Math 7393 - Moduli theory of sheaves and Vafa-Witten theory

Lecture 1 - January 18 - Alina. Introduction to modular forms, outline of the basic phenomenon studied in the course. Plan of the course, discussion of bibliography.

Lecture 2 - January 25 - Alina. The Quot functor and its representability. Main steps in the construction of the Quot scheme: uniform boundedness of quotients of a fixed coherent sheaf, flattening stratification, the Quot scheme as a stratum inside the Grassmannian; properness. Basic deformation theory of quotients: Zariski tangent space to the Quot scheme, obstruction space. Examples: the Quot scheme on P^1 is smooth, Hilbert schemes of points on surfaces are smooth.

References: “Fundamental Algebraic Geometry” - Chapters 5 and 6.

Lecture 3 - February 1 - Alina. The geometry of Hilbert schemes of points on surfaces: punctual Hilbert schemes and the Hilbert-Chow morphism. Torus actions, the Bialynicki-Birula decomposition. Topological calculations: the generating series for Euler characteristics of the Quot scheme on P^1, of the Hilbert schemes of points.

References: “Fundamental Algebraic Geometry” - Chapter 7. “On parametrized rational curves in Grassmann Varieties”, by S. A. Stromme. "On a cell decomposition of the Hilbert scheme of points in the plane,” by G. Ellingsrud and S. A. Stromme, Invent. Math. 91, 365 - 370 (1988).

Lecture 4 - February 8 - Mischa Mironov. Notion of stability, examples of stable and unstable sheaves, Harder-Narasimhan filtration, uniform boundedness results for semistable sheaves.

Reference: “The geometry of moduli spaces of sheaves”, by D. Huybrechts, M. Lehn -- Chapter1.

Lecture 5 - February 15 - Alina. Moduli of semistable sheaves — the moduli functor M. Notion of S-equivalence for semistable sheaves. Reductive group actions and basic elements of GIT. Construction of the moduli space of semistable sheaves as a GIT quotient of an open subscheme of Quot. The moduli space corepresents the functor M.

Reference: “The geometry of moduli spaces of sheaves”, by D. Huybrechts, M. Lehn -- Chapter 4.

Lecture 6 - February 22 - Alina. Discussion of the existence of universal and quasiuniversal families; descent conditions. Examples of moduli spaces of sheaves: moduli of semistable bundles over elliptic curves using Fourier-Mukai techniques.

References: “The geometry of moduli spaces of sheaves”, by D. Huybrechts, M. Lehn -- Chapter 4. “Abelian varieties” by A. Polishchuk. “Fourier-Mukai transforms and stable bundles on elliptic curves,” by G. Hein and D. Ploog, Contributions to Algebra and Geometry Volume 46 (2005), 423 - 434. “ Vector bundles over an elliptic curve,” by M. Atiyah, Proceedings of the London Math. Soc. 7 (1957), 414 - 452.

Lecture 7 - March 15 - Mohamed Elbehiry. Series of motivic invariants for moduli spaces of semistable sheaves over ruled surfaces.

References: "The Betti numbers of the moduli space of stable sheaves of rank 2 on a ruled surface", by K. Yoshioka, Math. Ann. 302 (1995), no. 3, 519–540. "Invariants of moduli spaces of stable sheaves on ruled surfaces", by S. Mozgovoy, arXiv:1302.4134.

Lecture 8 - March 26 (rescheduled from March 22) - Alina. On the cohomology of moduli spaces of sheaves.

Lecture 9 - March 29 - Mohamed. Motivic invariants for moduli spaces of sheaves over ruled surfaces - continuation of Lecture 7.

Lecture 10 - April 5. Part 1 - Alina: Equivariant cohomology and the Atiyah-Bott localization formula. Simple examples. Part 2 - Zhuang He: Spectral curves, the Hitchin system, the Higgs moduli space on curves and surfaces.

References for the lectures on April 5 and 12: "The moment map and equivariant cohomology", by M. Atiyah and R. Bott, Topology 23 (1984), 1-28. "Equivariant intersection theory", by D.Edidin and W. Graham, Invent. Math. 131 (1998), 595-634. Chapter 9 of "Mirror Symmetry and Algebraic Geometry", by D. Cox and S. Katz. Chapter 4 of "Mirror Symmetry", a Clay Mathematics monograph.

"Stable bundles and integrable systems", by N. Hitchin, Duke Math. J. 54 (1987), 91-114. "Spectral curves and the generalized theta divisor," by A. Beauville, M. Narasimhan, S. Ramanan, J. reine angew. Math. 398 (1989), 169-179.

Lecture 11 - April 12. Part 1 - Alina: Equivariant localization in examples. The count of lines on a general quintic threefold. Part 2 - Zhuang: The Higgs moduli space, continuation of April 5 lecture.

Lecture 12 - April 19 - Alina. Part 1: Perfect obstruction theories and virtual cycles. Localization formula for virtual cycles. Part 2: Vafa-Witten invariants defined via Higgs pairs on surfaces. Fixed loci in the monopole branch.

References: "The intrinsic normal cone", by K. Behrend and B. Fantechi, Invent. Math. 128 (1997), 45-88. "Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties", by J. Li and G. Tian, JAMS 11 (1998), 119-174. "Localization of virtual classes", by T. Graber and R. Pandharipande, Invent. Math. 135 (1999), 487-518.

"Vafa-Witten invariants for projective surfaces I: stable case", by Y. Tanaka and R. Thomas, arXiv:1702.08487v3.