AG Seminar Winter 2018
Student No.:40
Time:Thu 15:30-16:30, Nov.1-Jan.25
Instructor:Eduard Looijenga, Fu Lei, Xu Quan  
Place:Lecture Hall, Jin Chun Yuan West Bldg.
Starting Date:2018-11-1
Ending Date:2019-1-25

Welcome to the AG seminar in YMSC, Tsinghua University. We will continue the AG seminar this semester.

We will invite the newly faculties and post-docs, also visitors to give a talk so that we can have more knowledge to each other and academic communication.


Speaker: Enlin Yang (Peking University)

Title: Characteristic class and the epsilon factor of a constructible etale sheaf

In this talk, we will firstly recall the definitions and the properties of singular support and characteristic cycle of a constructible etale sheaf on a smooth variety. The singular support, defined by Beilinson, is a closed conical subset of the cotangent bundle. The characteristic cycle, constructed by Saito, is a linear combination of irreducible components of the singular support.
This theory is an algebraic analogue of that studied by Kashiwara and Schapira in a transcendental setting.
Using Beilinson and T. Saito's theory, we prove a twist formula for the epsilon factor of a constructible sheaf on a projective smooth variety over a finite field. This formula is a modified version of a conjecture by Kato and T. Saito. We also propose a relative version of the twist formula and discuss some applications. This is a joint work with Naoya Umezaki and Yigeng Zhao


Speaker: Qijun Yan, YMSC (Tsinghua University)

Title: A relation between Ekedahl-Oort stratifications of Shimura varieties and loop groups

I will first talk about the Ekedahl-Oort stratification for Shmura varieties of good reduction and then present a map from the reduction modulo p of a Shimura variety to its associated loop group.


Speaker: Shuhang Yang (YMSC, Tsinghua University)

Title: An Example of Elliptic Threefold over P^2

Let $E_0$ be the elliptic curve with j-invariant equal to 0. Consider the diagonal $\mu_3-$action on $E_0^3$. The quotient space $E_0^3/\mu_3$ is a Calabi-Yau threefold with terminal singularities. We construct a smooth model $X$ of $E_0^3/\mu_3$. There is an isotivial elliptic fibration $\pi: X\rightarrow \mathbb{P}^2$ with generic fibre isomorphic to $E_0$.
We also show that the discriminant locus $\Sigma_{X/\mathbb{P}^2}$ is a Hesse arrangement in $\mathbb{P}^2$, which is the set of 9 lines and 12 multiple points of order 3. The Mordell-Weil rank of an elliptic fibration over a rational surface is related to the Alexander polynomial of the discriminant locus by the work of Cogolludo and Libgober. In this way, we can determine the Mordell-Weil rank of $X$.

2018-12-13 15:00-17:00

Speaker: Dawei Chen (Boston College)

Title: Volumes of moduli spaces of Abelian differentials and Witten's conjecture

Computing volumes of moduli spaces has been a central topic that connects many branches of mathematics. For instance, the celebrated Witten’s conjecture regarding intersection numbers on moduli spaces of curves has a fascinating connection to the Weil-Peterson volume, which motivated Mirzakhani to give a proof via Teichmueller theory, hyperbolic geometry, and symplectic geometry. The first two other proofs of Witten’s conjecture given by Kontsevich and by Okounkov-Pandharipande also used various ideas in ribbon graphs, Gromov-Witten theory, and Hurwitz theory. In this lecture I will introduce an analogue of Witten’s intersection numbers on moduli spaces of Abelian differentials that compute their Masur-Veech volumes. In particular, I will explain its connections to the three proofs of Witten’s conjecture by Kontsevich, by Mirzakhani, and by Okounkov-Pandharipande. This is joint work with Martin Moeller and Adrien Sauvaget.


Speaker: Jingshan Chen (YMSC, Tsinghua University)

Title: Gorenstein stable log surfaces with $(K_X+\Lambda)^2=p_g(X,\Lambda)-1$

It is known that $(K_X+\Lambda)^2\ge p_g(X,\Lambda)-2$ for any connected Gorenstein KSBA stable log surface $(X,\Lambda)$. A log surface with $(K_X+\Lambda)^2=p_g(X,\Lambda)-2$ is known to be a tree of rational surfaces glued along lines. In this talk, I will report my work on the classification of Gorenstein stable log surfaces with $(K_X+\Lambda)^2=p_g(X,\Lambda)-1$.


Speaker: Jian Xiao, YMSC, Tsinghua University

Title: Hodge-index type inequalities, hyperbolic polynomials and complex Hessian equations

We discuss some relations between positivity, geometric PDEs and Hodge-index type theorems. It is noted that using complex Hessian equations and the concavity inequalities for elementary symmetric polynomials implies a generalized form of Hodge index inequality. Inspired by this result, using Garding's theory for hyperbolic polynomials, we obtain a mixed Hodge-index type theorem for classes of type (1,1). The new feature is that this Hodge-index type theorem holds with respect to mixed polarizations in which some satisfy particular positivity condition, but could be degenerate and even negative along some directions.


Speaker: Bingyu Xia, AMSS, CAS

Title: Hilbert scheme of twisted cubics as simple wall-crossings

Hilbert scheme is introduced by Grothendieck and it played an important role in algebraic geometry. Hilbert scheme of twisted cubics in the projective space P^3 is one of the easiest but nontrivial Hilbert scheme, its geometric structure was first described by Piene and Schlessinger in 1985. In this talk, I will introduce Bridgeland stability conditions on the derived category of the projective space P^3, and use wall-crossing phenomena of stability conditions to reprove Piene and Schlessinger's result.


Speaker: Will Donovan, YMSC, Tsinghua University

Title: Perverse sheaves of categories for mirror moduli spaces

The moduli spaces associated to the A-side and B-side of mirror symmetry carry much-studied bundles with connections, which may be related by a mirror map. I explain a categorification of such structures for certain examples, using perverse sheaves of categories on partial compactifications of the moduli spaces. This talk will discuss joint work with T Kuwakagi, and joint work with M Wemyss.


Speaker: Yitwah Cheung, YMSC, Tsinghua University

Title: Square-integrability of Siegel-Veech transforms

Motivated by counting problems for polygonal billiards and more generally for linear flows on surfaces, Veech (Annals 1998) introduced what is now known as the Siegel-Veech transform on the moduli space of holomorphic 1-forms, in analogy with the Siegel transform arising from the space of unimodular lattices. In that paper Veech established an (L^1) integral formula for this transform, a version of the classical Siegel integral formula. In this talk, I will describe some of the main ideas that allow us to extend Veech's result to establish that the Siegel-Veech transform of any bounded, compactly supported function is square integrable. This work is joint with Jayadev Athreya and Howard Masur.


Speaker: Gerard Van der Geer (Amsterdam University)

Title: Algebraic curves and modular forms of degree two

Siegel modular forms of degree 2 are intimately connected with moduli of curves of genus 2. We show that classical invariant theory yields an effective way to construct modular forms and then give attention to modular forms of low weight. This is joint work with Clery and Faber.