A key example (#):
Assume that we are given a family of curves C-> B of arithmetic genus g, such that each geometric fiber C_b has at worst planer singularities. We can consider the compactified Jacobian fibraiton f: J -> B. For convenience let's assume that both J and B are smooth (but f can have singular fibers!), and you are allowed to add some further mild assumptions when needed.
The question is: What we can say about the cohomology of J?
[The reason that we focus on such an example is because it appears in many branches of mathematics: Hitchin system, Lagrangian fibration of compact hyperkähler varieties, enumerative geometry, affine Springer thory, knot invariants, etc]
Tuesday. (A) Hodge theory and Decomposition Theorem (Francesca, Jacob)
Toy model of the decomposition theorem (where the map is proper and smooth); explain that it is from Hodge theory! [4, Theorem 1.5.3, and its references]
Review some basics of perverse sheaves, and state the decomposition theorem of Beilinson-Bernsterin-Deligne-Gabber, [1], [4 Lecture 1]
Explain the idea of the proof of de Cataldo-Migliorini which is to use linear algebra obtained from Hodge theory in a very complicated way (in particular the role played by the Riemann bilinear relation); focus on the semismall case. [2] (see also some discussions of [1])
If time permits (which is quite unlikely), explain more about the general case [3]
References:
[1] Williamson: The Hodge Theory of the Decomposiiton Theorem (Bourbaki talk after [3]).
[2] de Cataldo, Migliorini: The Hard Lefschetz Theorem and the topology of semismall maps.
[3] de Cataldo, Migliorini: The Hodge theory of algbraic maps.
[4] de Cataldo: perverse sheaves and topology of algebraic varieties (PCMI notes).
Wednesday. (B) The Ngô support theorem (Andres, Younghan)
Statement of the Ngô support theorem. [5, Section 7] or the survey article, see also [6, Section 1]
Explain what if we apply it to the compactified Jacobian example (#). (e..g. [8, Theorem 2.4])
Ideas of the proof; explain the Goresky-MacPherson inequality and how to inprove it for abelian fibrations.
If time permits, explain the proof of the \chi-independence theorem (again, for convenience, one can first focus on the integral curves. [6,7]
References:
[5] Ngo: Le lemme fondamental pour les algèbres de Lie. See also the survey article of Ngô.
[6] Maulik, Shen: Cohomological \chi-independence for moduli of one-dimensional sheaves and moduli of Higgs bundles.
[7] Chaudouard, Laumon: Un théorème du support pour la fibration de Hitchin
Thursday. Free afternoon.
Friday. (C) Applications in enumerative geometry （Alessio, Yifan）
The purpose of today is to understand the decomposition theorem for compactified Jacobian (#) in terms of Hilbert schemes. [1]; see also [2]
Discuss briefly the Maulik-Toda proposal for Gopakumar-Vafa invariants; explain that the Jacobian-Hilbert scheme correspondence above can be viewed as a toy model for the Gopakumar-Vafa/Pandharipande-Thomas invariants correspondence. [10]
References:
[8] Maulik, Yun: Macdonald formula for curves with planar singularities.
[9] Migliorini, Shende: A support theorem for Hilbert schemes of planar curves.
[10] Maulik, Toda: Gopakumar-Vafa invariants via vanishing cycles.
Satuarday. (D) Decomposition theorem, derived categories, and algebraic cycles (Shengxuan, Anibal)
The purpose of today is to understand the motivic enhancement of the decomposition theorem for compactified Jacobian fibration (#) using Arinkin's Poincaré sheaf. [12, Theorem 0.3]
Explain first a toy model of this: the motivic decomposition holds for abelian schemes (without singular fibers) via Mukai's Poincaré line bundle. [11]
If time permits, discuss the general conjecture and philosophy of Corti-Hanamura. [13]
References:
[11] Deninger, Murre: Motivic decomposition of abelian schemes and the Fourier transform.
[12] Maulik, Shen, Yin: Perverse filtrations and Fourier transforms.
[13] Corti, Hanamura: Motivic decomposition and intersection Chow motives.
