Algebraic Geometry Seminar

Abstract: It has been a long history, since Schwarz, that one of ways to search for a complex hyperbolic lattice is closely related to understanding the monodromy group of a hypergeometric system. In this talk, I will review the work of Schwarz and Klein on the classical Euler-Gauss hypergeometric functions, as well as the Deligne-Mostow theory on Lauricella hypergeometric functions. Then I will discuss a recent progress on this direction, inspired by the root system hypergeometric functions constructed by Heckman-Opdam.
 

Abstract: Let X be a surface of degree n, which is considered as a branch cover of CP^2 with respect to a generic projection. The surface X has a natural Galois cover with Galois group S_n. In this talk, we will use Moishezon-Teicher’s method to compute the fundamental group of the Galois cover of certain algebraic surfaces that can degenerate to a union of n projective planes such that no three planes meet in a line. This is joint work with M. Amram, C. Gong, M. Teicher and S.-L. Tan.
 

Abstract: In this talk, we will present some conjectural relations between scrollar syzygy resolution of algebraic curves and the classical theory of resolvents for solving algebraic equations. We will give a Galois theoretic perspective of the syzygy theory. If time permits, some possible number theory applications will also be discussed. This is a joint work with Wouter Castryck.
 

Abstract: I will give a brief introduction to the classical theory of birational geometry, in particular the minimal model program(MMP). Singularities play a crucial role in birational geometry and MMP. They are often measured by numbers called log discrepancies, and the “worst” type of singularities for which MMP works are called Log Canonical, or LC for short. I will then discuss some recent progresses in birational geometry in the LC category, including the finite generation of a log canonical ring and the boundedness on log Fano 3-folds.
 

Abstract: The notion of generalized polarized pairs was introduced by Birkar and Zhang to deal with the effectivity of Iitaka fibration. Its prototype already appeared in the treatment of the canonical bundle formula for elliptic fibrations. The notion has since found more and more applications in solving problems for usual pairs.
  In the other direction, efforts have been made by several authors to extend results about pairs to the generalized setting. In this talk, after introducing the basic definitions of generalized polarized pairs, I will report on a recent joint work with HAN Jingjun on the (numerical) nonvanishing conjecture for generalized log canonical pairs.
 

Abstract: The BCOV invariant is an invariant for Calabi-Yau manifolds. In this talk, we extend the BCOV invariant to Calabi-Yau pairs, i.e., a Kahler manifold equipped with a pluricanonical divisor. In certain case, this BCOV invariant is equivalent to an equivariant BCOV invariant introduced by Yoshikawa. Moreover, this BCOV invariant is well-behaved under birational equivalence. We expect that these constructions lead to a positive answer to the following conjecture: the BCOV invariant for Calabi-Yau threefold is a birational invariant.
 

Abstract: Deligne-Illusie in 80s opened the Hodge theory in positive characteristic, by providing an algebraic and elementary proof of the E_1 degeneration theorem and Kodaira's vanishing theorem over a characteristic zero field. The key result in their work is the so-called decomposition theorem for de Rham complexes. In this talk, I shall report our recent work on the decomposition theorem for intersection de Rham complexes and some consequences. This is a joint work with Zhang Zebao.
 

Abstract: I shall report some progress in a joint project with Steven Lu and Ruiran Sun in the study of a class of distributions on moduli spaces of minimal varieties. The ultimate goal is to show Borel hyperbolicity of those type moduli spaces.
 

Abstract: we discuss a criterion for rigid families of projective surfaces of minimal model. We propose a conjecture on the decomposition of fibers of non-rigid families.
 

Abstract: Noma developed the inner projection method and gave a classification of projective varieties in a sense. As an important consequence, he established a sharp linear bound for Castelnuovo-Mumford regularity of structure sheaves on smooth projective varieties of arbitrary dimension. I will show how blending Noma’s classification with multiplier ideal theory leads to the same bound for normal projective varieties with at worst isolated Q-Gorenstein singularities. By contrast, it is known only recently that such a bound fails for varieties with arbitrary singularities. This is based on a joint work with J. Moraga and J. Park.
 

Abstract: The Picard number of a rational surface equipped with a relatively minimal trigonal fibration is bounded in terms of the genus $g$ of a general fibre. When the Picard number attains the maximum for $g\geq3$, we describe such a fibred surface whose Mordell-Weil group is trivial. This is a joint work with Cheng Gong (Soochow Univ.) and Jun Lu (East China Normal Univ.).
 

Abstract: The P=W conjecture of de Cataldo, Hausel, and Migliorini relates the topology of Hitchin fibrations to the Hodge theory of character varieties via Simpson’s nonabelian Hodge theory. In recent joint work with Junliang Shen, we proved a compact analog of this conjecture, which relates the topology of Lagrangian fibrations to the Hodge theory of compact hyper-Kaehler manifolds. I will present circle of ideas as well as some applications of our result.
 

Abstract: We determine the minimum positive entropy of complex Enriques surface automorphisms. This together with McMullen's work completes the determination of the minimum positive entropy of complex surface automorphisms in each class of Enriques-Kodaira classification of complex surfaces. As in McMullen's work, we finally reduce the problem to computer algebra. In this talk, after recalling known results and differences from Enriques case, I would like to explain how one can reduce this problem to finite computational problems which can be done by computer. This is a joint work with Professor Keiji Oguiso.

Abstract: We will explain work in progress with Colmez, Hauseux and Niziol, describing the compactly supported p-adic étale cohomology of many p-adic period domains. The proof is an adaptation of Orlik's computation of the l-adic cohomology, combined with a vanishing result for extensions between mod p generalized Steinberg representations of p-adic reductive groups.

Abstract: In this talk, after briefly describing the geometry of the 4-nodal cubic surface, I will introduce the Keum-Naie-Mendes Lopes-Pardini surfaces. I will talk about the bicanonical map, the deformation and moduli of these surfaces.

Abstract: We consider algebraic varieties defined over a number field. We will define strong and weak approximation properties (with or without Brauer-Manin obstruction) for rational points on varieties. We will talk about some results on the non-invariance of such properties under the extension of the ground field.

Abstract: We introduce a local descent construction for inner forms of odd special orthogonal group in supercuspidal case, which can recover the local Vogan packets for odd special orthogonal groups parametrized by simple L-parameters. We will also talk about its relation to some other topics, such as local Gan-Gross-Prasad conjecture and local theta lifting.

Abstract: First I will report the recent progresses in minimal model theory in characteristic p, second I will introduce some open problems in this area, finally I will focus on the effectivity problem, discuss the possible approaches and explain the difficulties.

Abstract: This talk contains two parts. In the first part, I will give a review of various fundamental group schemes from topological fundamental group to fundamental groups schemes over field of arbitrary characteristics, such as Etale fundamental group scheme, Local fundamental group scheme, Nori fundamental group scheme, F-fundamental group scheme and so on. In the second part, I will give a proof of birational invariance of F-fundamental group schemes.

Abstract: In this talk, we would like to introduce how we can connect Langlands programme with noncommutative geometry. Indeed we apply the dual space of reductive Lie groups to construct a link between local Langlands correspondences and the reduced C*-algebras. This is a joint work with Hang Wang.

Abstract: Harder-Narasimhan filtration is a classic construction in the geometry of vector bundles over curves. In arithmetic geometry, this notion has an analogue in the setting of Euclidean lattices. However, in many situations of arithmetic geometry, lattices in non-Euclidean normed spaces are more intrinsic objects, on which the arithmetic analogue of Harder-Narasimhan theory does not work well. In this talk, I will explain a new approach to this problem, based on slope inequalities. This is a joint work with Atsushi Moriwaki.

Abstract: Let $X$ be a smooth projective variety defined over an algebraically closed field of arbitrary characteristic, and $f\colon X \to X$ a surjective morphism. The $i$-th cohomological dynamical degree $\chi_i(f)$ of $f$ is defined as the spectral radius of the pullback $f^*$ on the \'etale cohomology group $H^i_{et}(X, \bQ_\ell)$ and the $k$-th numerical dynamical degree $\lambda_k(f)$ as the spectral radius of the pullback $f^*$ on the vector space $N^k(X)_\bR$ of real algebraic cycles of codimension $k$ modulo numerical equivalence. Truong conjectured that $\chi_{2k}(f) = \lambda_k(f)$ for any $1 \le k \le \dim X$. When the ground field is the complex number field, the equality follows from the positivity property inside the de Rham cohomology of the ambient complex manifold $X(\bC)$. We prove this conjecture in the case of abelian varieties. The proof relies on a new result on the eigenvalues of self-maps of abelian varieties in prime characteristic, which is of independent interest.

Abstract: In this talk, we will introduce the recent progress on Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives of arbitrary rank. We will discuss an important technique used in the proof, namely, the arithmetic level raising for unitary groups of even rank. This is based on a joint work with Y. Tian, L. Xiao, W. Zhang, and X. Zhu.

Abstract: The set of irreducible components of an affine Deligne-Lusztig variety is interesting for many applications related to Shimura varieties, etc. A natural symmetry group J acts on this set, and it is desirable to determine the orbits and the stabilizers of this action. In joint work with Rong Zhou, we prove a formula for the number of orbits, earlier conjectured by Miaofen Chen and Xinwen Zhu. In joint work in progress with Xuhua He and Rong Zhou, we show that all the stabilizers are "very special parahorics". In many cases this already characterizes the stabilizers.

Abstract: We will will elaborate the notion of ‘int-amplified’ endomorphism f of a normal projective variety X, a property weaker than ‘polarized’ yet preserved by products. We will show that the existence of such a single f guarantees that every Minimal Model Program (MMP) is equivariant w.r.t. a finite-index submonoid of the whole monoid SEnd(X) of all surjective endomorphisms of X. Applications of the equivariant MMP are discussed: Kawaguchi-Silverman conjecture on the equivalence of arithmetic and dynamic degrees of an endomorphism, and characterization of a subvariety with Zariski dense periodic points. Some parts are based on joint work with Meng.

Abstract: In this talk, we give a sharp bound guaranteeing that a line bundle on a projective toric variety is k-jet ample in terms of its intersection numbers with invariant curves. In particular, we show that for any ample divisor L on a projective toric variety X of dimension n>=2, mL is k-jet ample if m is greater or equal to k+n-2.

Abstract: This work is encouraged by Professor De-Qi Zhang. We consider an arbitrary int-amplified surjective endomorphism f of a normal projective variety X over C and its f^(-1)-stable prime divisors. We extend the early result for the case of polarized endomorphisms to the case of int-amplified endomorphisms. Assume further that X has at worst Kawamata log terminal singularities. We prove that the total number of f^(-1)-stable prime divisors has an optimal upper bound dimX+ρ(X), where ρ(X) is the Picard number. Also, we give a sufficient condition for X to be rationally connected and simply connected. Finally, by running the minimal model program (MMP), we prove that, under some extra conditions, the end product of the MMP can only be an elliptic curve or a single point

Abstract: Let X be a normal projective variety of dimension n≥3 and G an abelian subgroup of Aut(X) such that all elements of G\{id} are of positive entropy. Dinh and Sibony proved that G is a free abelian group with rank(G)≤ n-1. The maximal rank case has been well understood by Zhang. We characterize the pair (X, G) such that rank(G) = n-2. This is based on joint work with Fei Hu.