textendash1778},
abstract = {The 7-dimensional link K of a weighted homogeneous hypersurface on the round 9-sphere in C5 has a nontrivial null Sasakian structure which is contact Calabi-Yau, in many cases. It admits a canonical co-closed G2-structure φ induced by the Calabi-Yau 3-orbifold basic geometry. We distinguish these pairs (K,φ) by the Crowley-Nordstr\"{o}m Z48-valued ν invariant, for which we prove odd parity and provide an algorithmic formula. We describe moreover a natural Yang-Mills theory on such spaces, with many important features of the torsion-free case, such as a Chern-Simons formalism and topological energy bounds. In fact compatible G2-instantons on holomorphic Sasakian bundles over K are exactly the transversely Hermitian Yang-Mills connections. As a proof of principle, we obtain G2-instantons over the Fermat quintic link from stable bundles over the smooth projective Fermat quintic, thus relating in a concrete example the Donaldson-Thomas theory of the quintic threefold with a conjectural G2-instanton count.},
keywords = {Gauge Theory},
pubstate = {published},
tppubtype = {article}
}
The 7-dimensional link K of a weighted homogeneous hypersurface on the round 9-sphere in C5 has a nontrivial null Sasakian structure which is contact Calabi-Yau, in many cases. It admits a canonical co-closed G2-structure φ induced by the Calabi-Yau 3-orbifold basic geometry. We distinguish these pairs (K,φ) by the Crowley-Nordström Z48-valued ν invariant, for which we prove odd parity and provide an algorithmic formula. We describe moreover a natural Yang-Mills theory on such spaces, with many important features of the torsion-free case, such as a Chern-Simons formalism and topological energy bounds. In fact compatible G2-instantons on holomorphic Sasakian bundles over K are exactly the transversely Hermitian Yang-Mills connections. As a proof of principle, we obtain G2-instantons over the Fermat quintic link from stable bundles over the smooth projective Fermat quintic, thus relating in a concrete example the Donaldson-Thomas theory of the quintic threefold with a conjectural G2-instanton count.
BARCO, Viviana DEL; MOROIANU, Andrei
Symmetric Killing tensors on nilmanifolds Journal Article
Em: BULLETIN DE LA SMF, vol. 3, iss. 148, pp. 411-438, 2020.
Resumo | Links | BibTeX | Tags: nilmanifolds
@article{nokey,
title = {Symmetric Killing tensors on nilmanifolds},
author = {Viviana DEL BARCO and Andrei MOROIANU},
url = {https://smf.emath.fr/publications/tenseurs-de-killing-symetriques-sur-les-nilvarietes},
doi = {10.24033/bsmf.2811},
year = {2020},
date = {2020-01-01},
journal = {BULLETIN DE LA SMF},
volume = {3},
issue = {148},
pages = {411-438},
abstract = {We study left-invariant symmetric Killing 2-tensors on two-step nilpotent Lie groups endowed with a left-invariant Riemannian metric and construct genuine examples, which are not linear combinations of parallel tensors and symmetric products of Killing vector fields.},
keywords = {nilmanifolds},
pubstate = {published},
tppubtype = {article}
}
We study left-invariant symmetric Killing 2-tensors on two-step nilpotent Lie groups endowed with a left-invariant Riemannian metric and construct genuine examples, which are not linear combinations of parallel tensors and symmetric products of Killing vector fields.
2019
del Barco, Viviana; Moroianu, Andrei
Killing Forms on 2-Step Nilmanifolds Journal Article
Em: The Journal of Geometric Analysis, vol. 31, pp. 863–887, 2019.
Resumo | Links | BibTeX | Tags: 2-step nilpotent Lie groups, Killing forms, Naturally reductive homogeneous spaces
@article{nokey,
title = {Killing Forms on 2-Step Nilmanifolds},
author = {Viviana del Barco and Andrei Moroianu },
url = {https://link.springer.com/article/10.1007/s12220-019-00304-1},
doi = {https://doi.org/10.1007/s12220-019-00304-1},
year = {2019},
date = {2019-11-05},
journal = {The Journal of Geometric Analysis},
volume = {31},
pages = {863\textendash887},
abstract = {We study left-invariant Killing k-forms on simply connected 2-step nilpotent Lie groups endowed with a left-invariant Riemannian metric. For k=2,3, we show that every left-invariant Killing k-form is a sum of Killing forms on the factors of the de Rham decomposition. Moreover, on each irreducible factor, non-zero Killing 2-forms define (after some modification) a bi-invariant orthogonal complex structure and non-zero Killing 3-forms arise only if the Riemannian Lie group is naturally reductive when viewed as a homogeneous space under the action of its isometry group. In both cases, k=2 or k=3, we show that the space of left-invariant Killing k-forms of an irreducible Riemannian 2-step nilpotent Lie group is at most one-dimensional.},
keywords = {2-step nilpotent Lie groups, Killing forms, Naturally reductive homogeneous spaces},
pubstate = {published},
tppubtype = {article}
}
We study left-invariant Killing k-forms on simply connected 2-step nilpotent Lie groups endowed with a left-invariant Riemannian metric. For k=2,3, we show that every left-invariant Killing k-form is a sum of Killing forms on the factors of the de Rham decomposition. Moreover, on each irreducible factor, non-zero Killing 2-forms define (after some modification) a bi-invariant orthogonal complex structure and non-zero Killing 3-forms arise only if the Riemannian Lie group is naturally reductive when viewed as a homogeneous space under the action of its isometry group. In both cases, k=2 or k=3, we show that the space of left-invariant Killing k-forms of an irreducible Riemannian 2-step nilpotent Lie group is at most one-dimensional.
del Barco, Viviana
Symplectic structures on free nilpotent Lie algebras Journal Article
Em: Proc. Japan Acad. Ser. A Math. Sci., vol. 95, pp. 88-90, 2019.
Resumo | Links | BibTeX | Tags: Free nilpotent Lie algebras, Symplectic structures
@article{nokey,
title = {Symplectic structures on free nilpotent Lie algebras},
author = {Viviana del Barco},
url = {https://projecteuclid.org/journals/proceedings-of-the-japan-academy-series-a-mathematical-sciences/volume-95/issue-8/Symplectic-structures-on-free-nilpotent-Lie-algebras/10.3792/pjaa.95.88.full},
doi = {https://doi.org/10.3792/pjaa.95.88},
year = {2019},
date = {2019-10-02},
journal = {Proc. Japan Acad. Ser. A Math. Sci.},
volume = {95},
pages = {88-90},
abstract = {In this short note we show a necessary and sufficient condition for the existence of symplectic structures on free nilpotent Lie algebras and their one-dimensional trivial extensions.},
keywords = {Free nilpotent Lie algebras, Symplectic structures},
pubstate = {published},
tppubtype = {article}
}
In this short note we show a necessary and sufficient condition for the existence of symplectic structures on free nilpotent Lie algebras and their one-dimensional trivial extensions.
de Freitas Leão, Rafael; Wainer, Samuel Augusto
Immersion in SSn by Complex Spinors Journal Article
Em: Advances in Applied Clifford Algebras volume, vol. 29, 2019.
Resumo | Links | BibTeX | Tags:
@article{nokey,
title = {Immersion in SSn by Complex Spinors},
author = {Rafael de Freitas Le\~{a}o and Samuel Augusto Wainer},
url = {https://link.springer.com/article/10.1007/s00006-019-0986-8},
doi = {https://doi.org/10.1007/s00006-019-0986-8},
year = {2019},
date = {2019-07-17},
journal = {Advances in Applied Clifford Algebras volume},
volume = {29},
abstract = {Since the first work of Thomas Friedrich showing that isometric immersions of Riemann surfaces are related to spinors and the Dirac equation, various works appeared generalizing this approach to more general Spin-manifolds, in particular the case of submanifolds of Spin-manifolds of constant curvature. In the present work we investigate the case of submanifolds of SpinC-manifolds of constant curvature.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
Since the first work of Thomas Friedrich showing that isometric immersions of Riemann surfaces are related to spinors and the Dirac equation, various works appeared generalizing this approach to more general Spin-manifolds, in particular the case of submanifolds of Spin-manifolds of constant curvature. In the present work we investigate the case of submanifolds of SpinC-manifolds of constant curvature.
del Barco, Viviana; Martin, Luiz Antonio Barrera San
De Rham 2-Cohomology of Real Flag Manifolds Journal Article
Em: SIGMA, vol. 15, 2019.
Resumo | Links | BibTeX | Tags: cellular homology, characteristic classes, de Rham cohomology, Flag Manifolds, Schubert cell
@article{nokey,
title = {De Rham 2-Cohomology of Real Flag Manifolds},
author = {Viviana del Barco and Luiz Antonio Barrera San Martin},
url = {https://www.emis.de/journals/SIGMA/2019/051/},
doi = {https://doi.org/10.3842/SIGMA.2019.051},
year = {2019},
date = {2019-07-05},
journal = {SIGMA},
volume = {15},
abstract = {Let FΘ=G/PΘ be a flag manifold associated to a non-compact real simple Lie group G and the parabolic subgroup PΘ. This is a closed subgroup of G determined by a subset Θ of simple restricted roots of g=Lie(G). This paper computes the second de Rham cohomology group of FΘ. We prove that it is zero in general, with some rare exceptions. When it is non-zero, we give a basis of H2(FΘ,R) through the Weil construction of closed 2-forms as characteristic forms of principal fiber bundles. The starting point is the computation of the second homology group of FΘ with coefficients in a ring R.},
keywords = {cellular homology, characteristic classes, de Rham cohomology, Flag Manifolds, Schubert cell},
pubstate = {published},
tppubtype = {article}
}
Let FΘ=G/PΘ be a flag manifold associated to a non-compact real simple Lie group G and the parabolic subgroup PΘ. This is a closed subgroup of G determined by a subset Θ of simple restricted roots of g=Lie(G). This paper computes the second de Rham cohomology group of FΘ. We prove that it is zero in general, with some rare exceptions. When it is non-zero, we give a basis of H2(FΘ,R) through the Weil construction of closed 2-forms as characteristic forms of principal fiber bundles. The starting point is the computation of the second homology group of FΘ with coefficients in a ring R.
Moreno, Andrés J.; Earp, Henrique N. Sá
Explicit soliton for the Laplacian co-flow on a solvmanifold Journal Article
Em: São Paulo Journal of Mathematical Sciences, vol. 15, pp. 280–292, 2019.
Resumo | Links | BibTeX | Tags: solvmanifold
@article{nokey,
title = {Explicit soliton for the Laplacian co-flow on a solvmanifold},
author = {Andr\'{e}s J. Moreno and Henrique N. S\'{a} Earp },
url = {https://link.springer.com/article/10.1007/s40863-019-00134-7},
doi = {http://dx.doi.org/10.1007/s40863-019-00134-7},
year = {2019},
date = {2019-05-17},
journal = {S\~{a}o Paulo Journal of Mathematical Sciences},
volume = {15},
pages = {280\textendash292},
abstract = {We apply the general Ansatz proposed by Lauret (Rend Semin Mat Torino 74:55\textendash93, 2016) for the Laplacian co-flow of invariant G2-structures on a Lie group, finding an explicit soliton on a particular almost Abelian 7\textendashmanifold. Our methods and the example itself are different from those presented by Bagaglini and Fino (Ann Mat Pura Appl 197(6):1855\textendash1873, 2018).},
keywords = {solvmanifold},
pubstate = {published},
tppubtype = {article}
}
We apply the general Ansatz proposed by Lauret (Rend Semin Mat Torino 74:55–93, 2016) for the Laplacian co-flow of invariant G2-structures on a Lie group, finding an explicit soliton on a particular almost Abelian 7–manifold. Our methods and the example itself are different from those presented by Bagaglini and Fino (Ann Mat Pura Appl 197(6):1855–1873, 2018).
Franco, Emilio; Jardim, Marcos; Menet, Grégoire
Brane involutions on irreducible holomorphic symplectic manifolds Journal Article
Em: Kyoto J. Math., vol. 59, iss. 1, pp. 195 - 235, 2019.
Resumo | Links | BibTeX | Tags: mirror symmetry
@article{nokey,
title = {Brane involutions on irreducible holomorphic symplectic manifolds},
author = {Emilio Franco and Marcos Jardim and Gr\'{e}goire Menet},
url = {https://projecteuclid.org/journals/kyoto-journal-of-mathematics/volume-59/issue-1/Brane-involutions-on-irreducible-holomorphic-symplectic-manifolds/10.1215/21562261-2018-0009.short},
doi = {10.1215/21562261-2018-0009},
year = {2019},
date = {2019-04-01},
journal = {Kyoto J. Math.},
volume = {59},
issue = {1},
pages = {195 - 235},
abstract = {In the context of irreducible holomorphic symplectic manifolds, we say that (anti)holomorphic (anti)symplectic involutions are brane involutions since their fixed point locus is a brane in the physicists’ language, that is, a submanifold which is either a complex or Lagrangian submanifold with respect to each of the three K\"{a}hler structures of the associated hyper-K\"{a}hler structure. Starting from a brane involution on a
K3
or Abelian surface, one can construct a natural brane involution on its moduli space of sheaves. We study these natural involutions and their relation with the Fourier\textendashMukai transform. Later, we recall the lattice-theoretical approach to mirror symmetry. We provide two ways of obtaining a brane involution on the mirror, and we study the behavior of the brane involutions under both mirror transformations, giving examples in the case of a
K3
surface and
K3
[
2
]
-type manifolds.},
keywords = {mirror symmetry},
pubstate = {published},
tppubtype = {article}
}
In the context of irreducible holomorphic symplectic manifolds, we say that (anti)holomorphic (anti)symplectic involutions are brane involutions since their fixed point locus is a brane in the physicists’ language, that is, a submanifold which is either a complex or Lagrangian submanifold with respect to each of the three Kähler structures of the associated hyper-Kähler structure. Starting from a brane involution on a
K3
or Abelian surface, one can construct a natural brane involution on its moduli space of sheaves. We study these natural involutions and their relation with the Fourier–Mukai transform. Later, we recall the lattice-theoretical approach to mirror symmetry. We provide two ways of obtaining a brane involution on the mirror, and we study the behavior of the brane involutions under both mirror transformations, giving examples in the case of a
K3
surface and
K3
[
2
]
-type manifolds.
K3
or Abelian surface, one can construct a natural brane involution on its moduli space of sheaves. We study these natural involutions and their relation with the Fourier–Mukai transform. Later, we recall the lattice-theoretical approach to mirror symmetry. We provide two ways of obtaining a brane involution on the mirror, and we study the behavior of the brane involutions under both mirror transformations, giving examples in the case of a
K3
surface and
K3
[
2
]
-type manifolds.
Fonseca, Tiago J.
Algebraic independence for values of integral curves Apresentação
23.03.2019.
Resumo | Links | BibTeX | Tags:
@misc{nokey,
title = {Algebraic independence for values of integral curves},
author = {Tiago J. Fonseca},
url = {https://msp.org/ant/2019/13-3/p03.xhtml},
doi = {10.2140/ant.2019.13.643},
year = {2019},
date = {2019-03-23},
journal = {Algebra and Number Theory},
volume = {13},
issue = {3},
pages = {643\textendash694},
abstract = {We prove a transcendence theorem concerning values of holomorphic maps from a disk to a quasiprojective variety over
¯¯¯¯
Q
that are integral curves of some algebraic vector field (defined over
¯¯¯¯
Q
). These maps are required to satisfy some integrality property, besides a growth condition and a strong form of Zariski-density that are natural for integral curves of algebraic vector fields.
This result generalizes a theorem of Nesterenko concerning algebraic independence of values of the Eisenstein series
E
2
,
E
4
,
E
6
. The main technical improvement in our approach is the replacement of a rather restrictive hypothesis of polynomial growth on Taylor coefficients by a geometric notion of moderate growth formulated in terms of value distribution theory.},
keywords = {},
pubstate = {published},
tppubtype = {presentation}
}
We prove a transcendence theorem concerning values of holomorphic maps from a disk to a quasiprojective variety over
¯¯¯¯
Q
that are integral curves of some algebraic vector field (defined over
¯¯¯¯
Q
). These maps are required to satisfy some integrality property, besides a growth condition and a strong form of Zariski-density that are natural for integral curves of algebraic vector fields.
This result generalizes a theorem of Nesterenko concerning algebraic independence of values of the Eisenstein series
E
2
,
E
4
,
E
6
. The main technical improvement in our approach is the replacement of a rather restrictive hypothesis of polynomial growth on Taylor coefficients by a geometric notion of moderate growth formulated in terms of value distribution theory.
¯¯¯¯
Q
that are integral curves of some algebraic vector field (defined over
¯¯¯¯
Q
). These maps are required to satisfy some integrality property, besides a growth condition and a strong form of Zariski-density that are natural for integral curves of algebraic vector fields.
This result generalizes a theorem of Nesterenko concerning algebraic independence of values of the Eisenstein series
E
2
,
E
4
,
E
6
. The main technical improvement in our approach is the replacement of a rather restrictive hypothesis of polynomial growth on Taylor coefficients by a geometric notion of moderate growth formulated in terms of value distribution theory.
Correa, Eder M.; Grama, Lino
Calabi-Yau metrics on canonical bundles of complex flag manifolds Journal Article
Em: Journal of Algebra, vol. 527, iss. 1, pp. 109-135, 2019.
Resumo | Links | BibTeX | Tags: Calabi–Yau
@article{nokey,
title = {Calabi-Yau metrics on canonical bundles of complex flag manifolds},
author = {Eder M. Correa and Lino Grama},
url = {https://www.sciencedirect.com/science/article/abs/pii/S0021869319301164?via%3Dihub},
doi = {https://doi.org/10.1016/j.jalgebra.2019.02.027},
year = {2019},
date = {2019-03-13},
journal = {Journal of Algebra},
volume = {527},
issue = {1},
pages = {109-135},
abstract = {In the present paper we provide a description of complete Calabi-Yau metrics on the canonical bundle of generalized complex flag manifolds. By means of Lie theory we give an explicit description of complete Ricci-flat K\"{a}hler metrics obtained through the Calabi Ansatz technique. We use this approach to provide several explicit examples of noncompact complete Calabi-Yau manifolds, these examples include canonical bundles of non-toric flag manifolds (e.g. Grassmann manifolds and full flag manifolds).},
keywords = {Calabi\textendashYau},
pubstate = {published},
tppubtype = {article}
}
In the present paper we provide a description of complete Calabi-Yau metrics on the canonical bundle of generalized complex flag manifolds. By means of Lie theory we give an explicit description of complete Ricci-flat Kähler metrics obtained through the Calabi Ansatz technique. We use this approach to provide several explicit examples of noncompact complete Calabi-Yau manifolds, these examples include canonical bundles of non-toric flag manifolds (e.g. Grassmann manifolds and full flag manifolds).
Jacob, Adam; Earp, Henrique N. Sá; Walpuski, Thomas
Tangent cones of Hermitian Yang–Mills connections with isolated singularities Journal Article
Em: Mathematical Research Letters, vol. 25, iss. 5, pp. 1429 – 1445, 2019.
Resumo | Links | BibTeX | Tags: Hermitian Yang–Mills
@article{nokey,
title = {Tangent cones of Hermitian Yang\textendashMills connections with isolated singularities},
author = {Adam Jacob and Henrique N. S\'{a} Earp and Thomas Walpuski},
url = {https://www.intlpress.com/site/pub/pages/journals/items/mrl/content/vols/0025/0005/a004/},
doi = {https://dx.doi.org/10.4310/MRL.2018.v25.n5.a4},
year = {2019},
date = {2019-02-01},
urldate = {2019-02-01},
journal = {Mathematical Research Letters},
volume = {25},
issue = {5},
pages = {1429 \textendash 1445},
abstract = {We give a simple direct proof of uniqueness of tangent cones for singular projectively Hermitian Yang\textendashMills connections on reflexive sheaves at isolated singularities modelled on a sum of -stable holomorphic bundles over .},
keywords = {Hermitian Yang\textendashMills},
pubstate = {published},
tppubtype = {article}
}
We give a simple direct proof of uniqueness of tangent cones for singular projectively Hermitian Yang–Mills connections on reflexive sheaves at isolated singularities modelled on a sum of -stable holomorphic bundles over .
2018
do Prado, Rafaela F.; Grama, Lino
Variational aspects of homogeneous geodesics on generalized flag manifolds and applications Journal Article
Em: Annals of Global Analysis and Geometry , vol. 55, pp. 451–477, 2018.
Resumo | Links | BibTeX | Tags:
@article{nokey,
title = {Variational aspects of homogeneous geodesics on generalized flag manifolds and applications},
author = {Rafaela F. do Prado and Lino Grama},
url = {https://link.springer.com/article/10.1007/s10455-018-9635-z},
doi = {https://doi.org/10.1007/s10455-018-9635-z},
year = {2018},
date = {2018-11-03},
journal = {Annals of Global Analysis and Geometry },
volume = {55},
pages = {451\textendash477},
abstract = {We study conjugate points along homogeneous geodesics in generalized flag manifolds. This is done by analyzing the second variation of the energy of such geodesics. We also give an example of how the homogeneous Ricci flow can evolve in such way to produce conjugate points in the complex projective space CP2n+1=Sp(n+1)/(U(1)×Sp(n)).},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We study conjugate points along homogeneous geodesics in generalized flag manifolds. This is done by analyzing the second variation of the energy of such geodesics. We also give an example of how the homogeneous Ricci flow can evolve in such way to produce conjugate points in the complex projective space CP2n+1=Sp(n+1)/(U(1)×Sp(n)).
Calvo-Andrade, Omegar; Corrêa, Maurício; Jardim, Marcos
Codimension One Holomorphic Distributions on the Projective Three-space Journal Article
Em: International Mathematics Research Notices, vol. 2020, iss. 23, 2018.
Resumo | Links | BibTeX | Tags:
@article{nokey,
title = {Codimension One Holomorphic Distributions on the Projective Three-space},
author = {Omegar Calvo-Andrade and Maur\'{i}cio Corr\^{e}a and Marcos Jardim},
url = {https://academic.oup.com/imrn/article-abstract/2020/23/9011/5144735?redirectedFrom=fulltext},
doi = {https://doi.org/10.1093/imrn/rny251},
year = {2018},
date = {2018-10-25},
journal = {International Mathematics Research Notices},
volume = {2020},
issue = {23},
abstract = {We study codimension one holomorphic distributions on the projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2 with locally free tangent sheaves and show that codimension one distributions of arbitrary degree with only isolated singularities have stable tangent sheaves. Furthermore, we describe the moduli space of distributions in terms of Grothendieck’s Quot-scheme for the tangent bundle. In certain cases, we show that the moduli space of codimension one distributions on the projective space is an irreducible, nonsingular quasi-projective variety. Finally, we prove that every rational foliation and certain logarithmic foliations have stable tangent sheaves.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We study codimension one holomorphic distributions on the projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2 with locally free tangent sheaves and show that codimension one distributions of arbitrary degree with only isolated singularities have stable tangent sheaves. Furthermore, we describe the moduli space of distributions in terms of Grothendieck’s Quot-scheme for the tangent bundle. In certain cases, we show that the moduli space of codimension one distributions on the projective space is an irreducible, nonsingular quasi-projective variety. Finally, we prove that every rational foliation and certain logarithmic foliations have stable tangent sheaves.
Henni, Abdelmoubine A.; Jardim, Marcos
Commuting matrices and the Hilbert scheme of points on affine spaces Journal Article
Em: Advances in Geometry, vol. 18, iss. 4, 2018.
Resumo | Links | BibTeX | Tags: Hilbert scheme of points
@article{nokey,
title = {Commuting matrices and the Hilbert scheme of points on affine spaces},
author = {Abdelmoubine A. Henni and Marcos Jardim},
url = {https://www.degruyter.com/document/doi/10.1515/advgeom-2018-0011/html},
doi = {https://doi.org/10.1515/advgeom-2018-0011},
year = {2018},
date = {2018-07-20},
journal = {Advances in Geometry},
volume = {18},
issue = {4},
abstract = {We give linear algebraic and monadic descriptions of the Hilbert scheme of points on the affine space of dimension n which naturally extends Nakajima’s representation of the Hilbert scheme of points on the plane. As an application of our ideas and recent results from the literature on commuting matrices, we show that the Hilbert scheme of c points on ℂ3 is irreducible for c ≤ 10.},
keywords = {Hilbert scheme of points},
pubstate = {published},
tppubtype = {article}
}
We give linear algebraic and monadic descriptions of the Hilbert scheme of points on the affine space of dimension n which naturally extends Nakajima’s representation of the Hilbert scheme of points on the plane. As an application of our ideas and recent results from the literature on commuting matrices, we show that the Hilbert scheme of c points on ℂ3 is irreducible for c ≤ 10.
del Barco, Viviana; Grama, Lino
On generalized G2-structures and T-duality Journal Article
Em: Journal of Geometry and Physics, vol. 132, pp. 109-113, 2018.
Resumo | Links | BibTeX | Tags: G2-Structures, T-duality
@article{nokey,
title = {On generalized G2-structures and T-duality},
author = {Viviana del Barco and Lino Grama},
url = {https://www.sciencedirect.com/science/article/pii/S0393044018303413?via%3Dihub},
doi = {https://doi.org/10.1016/j.geomphys.2018.05.021},
year = {2018},
date = {2018-06-18},
journal = {Journal of Geometry and Physics},
volume = {132},
pages = {109-113},
abstract = {This is a short note on generalized -structures obtained as a consequence of a -dual construction given in del Barco et al. (2017). Given classical -structure on certain seven dimensional manifolds, either closed or co-closed, we obtain integrable generalized -structures which are no longer a usual one, and with non-zero three form in general. In particular we obtain manifolds admitting closed generalized -structures not admitting closed (usual) -structures.},
keywords = {G2-Structures, T-duality},
pubstate = {published},
tppubtype = {article}
}
This is a short note on generalized -structures obtained as a consequence of a -dual construction given in del Barco et al. (2017). Given classical -structure on certain seven dimensional manifolds, either closed or co-closed, we obtain integrable generalized -structures which are no longer a usual one, and with non-zero three form in general. In particular we obtain manifolds admitting closed generalized -structures not admitting closed (usual) -structures.
del Barco, Viviana; Grama, Lino; Soriani, Leonardo
T-duality on nilmanifolds Journal Article
Em: Journal of High Energy Physics , vol. 153, 2018.
Resumo | Links | BibTeX | Tags: T-duality
@article{nokey,
title = {T-duality on nilmanifolds},
author = {Viviana del Barco and Lino Grama and Leonardo Soriani },
url = {https://link.springer.com/article/10.1007/JHEP05(2018)153#citeas},
doi = {https://doi.org/10.1007/JHEP05(2018)153},
year = {2018},
date = {2018-05-24},
journal = {Journal of High Energy Physics },
volume = {153},
abstract = {We study generalized complex structures and T-duality (in the sense of Bouwknegt, Evslin, Hannabuss and Mathai) on Lie algebras and construct the corresponding Cavalcanti and Gualtieri map. Such a construction is called Infinitesimal T -duality. As an application we deal with the problem of finding symplectic structures on 2-step nilpotent Lie algebras. We also give a criteria for the integrability of the infinitesimal T-duality of Lie algebras to topological T-duality of the associated nilmanifolds.},
keywords = {T-duality},
pubstate = {published},
tppubtype = {article}
}
We study generalized complex structures and T-duality (in the sense of Bouwknegt, Evslin, Hannabuss and Mathai) on Lie algebras and construct the corresponding Cavalcanti and Gualtieri map. Such a construction is called Infinitesimal T -duality. As an application we deal with the problem of finding symplectic structures on 2-step nilpotent Lie algebras. We also give a criteria for the integrability of the infinitesimal T-duality of Lie algebras to topological T-duality of the associated nilmanifolds.
Ballico, E.; Barmeier, S.; Gasparim, E.; Grama, Lino; Martin, L. A. B. San
A Lie theoretical construction of a Landau–Ginzburg model without projective mirrors Journal Article
Em: manuscripta mathematica, vol. 158, pp. 85–101, 2018.
Resumo | Links | BibTeX | Tags: Landau–Ginzburg
@article{nokey,
title = {A Lie theoretical construction of a Landau\textendashGinzburg model without projective mirrors},
author = {E. Ballico and S. Barmeier and E. Gasparim and Lino Grama and L. A. B. San Martin },
url = {https://link.springer.com/article/10.1007/s00229-018-1024-1},
doi = {https://doi.org/10.1007/s00229-018-1024-1},
year = {2018},
date = {2018-05-22},
journal = {manuscripta mathematica},
volume = {158},
pages = {85\textendash101},
abstract = {We describe the Fukaya\textendashSeidel category of a Landau\textendashGinzburg model LG(2) for the semisimple adjoint orbit of sl(2,C). We prove that this category is equivalent to a full triangulated subcategory of the category of coherent sheaves on the second Hirzebruch surface. We show that no projective variety can be mirror to LG(2), and that this remains so after compactification.},
keywords = {Landau\textendashGinzburg},
pubstate = {published},
tppubtype = {article}
}
We describe the Fukaya–Seidel category of a Landau–Ginzburg model LG(2) for the semisimple adjoint orbit of sl(2,C). We prove that this category is equivalent to a full triangulated subcategory of the category of coherent sheaves on the second Hirzebruch surface. We show that no projective variety can be mirror to LG(2), and that this remains so after compactification.
Freitas, Ana P. C.; del Barco, Viviana; Martin, Luiz A. B. San
Invariant almost complex structures on real flag manifolds Journal Article
Em: Annali di Matematica Pura ed Applicata, vol. 197, pp. 1821–1844, 2018.
Resumo | Links | BibTeX | Tags:
@article{nokey,
title = {Invariant almost complex structures on real flag manifolds},
author = {Ana P. C. Freitas and Viviana del Barco and Luiz A. B. San Martin },
url = {https://link.springer.com/article/10.1007/s10231-018-0751-y},
doi = {https://doi.org/10.1007/s10231-018-0751-y},
year = {2018},
date = {2018-04-27},
journal = {Annali di Matematica Pura ed Applicata},
volume = {197},
pages = {1821\textendash1844},
abstract = {In this work, we study the existence of invariant almost complex structures on real flag manifolds associated to split real forms of complex simple Lie algebras. We show that, contrary to the complex case where the invariant almost complex structures are well known, some real flag manifolds do not admit such structures. We check which invariant almost complex structures are integrable and prove that only some flag manifolds of the Lie algebra Cl admit complex structures.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
In this work, we study the existence of invariant almost complex structures on real flag manifolds associated to split real forms of complex simple Lie algebras. We show that, contrary to the complex case where the invariant almost complex structures are well known, some real flag manifolds do not admit such structures. We check which invariant almost complex structures are integrable and prove that only some flag manifolds of the Lie algebra Cl admit complex structures.
de Freitas Leão, Rafael; Wainer, Samuel Augusto
Immersion in Rn by Complex Spinors Journal Article
Em: Advances in Applied Clifford Algebras, vol. 28, 2018.
Resumo | Links | BibTeX | Tags:
@article{nokey,
title = {Immersion in Rn by Complex Spinors},
author = {Rafael de Freitas Le\~{a}o and Samuel Augusto Wainer},
url = {https://link.springer.com/article/10.1007/s00006-018-0856-9},
doi = {https://doi.org/10.1007/s00006-018-0856-9},
year = {2018},
date = {2018-04-26},
journal = {Advances in Applied Clifford Algebras},
volume = {28},
abstract = {A beautiful solution to the problem of isometric immersions in Rn using spinors was found by Bayard et al. (Pac J Math, arXiv:1505.02935v4 [math-ph], 2016). However to use spinors one must assume that the manifold carries a Spin -structure and, especially for complex manifolds where is more natural to consider Spin C-structures, this hypothesis is somewhat restrictive. In the present work we show how the above solution can be adapted to Spin C-structures.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
A beautiful solution to the problem of isometric immersions in Rn using spinors was found by Bayard et al. (Pac J Math, arXiv:1505.02935v4 [math-ph], 2016). However to use spinors one must assume that the manifold carries a Spin -structure and, especially for complex manifolds where is more natural to consider Spin C-structures, this hypothesis is somewhat restrictive. In the present work we show how the above solution can be adapted to Spin C-structures.
2017
Jardim, Marcos; Maican, Mario; Tikhomirov, Alexander S.
Moduli spaces of rank 2 instanton sheaves on the projective space Journal Article
Em: Pacific Journal of Mathematics, vol. 291, no. 2, 2017.
Resumo | Links | BibTeX | Tags:
@article{nokey,
title = {Moduli spaces of rank 2 instanton sheaves on the projective space},
author = {Marcos Jardim and Mario Maican and Alexander S. Tikhomirov},
url = {https://msp.org/pjm/2017/291-2/p06.xhtml},
doi = {10.2140/pjm.2017.291.399},
year = {2017},
date = {2017-09-14},
journal = {Pacific Journal of Mathematics},
volume = {291},
number = {2},
abstract = {We study the irreducible components of the moduli space of instanton sheaves on
P
3
, that is,
μ
-semistable rank 2 torsion-free sheaves
E
with
c
1
(
E
)
=
c
3
(
E
)
=
0
satisfying
h
1
(
E
(
−
2
)
)
=
h
2
(
E
(
−
2
)
)
=
0
. In particular, we classify all instanton sheaves with
c
2
(
E
)
≤
4
, describing all the irreducible components of their moduli space. A key ingredient for our argument is the study of the moduli space
T
(
d
)
of stable sheaves on
P
3
with Hilbert polynomial
P
(
t
)
=
d
⋅
t
, which contains, as an open subset, the moduli space of rank 0 instanton sheaves of multiplicity
d
; we describe all the irreducible components of
T
(
d
)
for
d
≤
4
.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We study the irreducible components of the moduli space of instanton sheaves on
P
3
, that is,
μ
-semistable rank 2 torsion-free sheaves
E
with
c
1
(
E
)
=
c
3
(
E
)
=
0
satisfying
h
1
(
E
(
−
2
)
)
=
h
2
(
E
(
−
2
)
)
=
0
. In particular, we classify all instanton sheaves with
c
2
(
E
)
≤
4
, describing all the irreducible components of their moduli space. A key ingredient for our argument is the study of the moduli space
T
(
d
)
of stable sheaves on
P
3
with Hilbert polynomial
P
(
t
)
=
d
⋅
t
, which contains, as an open subset, the moduli space of rank 0 instanton sheaves of multiplicity
d
; we describe all the irreducible components of
T
(
d
)
for
d
≤
4
.
P
3
, that is,
μ
-semistable rank 2 torsion-free sheaves
E
with
c
1
(
E
)
=
c
3
(
E
)
=
0
satisfying
h
1
(
E
(
−
2
)
)
=
h
2
(
E
(
−
2
)
)
=
0
. In particular, we classify all instanton sheaves with
c
2
(
E
)
≤
4
, describing all the irreducible components of their moduli space. A key ingredient for our argument is the study of the moduli space
T
(
d
)
of stable sheaves on
P
3
with Hilbert polynomial
P
(
t
)
=
d
⋅
t
, which contains, as an open subset, the moduli space of rank 0 instanton sheaves of multiplicity
d
; we describe all the irreducible components of
T
(
d
)
for
d
≤
4
.
von Flach, Rodrigo A.; Jardim, Marcos
Moduli spaces of framed flags of sheaves on the projective plane Journal Article
Em: Journal of Geometry and Physics, vol. 118, pp. 138-168, 2017.
Resumo | Links | BibTeX | Tags:
@article{nokey,
title = {Moduli spaces of framed flags of sheaves on the projective plane},
author = {Rodrigo A. von Flach and Marcos Jardim},
url = {https://www.sciencedirect.com/science/article/pii/S0393044017300323?via%3Dihub},
doi = {https://doi.org/10.1016/j.geomphys.2017.01.019},
year = {2017},
date = {2017-06-09},
journal = {Journal of Geometry and Physics},
volume = {118},
pages = {138-168},
abstract = {We study the moduli space of framed flags of sheaves on the projective plane via an adaptation of the ADHM construction of framed sheaves. In particular, we prove that, for certain values of the topological invariants, the moduli space of framed flags of sheaves is an irreducible, nonsingular variety carrying a holomorphic pre-symplectic form.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
We study the moduli space of framed flags of sheaves on the projective plane via an adaptation of the ADHM construction of framed sheaves. In particular, we prove that, for certain values of the topological invariants, the moduli space of framed flags of sheaves is an irreducible, nonsingular variety carrying a holomorphic pre-symplectic form.
Franco, Emilio; Jardim, Marcos; Marchesi, Simone
Branes in the moduli space of framed sheaves Journal Article
Em: Bulletin des Sciences Mathématiques, vol. 141, iss. 4, pp. 353-383, 2017.
Resumo | Links | BibTeX | Tags:
@article{nokey,
title = {Branes in the moduli space of framed sheaves},
author = {Emilio Franco and Marcos Jardim and Simone Marchesi},
url = {https://www.sciencedirect.com/science/article/pii/S0007449717300386?via%3Dihub},
doi = {https://doi.org/10.1016/j.bulsci.2017.04.002},
year = {2017},
date = {2017-06-05},
journal = {Bulletin des Sciences Math\'{e}matiques},
volume = {141},
issue = {4},
pages = {353-383},
abstract = {In the physicist's language, a brane in a hyperk\"{a}hler manifold is a submanifold which is either complex or lagrangian with respect to three K\"{a}hler structures of the ambient manifold. By considering the fixed loci of certain involutions, we describe branes in Nakajima quiver varieties of all possible types. We then focus on the moduli space of framed torsion free sheaves on the projective plane, showing how the involutions considered act on sheaves, and proving the existence of branes in some cases.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
In the physicist's language, a brane in a hyperkähler manifold is a submanifold which is either complex or lagrangian with respect to three Kähler structures of the ambient manifold. By considering the fixed loci of certain involutions, we describe branes in Nakajima quiver varieties of all possible types. We then focus on the moduli space of framed torsion free sheaves on the projective plane, showing how the involutions considered act on sheaves, and proving the existence of branes in some cases.
Ballico, E.; Gasparim, E.; Grama, Lino; Martin, L. A. B. San
Some Landau–Ginzburg models viewed as rational maps Journal Article
Em: Indagationes Mathematicae, vol. 28, iss. 3, pp. 615-628, 2017.
Resumo | Links | BibTeX | Tags: Landau–Ginzburg
@article{nokey,
title = {Some Landau\textendashGinzburg models viewed as rational maps},
author = {E. Ballico and E. Gasparim and Lino Grama and L.A.B. San Martin},
url = {https://www.sciencedirect.com/science/article/pii/S0019357717300150?via%3Dihub},
doi = {https://doi.org/10.1016/j.indag.2017.01.007},
year = {2017},
date = {2017-05-18},
journal = {Indagationes Mathematicae},
volume = {28},
issue = {3},
pages = {615-628},
abstract = {Gasparim, Grama and San Martin (2016) showed that height functions give adjoint orbits of semisimple Lie algebras the structure of symplectic Lefschetz fibrations (superpotential of the LG model in the language of mirror symmetry). We describe how to extend the superpotential to compactifications. Our results explore the geometry of the adjoint orbit from 2 points of view: algebraic geometry and Lie theory.},
keywords = {Landau\textendashGinzburg},
pubstate = {published},
tppubtype = {article}
}
Gasparim, Grama and San Martin (2016) showed that height functions give adjoint orbits of semisimple Lie algebras the structure of symplectic Lefschetz fibrations (superpotential of the LG model in the language of mirror symmetry). We describe how to extend the superpotential to compactifications. Our results explore the geometry of the adjoint orbit from 2 points of view: algebraic geometry and Lie theory.
Grama, Lino; Negreiros, Caio J. C.; Oliveira, Ailton R.
Invariant almost complex geometry on flag manifolds: geometric formality and Chern numbers Journal Article
Em: Annali di Matematica Pura ed Applicata, vol. 196, pp. 165–200, 2017.
Resumo | Links | BibTeX | Tags: Chern numbers
@article{nokey,
title = {Invariant almost complex geometry on flag manifolds: geometric formality and Chern numbers},
author = {Lino Grama and Caio J. C. Negreiros and Ailton R. Oliveira },
url = {https://link.springer.com/article/10.1007/s10231-016-0568-5#citeas},
doi = {https://doi.org/10.1007/s10231-016-0568-5},
year = {2017},
date = {2017-02-01},
journal = {Annali di Matematica Pura ed Applicata},
volume = {196},
pages = {165\textendash200},
abstract = {In the first part of this paper, we study geometric formality for generalized flag manifolds, including full flag manifolds of exceptional Lie groups. In the second part, we deal with the problem of the classification of invariant almost complex structures on generalized flag manifolds using topological methods.},
keywords = {Chern numbers},
pubstate = {published},
tppubtype = {article}
}
In the first part of this paper, we study geometric formality for generalized flag manifolds, including full flag manifolds of exceptional Lie groups. In the second part, we deal with the problem of the classification of invariant almost complex structures on generalized flag manifolds using topological methods.