Caltech Algebra & Geometry Seminar 2022-2023

Mondays 4-5pm Linde Hall Room 387 OR on Zoom

Organizers: Tim-Henrik Buelles and Abhishek Oswal

Winter 2023

January 9, 2023

Coherent matrix factorizations and superconnections

Zhaoting Wei (Texas A&M University - Commerce)

Matrix factorization plays an important role in mathematics and it is interesting to consider matrix factorizations with coherent components over non-affine spaces. On difficulty is that it is quite complicated to describe quasi-isomorphisms between them. On the other hand, people showed that the category of flat anti-holomorphic superconnections is equivalent to the bounded derived category of coherent sheaves on complex manifolds, which assures us that superconnection is a useful tool in the study of coherent sheaves. In this talk I will describe an attempt to use (non-flat) anti-holomorphic superconnections to give resolutions of coherent matrix factorizations and some possible applications along this line.

January 30, 2023

On P^1 stabilization in unstable motivic homotopy theory

Aravind Asok (University of Southern California)

Unstable motivic homotopy theory gives tools for analyzing the structure of algebraic vector bundles on smooth varieties over a field.  For example, one can study the problem of when a vector bundle splits off trivial rank 1 summands.  I will discuss how an analog of Freudenthal suspension theorem in motivic homotopy theory allows one to completely analyze the preceding splitting problem just described.  

I will discuss an analog in motivic homotopy theory of the classical Freudenthal suspension theorem. 

February 6, 2023 

The Tate conjecture for h^{2,0}=1 varieties over finite fields

Xiaolei Zhao (UC Santa Barbara)

The past decade has witnessed a great advancement on the Tate conjecture for varieties with Hodge number h^{2, 0} = 1. Charles, Madapusi-Pera and Maulik completely settled the conjecture for K3 surfaces over finite fields, and Moonen proved the Mumford-Tate (and hence also Tate) conjecture for more or less arbitrary h^{2, 0} = 1 varieties in characteristic 0.

In this talk, I will explain that the Tate conjecture is true for mod p reductions of complex projective h^{2, 0} = 1 varieties when p is big enough, under a mild assumption on moduli. By refining this general result, we prove that in characteristic p at least 5 the BSD conjecture holds for a height 1 elliptic curve E over a function field of genus 1, as long as E is subject to the generic condition that all singular fibers in its minimal compacification are irreducible. We also prove the Tate conjecture over finite fields for a class of surfaces of general type and a class of Fano varieties. The overall philosophy is that the connection between the Tate conjecture over finite fields and the Lefschetz (1, 1)-theorem over the complex numbers is very robust for h^{2, 0} = 1 varieties, and works well beyond the hyper-Kähler world.

This is based on joint work with Paul Hamacher and Ziquan Yang.

February 13, 2023 

p-adic Borel hyperbolicity of Shimura varieties of abelian type

Abhishek Oswal (Caltech)

Let S be a Shimura variety such that the connected components of the set of complex points of S are quotients of Hermitian symmetric domains by torsion-free arithmetic subgroups. Borel then proved that any holomorphic map from a complex algebraic variety into S is in fact algebraic. In this talk, I’ll discuss a p-adic analogue of this algebraization result. This is joint work with Anand Patel, Ananth Shankar and Xinwen Zhu. 

Feb 27, 2023

On the D-module of an isolated singularity

Thomas Bitoun (University of Calgary)

Let Z be the germ of a complex hypersurface isolated singularity of equation f. We consider the family of analytic D-modules generated by the powers of 1/f and relate it to the pole order filtration on the top cohomology of the complement of \{f=0\}. This work builds on Vilonen’s characterization of the intersection homology D-module. Some other keywords are mixed Hodge modules, logarithmic de Rham complex and residues.

March 6, 2023

Calabi-Yau varieties of large index

Chengxi Wang (UCLA)

A projective variety X is called Calabi-Yau if its canonical divisor is Q-linearly equivalent to zero. The smallest positive integer m with mK_X​ linearly equivalent to zero is called the index of X. Using ideas from mirror symmetry, we construct Calabi-Yau varieties with index growing doubly exponentially with dimension. We conjecture they are the largest index in each dimension based on evidence in low dimensions. We also give Calabi-Yau varieties with large orbifold Betti numbers or small minimal log discrepancy. Joint work with Louis Esser and Burt Totaro.

Spring 2023

April 17, 2023

Tate semisimplicity over finite fields

Ananth Shankar (University of Wisconsin at Madison)

Tate proved (amongst other things) that the action of Frobenius on the Tate-module of an abelian variety over a finite field is semisimple. I will discuss the analogue of this question in the context of exceptional Shimura varieties. This is joint work with Ben Bakker and Jacob Tsimerman. 

May 8, 2023

Reduction of Brauer classes on K3 surfaces

Salim Tayou (Harvard University)

Given a Brauer class on a K3 surface defined over a number field, I will prove that there exists infinitely many primes where the reduction of the Brauer class vanishes, under certain technical hypotheses. This answers a question of Frei--Hassett--Várilly-Alvarado. The proof relies on Arakelov intersection theory on integral models of GSpin Shimura varieties. The result of this talk is joint work with Davesh Maulik. 

May 15, 2023

G-functions and Atypicality

David Urbanik (IHES and MSRI)

Many problems in the field of unlikely intersections and the study of "special" moduli require understanding heights of moduli points in terms of arithmetic data such as the degree of a corresponding field extension. In this talk we explain how the theory of G-functions can be used to control the heights of such moduli in the general setting of a degenerating family of smooth projective varieties.

June 4, 2023

Some cases of the Zilber-Pink conjecture for curves in A_g

Georgios Papas (Hebrew University of Jerusalem)

The Zilber-Pink conjecture is a far reaching and widely open conjecture in the field of unlikely intersections generalizing many previous results in the area such as the André-Oort conjecture. We discuss this conjecture and how some cases of it can be established for curves in $\mathcal{A}_g$, the moduli space of principally polarized g-dimensional abelian varieties, following the Pila-Zannier strategy and bounds for the values of the Weil height at certain exceptional points of the curve.