Algebraic Geometry Seminar

Abstract: Given a big line bundle L on a projective manifold, Lazarsfeld–Mustată and Kaveh–Khovanskii introduced method of constructing convex bodies associated with L. These convex bodies are known as Okounkov bodies. When L is endowed with a singular positive Hermitian metric h, I will explain how to construct smaller convex bodies from the data (L,h). These convex bodies play important roles in the study of the singularities of h.

Abstract: Foliation theory has been a crucial role in birational geometry over the past few decades, such as in Miyaoka's resolution of the abundance conjecture for complex projective varieties of dimension three. This talk aims to provide an overview of the theory of birational geometry of foliations and highlight some of its applications.
 

Abstract: The purpose of this talk is to provide an overview of the recent developments on the Minimal Model Program for algebraically integrable foliations over a complex projective variety. 
 

Abstract: Minimal model program for foliations is an analogue of the classical minimal model program. Foliated dlt foliations play the same role as dlt pairs in the classical minimal model program, making it a natural class of singularities to study in the theory of foliations. In this talk, we show that we can run an MMP on a Q-factorial foliated dlt algebraically integrable foliation. Numerous applications also will be presented. Joint work with Jingjun Han, Jihao Liu, and Lingyao Xie.

Abstract: Let X be a smooth complex projective variety of dimension d and f an automorphism of X. Suppose that the pullback f^* of f on the real Néron–Severi space N^1(X)_R is unipotent and denote the index of the eigenvalue 1 by k+1. We prove an upper bound for the polynomial volume growth plov(f) of f, or equivalently, for the Gelfand–Kirillov dimension of the twisted homogeneous coordinate ring associated with (X, f), as follows:
plov(f) \leq (k/2 + 1)d.
Combining with the inequality k \leq 2(d-1) due to Dinh–Lin–Oguiso–Zhang, we obtain an optimal inequality that
plov(f) \leq d^2,
which affirmatively answers questions of Cantat–Paris-Romaskevich and Lin–Oguiso–Zhang. This is joint work with Chen Jiang.

Abstract: The concept of generalized pairs, introduced by Birkar and Zhang, plays an important role in modern birational geometry. This concept has been particularly crucial in proving the Borisov-Alexeev-Borisov conjecture and the McKernan-Shokurov conjecture. In this talk, I will explain how the theory of generalized pairs is applied to foliation theory from various perspectives, including but not limited to: Bertini-type theorems, the canonical bundle formula, the minimal model program (MMP) with scaling, the base-point-freeness theorem, the existence of flips, flops between minimal models, and the explicit adjunction formula. The talk is based on several joint works and ongoing collaborations with one or more of the following scholars: Paolo Cascini, Guodu Chen, Yifei Chen, Jingjun Han, Yujie Luo, Fanjun Meng, Calum Spicer, Roberto Svaldi, Yanze Wang, and Lingyao Xie.

Abstract: I will show the singularities of the secant variety to a smooth projective subvariety embedded by a sufficiently positive (pluri-) adjoint linear series are Du Bois but not rational in general. Moreover, the secant variety can be p-Du Bois, in the sense of Shen-Venkatesh-Vo, if certain Hodge numbers are zero; however it is almost never pre-1-rational. So unlike local complete intersections, p-Du Bois does not imply (p-1)-rational for secant varieties. The talk is based on the joint work with C-C. Chou and with S. Olano and D. Raychaudhury.

Abstract: If the delta invariant of a Fano manifold is greater than one, then the Fano manifold is K-stable and admits a KE metric. In this case, it admits no nontrivial holomorphic vector field. For a Fano manifold with nontrivial holomorphic vector fields, we will introduce another "delta" invariant characterizing its K-polystability. Moreover, the g-weighted version of this invariant can be used to characterizing the existence of g-solitons on a Fano manifold. As an application, we will give a family of Fano threefolds admitting g-solitons for any weight function g.

Abstract: In the preprint by Sun-Zhang, a invariant called the weighted volume is introduced. This invariant combines two well-studied invariants: the normalized volume and the H-invariant, both prominent in the literature on K-stability. In this talk, we will discuss the weighted volume in the context of toric Fano fibrations. It is intrinsically linked to a convex optimization problem on an unbounded polyhedron.

Abstract: In this talk, we will propose a question of how to explicitly determine the optimal degenerations of the K-unstable Fano manifolds as predicted by the Hamilton-Tian conjecture. We answer this question for a family of K-unstable Fano threefolds (No 2.23 in Mori-Mukai's list), which has discrete automorphism groups and the normalized Kahler-Ricci flow develops Type II singularity. Our approach is based on a new method to check weighted K-stability, which generalizes Abban-Zhuang's theory to give an estimate of the weighted delta invariant by dimension induction. Some speculative relations between the delta invariant and the H invariant will also be discussed. This is based on a joint work with Linsheng Wang.

Abstract: Fano type fibrations include many central ingredients of birational geometry, such as Fano varieties, Mori fiber spaces, flipping and divisorial contractions, crepant models, and germs of singularities. In this talk, we will first review some background on Fano type fibrations, and then outline some ideas in Birkar’s proof of the boundedness of Fano type fibrations, which can be viewed as a relative version of the well-known BAB conjecture.

Abstract: We present two type curvature flows related to mean curvature and harmonic mean curvature in asymptotic flat space. As a consequence, we construct a different kind of foliation and define a geometric center of mass. This is a joint work with Prof. Yuqiao Li and Prof. Jun Sun.

Abstract: Using MacPherson's Euler obstruction function, one can identify the abelian group of constructible functions with the group of algebraic cycles on a smooth complex algebraic variety. Kashiwara's local index formula gives an alternative approach to this identification by using characteristic cycles for holonomic D-modules (they are Lagrangian cycles in the cotangent bundle). This identification then enables us to define Chern classes of algebraic cycles by using characteristic cycles. In this talk, I will first explain how to obtain Chern classes of the pushforward of Lagrangian cycles under an open embedding with normal crossing complement by using logarithmic cotangent bundles motivated by D-module theory. Then I will discuss applications of such Chern classes in understanding Chern-Mather classes of very affine varieties and in proving the Involution Conjecture of Huh and Sturmfels in likelihood geometry. This work is joint with Maxim, Rodriguez, and Wang.

Abstract: B. Bhatt and P. Scholze introduced the notion of the pro-étale fundamental group for a topologically Noetherian scheme X in their celebrated work "The pro-étale cohomology for schemes". This is a topological group that classifies the geometric covers of X. Under the Yoneda embedding, the geometric covers are identified with sheaves of sets which are locally constant sheaves for the pro-étale topology. In particular, the finite étale covers are geometric. Therefore, the pro-étale fundamental group refines Grothendieck's étale fundamental group which classifies only finite étale covers. There is a natural morphism from the pro-étale fundamental group to the étale fundamental group which realizes the étale fundamental group as the profinite completion of the pro-étale fundamental group. However, there has been no direct comparison between the topological and pro-étale fundamental groups. In this talk, we are going to present this comparison. We'll also introduce some comparison theorems in the p-adic setting.

Abstract: Unlike the reductive group compactifications(e.g toric varieties), equivariant compactification of unipotent algebraic groups is less known. In this talk，I will report some recent progress on this topic, especially for vector groups and Heisenberg groups. This is based on joint works in progress with Zhijun Luo(IBS,CCG).

Abstract: Algebraic varieties with nef anticanonical bundles exhibit homogeneous structures for their Albanese morphisms and maximal rationally chain-connected (MRC) fibrations. In this talk, we examine the relationship between automorphism groups and these homogeneous structures. Besides, a classical theorem by Nishi and Matsumura establishes a natural group homomorphism associated with the Albanese morphism, which induces the Chevalley decomposition of the automorphism group up to an isogeny. We extend this perspective by proving an analogous result for the MRC fibrations. This is joint work with Jinsong Xu.

Abstract: The study of rigidity of holomorphic maps originated from the work of Poincaré and later Alexander for maps sending one open piece of the sphere into another. Webster obtained rigidity for holomorphic maps between pieces of spheres of different dimension, proving that any such map between spheres in Cn and Cn+1 is totally geodesic. There are plenty of results about the holomorphic proper mapping between balls and generalized balls. The rigidity for maps between bounded symmetric domains are more complicated. Mok conjectures that If the rank r of D does not exceed the rank r' of D' and both ranks r,r'≥ 2, then the proper holomorphic maps is totally geodesic. There are some results about this conjecture. In 2007, Kim and Zaitsev proved the rigidity of locally defined CR embeddings between Shilov boundaries of general Cartan type I bounded symmetric domains of higher rank. In this talk, we will introduce a different method coming from algebraic geometry to study this kind of map and prove the rigidity of holomorphic mapping between bounded symmetric domains preserving Shilov boundaries from rank 1 to high rank. This is joint work with Sui-Chung Ng.

Abstract: In 1999, Verbitsky introduced the concept of hyperholomorphic vector bundles on higher-dimensional hyper-Kähler manifolds. Despite their good moduli properties that resemble those of simple sheaves on K3 surfaces, constructing examples beyond tangent bundles and their relatives is quite challenging. Recently, the notion of atomic sheaves, studied by Markman and Beckmann, has been helpful in this direction. In this talk, we provide new examples of atomic sheaves on an infinite series of families of projective hyper-Kähler manifolds. Furthermore, after applying derived equivalences and deformations, we construct (projectively) hyperholomorphic bundles on any hyper-Kähler manifold of $K3^{[n]}$-type for infinitely many $n$. This is based on joint work with Zhiyu Liu.

Abstract: Let Y be a smooth del Pezzo threefold of degree d, an instanton sheaf E on Y is a rank two stable sheaf such that certain cohomology vanishing condition is satisfied. The locus of instanton bundles on a smooth cubic threefolds parametrized all index one degree 14 prime Fano threefolds sharing the same intermediate Jacobians. In my talk I will first give a Bridgeland moduli theoretical construction of moduli space of instanton sheaves on cubic threefolds. Then I will identify the fiber of the (categorical) period map for a smooth and proper subcategory of Kuznetsov components of a 1-nodal maximally non-factorial prime Fano threefold of degree 14 with the locus of Bridgeland stable objects which are not locally free. Then I will generalize this identification to other genus. Finally I will propose a conjectural identification of the fiber of categorical period map for at most 1-nodal Fano threefolds with Bridgeland moduli spaces in Kuznetsov components of del Pezzo threefolds. If time allows, I will talk about an interesting application: Kuznetsov's conjecture on jumping lines for instanton bundles.

Abstract: In the past few years, wall crossing for K-stability has been studied in linear setting and many explicit examples are worked out. Here linear setting means we only consider the log pair where the ambient space is Fano and the boundary is proportional to the anti-canonical bundle. In this talk, I will present a non-linear wall crossing theory for K-stability, where the ambient space is not necessarily Fano and the boundary is not necessarily proportional to the anti-canonical bundle. In this theory, a proper K-moduli is constructed for K-semistable log Fano pairs with real coefficients (not just rational ones). Joint with Yuchen Liu.