Spring 2023
January 25: Arnab Kundu (Université Paris-Saclay)
Location: Zoom (Special Time: 11:30 AM EST)
Title: Torsors on Valuation Rings
Abstract: A conjecture of Grothendieck and Serre states that a torsor under a reductive group over a Noetherian regular scheme X is Zariski locally trivial if it is generically trivial. Recently, this conjecture has seen progress through the work of Fedorov, Panin and Česnavičius. We shall see the historical background of this conjecture, followed by the techniques that go into the proof of the quasi-split case in the analogous situation when X is a smooth scheme over a valuation ring of rank one.
February 8: Eric Riedl (University of Notre Dame)
Location: REC 112
Title: Non-free curves and Geometric Manin's Conjecture
Abstract: When studying the space of maps from a smooth curve B to X, the free curves, that is, those whose restricted tangent bundle has vanishing higher cohomology, are particularly nice. They are smooth points of the space of morphisms, and lie on a unique component of the expected dimension. Nonfree curves, on the other hand, can be much more difficult to understand. The components of the space of maps that consist entirely of nonfree curves are particularly difficult to understand. In this talk, we provide a geometric classification of the ways that a family of curves can be nonfree. In the more general setting of a Fano fibration, we show that these curves come from a-covers of X, and we show that the set of such a-covers up to equivalence form a bounded family. This proves the first of Batyrev's heuristics that make up Geometric Manin's Conjecture for Fano fibrations over arbitrary base curves. This is joint work with Brian Lehmann and Sho Tanimoto.
February 22: Deepam Patel (Purdue University)
Location: Zoom
Title: Enriched Hodge Structures
Abstract: An important application of the theory of mixed Hodge structures is in the study of algebraic cycles or K-theory (for example via the Hodge or Bloch-Beilinson conjectures) of smooth projective varieties. On the other hand, mixed hodge theory is often insufficient for the study of cycles/K-theory in the open or singular settings. I will discuss an enrichment of the category of mixed Hodge structures and some results on the existence of natural objects in this category which allow for the possibility of understanding cycles on complex analytic links. If there is time, I'll discuss some analogous results in the setting of formal schemes (where previously even the existence of a MHS was unknown). This is partially based on joint work with M. Nori and V. Srinivas.
March 1: Yueqiao Wu (University of Michigan)
Location: REC 112
Title: A non-Archimedean characterization of local K-stability
Abstract: Log Fano cone singularities are generalizations of cones over log Fano varieties, and have a local K-stability theory extending the one for log Fano varieties. In this talk, we aim to give a characterization for local K-stability from a non-Archimedean point of view. As a consequence of this characterization, we can show that a log Fano cone singularity is K-polystable with respect to a larger class of test configurations if it admits a Ricci-flat Kähler cone metric, strengthening earlier results of Collins-Székelyhidi and Li.
March 22: Fumiaki Suzuki (University of California, Los Angeles)
Location: REC 112
Title: Two coniveau filtrations and algebraic equivalence over finite fields
Abstract: Over the complex numbers, the integral cohomology of a smooth projective variety is endowed with the coniveau and strong coniveau filtrations. The two filtrations differ in general as recently shown by Benoist and Ottem, and this result may be exnteded to the l-adic setting over any algebraically closed field of characteristic not 2. In this talk. I would like to discuss some consequences of the equality of the two filtrations for algebraic equivalence for codimension 2 cycles over finite fields. As an application, we show the vanishing of the third unramified cohomology for a large class of rationally chain connected threefolds over finite fields, confirming a conjecture of Colliot-Thelene and Kahn. This is a joint work with Federico Scavia.
March 28 (Colloquium, Special Date): Neena Gupta (Indian Statistical Institute)
Location: Zoom, 3:30-4:30PM (https://purdue-edu.zoom.us/j/92482721838?pwd=YTBrN1owSHRjM3pkaXBNQXFvcVprQT09)
Title: Some Problems on Polynomial ring
Abstract:
``Polynomials and power series
May they forever rule the world."
Thus begins a poem composed by Shreeram S. Abhyankar in 1970. Polynomials are introduced at a very early stage in our studies. Yet there are many interesting fundamental problems on polynomial rings which are easy to state but difficult to approach. In this talk we shall discuss a few problems on polynomial rings, and some recent progress on them using group action of the additive group (k, +) for a field k.
March 29: Mads Bach Villadsen (Stony Brook University)
Location: REC 112
Title: Generic vanishing and Chen-Jiang decompositions
Abstract: The Generic Vanishing theorem of Green and Lazarsfeld describes the behaviour of the cohomology of direct images of canonical bundles to abelian varieties when twisted by line bundles of degree zero. I will discuss Chen-Jiang decompositions of these direct images, first introduced by J. Chen and Z. Jiang for generically finite morphisms to abelian varieties, which explain their generic vanishing behaviour and certain positivity properties in detail.
April 5: Wenliang Zhang (University of Illinois Chicago)
Location: Zoom
Title: Holonomic D-modules in prime characteristic
Abstract: Over the complex numbers, the notion of holonomic D-modules is central to many areas including the Riemann-Hilbert correspondence. I would like to discuss an approach to holonomic D-modules in prime characteristic p, based on an ongoing joint project with Gennady Lyubeznik. No prior knowledge of D-modules will be assumed.
April 12: Nathan Chen (Columbia University)
Location: REC 112
Title: Fano hypersurfaces and differential forms via positive characteristic
Abstract: Holomorphic forms are an important birational invariant for studying the geometry of a variety. Surprisingly, Kollár showed that in positive characteristic certain (singular) Fano varieties admit many global (n-1)-forms, and he combined this with a specialization method to prove nonrationality of many complex Fano hypersurfaces. In this talk, we will revisit this construction and use it to study geometric questions about Fano hypersurfaces: what are their possible rational endomorphisms, and what does their birational automorphism group look like? Parts of this are joint with David Stapleton as well as with Lena Ji-Stapleton.
April 19: Rui-Jie Yang (Max Planck Institute for Mathematics, Bonn)
Location: Zoom (Special Time: 11:30 AM EDT)
Title: Higher multiplier ideals
Abstract: For any effective Q-divisor D on a complex manifold X, there is a multiplier ideal associated to the pair (X,D), which is an ideal sheaf measuring the singularity of the pair and has many applications in algebraic geometry and commutative algebra. In this talk, I will discuss the construction of a new family of ideal sheaves associated to (X,D), indexed by an integer indicating the Hodge level, such that the lowest level recovers the usual multiplier ideals. This family of ideals is closely related to, but different from, the theory of Hodge ideals developed by Popa and Mustata. Their local and global properties are established systematically using various types of D-modules. One key mechanism is the gluing of local nearby cycles along the divisor as ''twisted complex Hodge modules'' using the language of D-algebra by Bernstein and Beilinson and the theory of complex Hodge modules by Sabbah and Schnell. Some application to the Riemann-Schottky problem and minimal exponents will be given. This is based on the joint work with Christian Schnell.
April 26: Shizhuo Zhang (Max Planck Institute for Mathematics, Bonn)
Location: Zoom
Title: New perspective on categorical Torelli problems for del Pezzo threefold
Abstract: Let Y be a del Pezzo threefold of Picard rank one and degree d ≥ 2. We provide a Brill-Noether reconstruction of those del Pezzo threefolds as a subscheme of a Bridgeland moduli spaces in their Kuznetsov components. We show that any exact equivalence between their Kuznetsov components preserves a distinguished object up to some natural auto-equivalences of the Kuznetsov component. As a result, we give a uniform proof of categorical Torelli theorem for them. Further more, we compute the group of auto-equivalences of their Kuznetsov component by extending the exact equivalences to the whole bounded derived categories. If time permits I will also talk about the group of auto-equivalences of Kuznetsov component of index one prime Fano threefold where a different techniques are used. As an application we show that the group of automorphism of index one genus 8 prime Fano threefold is isomorphic to that of associated Phaffian cubic threefold, which is not known in the literature.