Algebraic Geometry Seminar

Abstract: In this talk I shall present an arithmetic Simpson correspondence between rank-2 logarithmic periodic Higgs bundles and motivic local systems on hyperbolic affine curves via $p$ to $\ell$ companions conjectured by Deligne and solved by Abe and Drinfeld’s work on Langlands correspondence over function fields. This is a joint work with Raju Krishnamoorthy and Jinbang Yang.
 

Abstract: In this talk, I will introduce the recent progress of Bogomolov's inequality in positive characteristic and Bogomolov type inequalities for threefolds. The related applications will be discussed.
 

Abstract: Coleman and Oort conjectured that when the genus g is large enough, every Shimura subvariety in the Siegel modular variety A_g should not be contained generically in the open Torelli locus T_g. We prove this for Shimura subvarieties of unitary type satisfying certain numerical conditions. This is a joint work with K. Zuo and X. Lu.
 

Abstract: Let $X$ be a projective variety of dimension $n\ge1$ over an algebraically closed field of arbitrary characteristic. We prove a Fujiki-Lieberman type theorem on the structure of the automorphism group of $X$. Let $G$ be a group of zero entropy automorphisms of $X$ and $G_0$ the set of elements in $G$ which are isotopic to the identity. We show that after replacing $G$ by a suitable finite-index subgroup, $G/G_0$ is a unipotent group of the derived length at most $n-1$. This result was first proved by Dinh, Oguiso and Zhang for compact K\"ahler manifolds.
 

Abstract: In this talk, I will first introduce the thesis of Akshay Venkatesh on Beyond Endoscopy for $\mathrm{Sym}^2$ $L$-functions on $\mathrm{GL}_2$ over $\mathbb{Q}$ or a totally real field. The idea follows a suggestion of Peter Sarnak on using the Kuznetsov relative trace formula instead of the Arthur-Selberg trace formula for the Beyond Endoscopy problem. I will then discuss how to generalize Venkatesh's work from totally real to arbitrary number fields. The main supplement is an integral formula for the Fourier transform of Bessel functions over $\mathbb{C}$.
 

Abstract: Serre remarked that his modularity conjecture could be viewed as part of a mod p Langlands philosophy. Indeed the weak conjecture can be viewed as asserting the existence of a global mod p Langlands correspondence for GL2/Q, and the strong conjecture can be viewed as a local-global compatibility statement. In this talk, I will explain a precise interpretation of this remark provided by Emerton and its generalization to totally real fields proposed by Buzzard-Diamond-Jarvis.
 

Abstract: Let $H$ be an ample line bundle over an $n$-dimensional projective manifold $X$. Fujita's conjecture says that $K_X+mH$ is basepoint free for $m\geq n+1$ and $K_X+mH$ is very ample for $m\geq n+2$. On the other hand, Demailly introduced the so-called Seshadri constant to measure the positivity of $H$ locally at a point. In this talk, I will consider the case where $X$ is a Fano manifold and $H$ is the fundamental line bundle of $X$. I will explain how the existence of good ladders can be applied to the study of Fujita conjecture and the Seshadri constants in this situation. In particular, some applications to Fano manifolds with large index will be presented.
 

Abstract: In this talk, we will discuss the following problem: For a fixed positive integer n and a fixed positive real number $\epsilon$, let $X_{a1a2...an} \to A^n$ be the weighted blowup at the origin with the weight $(a1, a2, ..., an)$, where $a1, a2, ..., an$ are coprime positive integers such that $a1 \leq a2 \leq ... \leq an$. If the log pair $(X_{a1a2...an},0)$ is $\epsilon$-lc, is $a1$ bounded? Precisely, for a fixed positive real number $\epsilon$, is there a positive integer $M$ depending only on $n$ and $\epsiln$, such that if $(X_{a1a2...an},0)$ is $\epsilon$-lc, then $a_1 \leq M$.
 

Abstract: For a real four manifold, the S-duality conjecture of Vafa-Witten (1994) predicts that the S-transformation sends the gauge group SU(r)-invariants counting instantons to the Langlands dual gauge group SU(r)/Z_r-invariants; and both of the invariants satisfy modularity properties. The SU(r)-Vafa-Witten invariants have been constructed by Tanaka-Thomas using the moduli space of semistable Higgs bundle or sheaves on a smooth projective surface. In this talk I will present the idea of using moduli space of twisted sheaves and twisted Higgs sheaves on a projective surface to define the Langlands dual gauge group SU(r)/Z_r-Vafa-Witten invariants, and prove the S-duality conjecture of Vafa-Witten for projective plane in rank two and K3 surfaces in prime ranks.
 

Abstract: We study vector bundles on flag varieties over algebraic closed field k. In the first part, we suppose G=Gk(d,n) (d≤n−d) to be the Grassmannian manifold parameterizing linear subspaces of dimension d in k^n, where k is an algebraic closed field of characteristic p>0. Let E be a uniform vector bundle on G of rank r≤d. We show that E is a direct sum of line bundles unless it is a twist of either the pull back of the universal bundle Hd or its dual under the m-th absolute Frobenius morphism, where(m≥0). In the second part, splitting properties of vector bundles on general flag varieties over characteristic 0 are considered.
 

Abstract: Virtual residue was first introduced by Chang and Li to interpret the correlations of the Landau-Ginzburg model. It is a generalization of the classical Grothendieck residue. We will discuss about the basic properties of the virtual residue, and the applications of it in algebraic geometry such as the Cayley-Bacharach theorems with excess vanishing.
 

Abstract: In this talk, I will give a survey on the study of minimal model program on moduli space of K3 surfaces. More precisely, I will discuss the relation between various compactifications of moduli of K3 surfaces via Briational Geometry. Those compactifications come from GIT, arithmetic and singularity theory.
 

Abstract: The cyclic triple cover of the three dimensional complex projective space branched along six hyperplanes in general position is a Calabi-Yau threefold. By varying the hyperplane arrangements, we get a complete family of Calabi-Yau threefolds. I will show the global Torelli theorem holds for this Calabi-Yau family. The main ingredients of the proof are Deligne-Mostow's results on branched cyclic covers of the projective line and a monodromy group computation. This is a joint work with Mao Sheng. 
 

Abstract: We will explain some decomposition theorems of pushforwards of pluricanonical bundles to abelian varieties due to Chen-Jiang, Pareschi-Popa-Schnell, and Lombardi-Popa-Schnell. Then we will explain how to extend these results to singular spaces. Finally, we shall give some geometric applications of these results.
 

Abstract: In this talk, I will introduce the hyperbolicity part of the Shafarevich Conjecture, which claims that a family of smooth curves of g>1 over C^* must be isotrivial, and its generalizations. In particular, I'll show our recent result, which shows that the log-Kodaira dimension of the base space of a family of smooth projective varieties is no less than the family's variation.