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Equivariant stable sheaves and toric GIT

Published online by Cambridge University Press:  07 January 2022

Andrew Clarke  and
Carl Tipler
Andrew Clarke
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, Av. Athos da Silveira Ramos 149, Rio de Janeiro, RJ, 21941-909, Brazil (andrew@im.ufrj.br)
Carl Tipler
Affiliation:
Univ Brest,UMR CNRS 6205, Laboratoire de Mathématiques de Bretagne Atlantique, France (carl.tipler@univ-brest.fr)
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Abstract

For $(X,\,L)$ a polarized toric variety and $G\subset \mathrm {Aut}(X,\,L)$ a torus, denote by $Y$ the GIT quotient $X/\!\!/G$. We define a family of fully faithful functors from the category of torus equivariant reflexive sheaves on $Y$ to the category of torus equivariant reflexive sheaves on $X$. We show, under a genericity assumption on $G$, that slope stability is preserved by these functors if and only if the pair $((X,\,L),\,G)$ satisfies a combinatorial criterion. As an application, when $(X,\,L)$ is a polarized toric orbifold of dimension $n$, we relate stable equivariant reflexive sheaves on certain $(n-1)$-dimensional weighted projective spaces to stable equivariant reflexive sheaves on $(X,\,L)$.

Keywords

Stable sheavesslope stabilityequivariant sheavesdescenttoric GIT
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 2 , April 2023 , pp. 385 - 416
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Angella, D. and Spotti, C.. Kähler-Einstein metrics: old and new. Complex Manifolds 4 (2017), 200–244.CrossRefGoogle Scholar
Cox, D., Little, J. and Schenck, H.. Toric Varieties, Providence, RI (American Mathematical Society (AMS), 2011).CrossRefGoogle Scholar
Danilov, V. I.. The geometry of toric varieties. Uspekhi Mat. Nauk 33 (1978), 85–134.Google Scholar
Dasgupta, J., Dey, A. and Khan, B., Stability of equivariant vector bundles over toric varieties. ArXiv preprint 1910.13964.Google Scholar
Donaldson, S., Kähler-Einstein metrics and algebraic geometry. In Current developments in mathematics 2015, pp. 1–25 (Int. Press, Somerville, MA, 2016).CrossRefGoogle Scholar
Futaki, A.. The Ricci curvature of symplectic quotients of Fano manifolds. Tohoku Math. J. (2) 39 (1987), 329–339.CrossRefGoogle Scholar
García-Prada, O.. Invariant connections and vortices. Comm. Math. Phys. 156 (1993), 527–546.CrossRefGoogle Scholar
Guenancia, H.: Semistability of the tangent sheaf of singular varieties. Algebr. Geom., 3(5) (2016), 508–542.CrossRefGoogle Scholar
Hering, M., Nill, B. and Suess, H., Stability of tangent bundles on smooth toric Picard-rank-2 varieties and surfaces. ArXiv preprint 1910.08848.Google Scholar
Huybrechts, D. and Lehn, M., The Geometry of Moduli Spaces of Sheaves, 2 ed., Cambridge Mathematical Library (Cambridge University Press, 2010).Google Scholar
Ilten, N. and Süss, H.. Equivariant vector bundles on $T$-varieties. Transform. Groups 20 (2015), 1043–1073.CrossRefGoogle Scholar
Klyachko, A. A.. Equivariant bundles over toric varieties. Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), 1001–1039.Google Scholar
Kobayashi, S., Differential geometry of complex vector bundles, volume 15 of Publications of the Mathematical Society of Japan. (Princeton University Press, Princeton, NJ; Princeton University Press, Princeton, NJ, 1987). Kanô Memorial Lectures, 5.Google Scholar
Kool, M.: Fixed point loci of moduli spaces of sheaves on toric varieties. Adv. Math., 227(4) (2011), 1700–1755.CrossRefGoogle Scholar
Lazarsfeld, R., Positivity in algebraic geometry. I, volume 48 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. (Springer-Verlag, Berlin, 2004). Classical setting: line bundles and linear series.Google Scholar
Mehta, V. B. and Ramanathan, A.. Restriction of stable sheaves and representations of the fundamental group. Invent. Math. 77 (1984), 163–172.CrossRefGoogle Scholar
Mumford, D., Fogarty, J. and Kirwan, F., Geometric invariant theory, volume 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)]. 3rd ed. (Springer-Verlag, Berlin, 1994).Google Scholar
Riera, I. M.. Parabolic vector bundles and equivariant vector bundles. Internat. J. Math. 13 (2002), 907–957.CrossRefGoogle Scholar
Nevins, T.. Descent of coherent sheaves and complexes to geometric invariant theory quotients. J. Algebra 320 (2008), 2481–2495.CrossRefGoogle Scholar
Perling, M.. Graded rings and equivariant sheaves on toric varieties. Math. Nachr. 264 (2004), 181–197.CrossRefGoogle Scholar
Schneider, R., Convex Bodies: the Brunn-Minkowski Theory, volume 44 of Encyclopedia of Mathematics and its Applications. 2nd. ed. (Cambridge University Press, 2013).Google Scholar
Thaddeus, M., Toric quotients and flips. In Topology, geometry and field theory, pp. 193–213 (World Sci. Publ., River Edge, NJ, 1994).Google Scholar