Algebraic Geometry Seminar

Log structures endow schemes with tropical/toric data in a natural way. The corresponding notion of ``differentials with log poles'' makes many mildly singular spaces appear smooth, including simple normal crossings and toric varieties. Stranger still, blowups are not only ``\'etale'' and proper, but ``monomorphisms'' with a natural functor of points! We will work out the example of log smooth curves in detail before seeing a number of payoffs this added structure has to offer. Time permitting, we will discuss an ongoing research direction concerning intersection theory in this setting.

A Fano threefold Y of Picard rank 1 and index 2 admits a canonical semiorthogonal decomposition of its derived category; this decomposition comes with a non-trivial component Ku(Y) — called the Kuznetsov component — that encodes most of the geometry of Y. I will present a joint work with M. Petkovic and F. Rota in which we describe certain moduli spaces of Bridgeland-stable objects on Ku(Y), via the stability conditions constructed by Bayer, Macrì, Lahoz and Stellari. Furthermore, in our work we study the behavior of the Abel-Jacobi map on these moduli. As an application in the case of degree d = 2, we prove a strengthening of a categorical Torelli Theorem by Bernardara and Tabuada.

A D-affine variety is such a variety X that the category of D_X-modules behaves like the category of O_X-modules of an affine variety. Beilinson and Bernstein showed that complex generalized flag varieties are D-affine. It is a folklore conjecture that any smooth projective D-affine variety is of this form. I will talk about current state of this problem. In positive characteristic the problem is related to a new generalization of Miyaoka's generic semipositivity theorem.

I will talk about an analogue of Hodge theory in positive characteristic. In particular, I will show analogues of Schmid’s nilpotent orbit theorem and nearby cycles in positive characteristic. As an application I will prove some strong semipositivity theorems for analogs of complex polarized variations of Hodge structures. This implies semipositivity for the relative canonical divisor of a semistable reduction and it also gives some new results over complex numbers.

I will discuss the asymptotic non-vanishing of syzygies for products of projective spaces, generalizing the monomial methods of Ein-Erman-Lazarsfeld. This provides the first example of how the asymptotic syzygies of a smooth projective variety whose embedding line bundle grows in a semi-ample fashion behave in nuanced and previously unseen ways.

It follows from an observation of A. Coble in 1919 that the automorphism group of an unnodal Enriques surface contains the 2-congruence subgroup of the Weyl group of the E_{10}-lattice. In this talk, I will explain how much bigger the automorphism group of an unnodal Enriques surface can be. Furthermore, I will determine the automorphism group of a generic Enriques surface with smooth K3 cover in arbitrary characteristic, improving the corresponding result of W. Barth and C. Peters for very general Enriques surfaces over the complex numbers.

Recently, the question of whether one can construct nicely behaved moduli spaces for Fano varieties using K-stability has attracted significant interest. More precisely, the Fano K-moduli Conjecture predicts that K-polystable Fano varieties with fixed volume and dimension form a projective good moduli space. In this talk, I will explain the proof of openness of K-semistability for Fano varieties which is one major step in the Fano K-moduli Conjecture. Our proof is a combination of valuative criterion for K-semistability due to Fujita and Li, boundedness of complements due to Birkar, and approximation techniques. This talk is based on joint work with Harold Blum and Chenyang Xu.

In classical differential geometry, Chern-Gauss-Bonnet theorem provides a beautiful index formula connecting geometry to topology. In the first half of the talk, I will discuss how it can be generalized flatly to all perverse sheaves on projective manifolds using D-modules, which is the Dubson-Kashiwara index formula. Then I will talk about generalizations of the Dubson-Kashiwara index formula for perverse sheaves on quasi-projective manifolds from a logarithmic point of view and how they related to Grothendieck-Riemann-Roch using the intersection theory. This is based on joint work with Peng Zhou. In the second half of the talk, I will talk a similar story but from a relative perspective. The topology of Milnor fibers associated to a holomorphic function f contains information of singularities of f. Using D-modules, one can construct the Bernstein-Sato polynomial for f. By a classical theorem of Kashiwara-Malgrange, the monodromy eigenvalues of Milnor fibers can be calculated by using roots of the Bernstein-Sato polynomial. I will talk about how this can be generalized for several holomorphic functions by using Alexander modules in the sense of Sabbah and Bernstein-Sato ideals. This is based on work joint with Nero Budur, Robin Veer and Peng Zhou.

We study the relative Gromov-Witten theory on T*P^1 \times P^1 and show that certain equivariant limits give us the relative invariants on P^1\times \P^1. By formulating the quantum multiplications on Hilb(T*P^1) computed by Devash Maulik and Alexei Oblomkov as vertex operators and computing the product expansion, we demonstrate how to get the insertion and tangency operators computed by Yaim Cooper and Rahul Pandharipande in the equivariant limits.

We associate to any loop-free matroid a motivic zeta function. If the matroid is representable by a complex hyperplane arrangement, then this coincides with the motivic Igusa zeta function of the arrangement. We show that this zeta function is rational and satisfies a functional equation. Moreover, it specializes to the topological zeta function of Robin van der Veer. We answer two questions of van der Veer about this topological zeta function. This is joint work with Dave Jensen and Jeremy Usatine.

The arc space of a variety is an infinite dimensional object. It relates to valuation theory, and has been used to define string theoretic invariants of singular Calabi-Yau varieties and to study singularities in the minimal model program. The infinitesimal structure of arc spaces present interesting features: the formal neighborhood of the arc space at k-rational arcs (where k is the ground field) is an infinite dimensional scheme, but a theorem of Drinfeld-Grinber-Kazhdan states that if the arc is not entirely contained in the singular locus of the variety then the singularities of the formal neighborhood are finite dimensional. The interest in this type of results originates from the expectation that there should be a well behaved theory of perverse sheaves on arc spaces. In this talk, I will present a novelle approach to these results which relies on a general notion of embedding codimension. The talk is based on joint work with Roi Docampo and Christopher Chiu.