Peking University Algebraic Geometry Seminar

A useful vanishing theorem for understanding characteristic zero singularities is Grauert–Riemenschneider vanishing, which asserts that if f : Y → X is a projective birational morphism and Y is smooth, then higher pushforwards of the sheaf of top forms on Y vanish. A remarkable consequence of this result is that characteristic zero klt singularities are rational.

As one might expect, this vanishing theorem fails over fields of positive characteristic. In this talk, we will explain how to prove a Witt vector version of Grauert–Riemenschneider vanishing, and consequences to the rationality of certain singularities in positive characteristic.

Let C be a complex integral curve with planar singularities. Let J be the compactified Jacobian of C. There are two filtrations on the cohomology group H^*(J). One is obtained by the nilpotent morphism defined by cupping a certain ample divisor on J, which we call the Lefschetz filtration. To obtain the other filtration, we put C into a family of curves 𝒞 → B so that J can be embedded into a family f : 𝒥 → B, and we let B, 𝒞, 𝒥 be smooth. Then Rf_*(ℚ_𝒥) decomposes into a direct sum of its (shifted) perverse cohomologies. Restricting this decomposition to fibers, we get a filtration on H^*(J) called the perverse filtration. We show that these two filtrations are opposite to each other as conjectured by Maulik–Yun.

The Enriques criterion for unirationality of conic bundles states that if X is a conic bundle over P², then X is unirational if and only if the conic bundle admits a rational multisection. Furthermore, if X is unirational but not rational, then such a rational multisection necessarily has even degree. In this talk, we report on joint work in progress with Jeffrey Diller and Eric Riedl studying when such a conic bundle X → P² admits a degree 2 rational multisection.

I will discuss resolution of singularities for foliations and present our approach via a weighted principalization theorem for ideals on smooth orbifolds equipped with a foliation. As an application, I will describe the resolution of certain foliations in arbitrary dimensions, including Darboux totally integrable foliations. This is joint work with D. Abramovich, M. Temkin, and J. Wlodarczyk.

Markman and O'Grady uncovered a deep relation between abelian fourfolds of Weil type with discriminant 1 and hyper-Kähler varieties of generalized Kummer type, at the level of Hodge theory and period domains. Markman was able to use this to prove the Hodge conjecture for these fourfolds; he later found also a different proof which works for Weil fourfolds with arbitrary discriminant. In my talk I will explain how Weil fourfolds with discriminant 1 are very closely related to certain hyper-Kähler varieties of OG6-type, in a direct and geometric way. As a consequence, we obtain another proof of the Hodge conjecture for Weil fourfolds with discriminant 1, as well as for many families of hyper-Kähler varieties of OG6-type which form loci of codimension 1 in their moduli spaces. The results that I will discuss are joint work with Lie Fu.

The period index problem has been suggested by Colliot-Thélène in early 2000s, and some of its low dimensional cases have been solved by de Jong, Lieblich and more recently Perry–Hotchkiss. We adapt the point of view of moduli of twisted sheaves, and by combining classical results in the theory of moduli of vector bundles, we realize the moduli of twisted vector bundles as the obstruction class of descending the determinantal line bundle; thus giving a bound on the period index problem for genus 2 curves over an arbitrary field.

In this talk, I will present a recent result regarding the volumes of foliations. We prove that for log canonical foliations which are birationally bounded by algebraically integrable families, the set of their volumes satisfies the DCC, answering a special case of a question posed by Cascini, Hacon, and Langer. As a key ingredient, I will discuss the deformation invariance of relative log canonical volumes for a family of weakly semistable morphisms, which can be viewed as a relative version of the result proved by Hacon, McKernan, and Xu.