14J33 18G80 14D20 14J45; Analysis; Theoretical Computer Science; Algebra and Number Theory; Statistics and Probability; Mathematical Physics; Geometry and Topology; Discrete Mathematics and Combinatorics; Computational Mathematics; 14J33; 18G80; 14D20; 14J45; Mathematics - Algebraic Geometry; Mathematics - Geometric Topology; Mathematics - Quantum Algebra; Mathematics - Symplectic Geometry
Résumé :
[en] We propose a conjectural semiorthogonal decomposition for the derived category of the moduli space of stable rank 2 bundles with fixed determinant of odd degree, independently formulated by Narasimhan. We discuss some evidence for and furthermore propose semiorthogonal decompositions with additional structure. We also discuss two other decompositions. One is a decomposition of this moduli space in the Grothendieck ring of varieties, which relates to various known motivic decompositions. The other is the critical value decomposition of a candidate mirror Landau-Ginzburg model given by graph potentials, which in turn is related under mirror symmetry to Muñoz's decomposition of quantum cohomology. This corresponds to an orthogonal decomposition of the Fukaya category. We discuss how decompositions on different levels (derived category of coherent sheaves, Grothendieck ring of varieties, Fukaya category, quantum cohomology, critical sets of graph potentials) are related and support each other.
Disciplines :
Mathématiques
Auteur, co-auteur :
BELMANS, Pieter ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)
Galkin, Sergey ; PUC-Rio, Departamento de Matemática, Rio de Janeiro, Brazil
Mukhopadhyay, Swarnava; School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India
Co-auteurs externes :
yes
Langue du document :
Anglais
Titre :
Decompositions of moduli spaces of vector bundles and graph potentials
The first author was partially supported by the FWO (Research Foundation–Flanders). The third author was partially supported by the Department of Atomic Energy, India, under project no. 12-R&D-TFR-5.01-0500 and also by the Science and Engineering Research Board, India (SRG/2019/000513).
S. del Baño, ' On the Chow motive of some moduli spaces ', J. Reine Angew. Math. 532 (2001), 105-132.
S. del Baño, ' On the motive of moduli spaces of rank two vector bundles over a curve ', Compositio Math. 131 (1) (2002), 1-30.
F. Bastianelli, P. Belmans, S. Okawa, and A. T. Ricolfi, 'Indecomposability of derived categories in families', (2020). arXiv:2007.00994v1.
P. Belmans, 'Semiorthogonal decompositions for moduli of sheaves on curves', Oberwolfach Workshop 1822, Interactions between Algebraic Geometry and Noncommutative Algebra vol. 15 (2018), 1473-1476.
P. Belmans, L. Fu, and T. Raedschelders, ' Derived categories of flips and cubic hypersurfaces ', Proc. Lond. Math. Soc. 125 (6) (2022), 1452-1482. arXiv:2002.04940v2.
P. Belmans, L. Fu, and T. Raedschelders, ' Hilbert squares: derived categories and deformations ', Selecta Math. New Ser. 25 (37) (2019). arXiv:1810.11873v2.
P. Belmans, S. Galkin, and S. Mukhopadhyay, 'Graph potentials and symplectic geometry of moduli spaces of vector bundles', (2022). arXiv:2206.11584.
P. Belmans, S. Galkin, and S. Mukhopadhyay, 'Graph potentials and topological quantum field theories', (2022). arXiv:2205.07244.
P. Belmans, and A. Krug, ' Derived categories of (nested) Hilbert schemes ', Michigan Mathematical Journal. arXiv:1909.04321v2.
P. Belmans, and S. Mukhopadhyay, ' Admissible subcategories in derived categories of moduli of vector bundles on curves ', Adv. Math. 351 (2019), 653-675. arXiv:1807.00216v1.
I. Biswas, T. Gómez, and K-S. Lee, ' Semi-orthogonal decomposition of symmetric products of curves and canonical system ', Rev. Mat. Iberoam. 37 (5) (2021), 1885-1896. arXiv:1807.10702v2.
F. Bittner, ' The universal Euler characteristic for varieties of characteristic zero ', Compos. Math. 140 (4) (2004), 1011-1032. arXiv:math/0111062v1.
A. I. Bondal, M. Larsen, and V. Lunts, ' Grothendieck ring of pretriangulated categories ', Int. Math. Res. Not. 29 (2004), 1461-1495. arXiv:math/0401009v1.
A. I. Bondal, and D. O. Orlov, ' Semiorthogonal decomposition for algebraic varieties ', (1995). arXiv:alg-geom/9506012.
D. Cheong, and C. Li, 'On the conjecture of GGI for ', Adv. Math. 306 (2017), 704-721. arXiv:1412.2475v5.
U. V. Desale, and S. Ramanan, ' Classification of vector bundles of rank on hyperelliptic curves ', Invent. Math. 38 (2) (1976), 161-185.
A. Fonarev, and A. Kuznetsov, ' Derived categories of curves as components of Fano manifolds ', J. Lond. Math. Soc. 97 (1) (2018), 24-46. arXiv:1612.02241v2.
S. Galkin, 'The conifold point', arXiv:1404.7388v1.
S. Galkin, V. Golyshev, and H. Iritani, ' Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures ', Duke Math. J. 165 (11) (2016), 2005-2077. arXiv:1404.6407v4.
T. L. Gómez, and K-S. Lee, 'Motivic decompositions of moduli spaces of vector bundles on curves', arXiv:2007.06067v1.
G. Harder, ' Eine Bemerkung zu einer Arbeit von P. E. Newstead ', J. Reine Angew. Math. 242 (1970), 16-25.
G. Harder, and M. S. Narasimhan, ' On the cohomology groups of moduli spaces of vector bundles on curves ', Math. Ann. 212 (1975), 215-248.
A. Hatcher, and W. Thurston, ' A presentation for the mapping class group of a closed orientable surface ', Topology 19 (3) (1980), 221-237.
F. Heinloth, ' A note on functional equations for zeta functions with values in Chow motives ', Ann. Inst. Fourier (Grenoble) 57 (6) (2007), 1927-1945. arXiv:math/0512237v1.
Q. Jiang, and N. C. Leung, ' Derived categories of projectivizations and flops ', Adv. Math. 396 (108169) (2022), 44.
M. M. Kapranov, 'The elliptic curve in the S-duality theory and Eisenstein series for Kac-Moody groups', arXiv:math/0001005v2.
A. Krug, D. Ploog, and P. Sosna, ' Derived categories of resolutions of cyclic quotient singularities ', Q. J. Math. 69 (2) (2018), 509-548. arXiv:1701.01331v2.
A. Krug, and P. Sosna, ' On the derived category of the Hilbert scheme of points on an Enriques surface ', Selecta Math. (N.S.) 21 (4) (2015), 1339-1360. arXiv:1404.2105v1.
A. Kuznetsov, ' Homological projective duality ', Publ. Math. Inst. Hautes Études Sci. 105 (2007), 157-220. arXiv:math/0507292v1.
A. Kuznetsov, ' Lefschetz decompositions and categorical resolutions of singularities ', Selecta Math. (N.S.) 13 (4) (2008), 661-696. arXiv:math/0609240v2.
A. Kuznetsov, 'Semiorthogonal decompositions in algebraic geometry', in Proceedings of the International Congress of Mathematicians -mdash;Seoul 2014. Vol. II. (Kyung Moon Sa, Seoul, 2014), 635-660. arXiv:1404.3143v3.
A. Kuznetsov, and M. Smirnov, ' On residual categories for Grassmannians ', Proc. Lond. Math. Soc. (3) 120 (5) (2020), 617-641. arXiv:1802.08097v3.
A. Kuznetsov, and M. Smirnov, ' Residual categories for (co)adjoint Grassmannians in classical types ', Compos. Math. 157 (6) (2021), 1172-1206. arXiv:2001.04148v3.
K-S. Lee, 'Remarks on motives of moduli spaces of rank 2 vector bundles on curves', arXiv:1806.11101v2.
K-S. Lee, and M. S. Narasimhan, 'Symmetric products and moduli spaces of vector bundles of curve', arXiv:2106.04872v1.
X. Lin, 'On nonexistence of semi-orthogonal decompositions in algebraic geometry', arXiv:2107.09564v1.
D. Mumford, and P. Newstead, ' Periods of a moduli space of bundles on curves ', Amer. J. Math. 90 (1968), 1200-1208.
V. Muñoz, ' Higher type adjunction inequalities for Donaldson invariants ', Trans. Amer. Math. Soc. 353 (7) (2001), 2635-2654. arXiv:math/9901046v2.
V. Muñoz, ' Quantum cohomology of the moduli space of stable bundles over a Riemann surface ', Duke Math. J. 98 (3) (1999), 525-540. arXiv:alg-geom/9711030v1.
V. Muñoz, 'Ring structure of the Floer cohomology of ', Topology 38 (3) (1999), 517-528. arXiv:dg-ga/9710029v1.
M. S. Narasimhan, 'Derived categories of moduli spaces of vector bundles on curves', J. Geom. Phys. 122 (2017), 53-58.
M. S. Narasimhan, ' Derived categories of moduli spaces of vector bundles on curves II ', in Geometry, Algebra, Number Theory, and Their Information Technology Applications vol. 251 (Springer Proc. Math. Stat. Springer, Cham, 2018), 375-382.
M. S. Narasimhan, and C. S. Seshadri, ' Stable and unitary vector bundles on a compact Riemann surface ', Ann. of Math. (2) 82 (1965), 540-567.
P. Newstead, ' Stable bundles of rank and odd degree over a curve of genus ', Topology 7 (1968), 205-215. url: https://doi.org/10.1016/0040-9383(68)90001-3.
P. Newstead, ' Topological properties of some spaces of stable bundles ', Topology 6 (1967), 241-262.
D. O. Orlov, 'Triangulated categories of singularities and D-branes in Landau-Ginzburg models', Tr. Mat. Inst. Steklova 246 (2004). Translation in Proc. Steklov Inst. Math. 246 (3) (2004), 227-248, 240-262. arXiv:math/0302304v2.
A. Samokhin, 'The Frobenius morphism on flag varieties, I', arXiv:1410.3742v3.
F. Sanda, and Y. Shamoto, 'An analogue of Dubrovin's conjecture', Ann. Inst. Fourier (Grenoble) 70 (2) (2020), 621-682. arXiv:1705.05989v2.
I. Smith, ' Floer cohomology and pencils of quadrics ', Invent. Math. 189 (1) (2012), 149-250. arXiv:1006.1099v3.
J. Tevelev, and S. Torres, 'The BGMN conjecture via stable pairs', (2021). arXiv:2108.11951.
M. Thaddeus, ' Stable pairs, linear systems and the Verlinde formula ', Invent. Math. 117 (2) (1994), 317-353. arXiv:alg-geom/9210007v1.
Y. Toda, ' Semiorthogonal decompositions of stable pair moduli spaces via d-critical flips ', J. Eur. Math. Soc. (JEMS) 23 (5) (2021), 1675-1725. arXiv:1805.00183v2.
K. Xu, and S-T. Yau, 'Semiorthogonal decomposition of ', arXiv:2108.13353v2.
