Jie Liu

Graded sequences of ideals, or filtrations on local rings, play an important role in commutative algebra, the theory of singularities and birational geometry. In this talk, I will introduce the saturation of a filtration and discuss several results about multiplicities of filtrations. The first is a theorem generalizing Rees' theorem on Hilbert-Samuel multiplicities of ideals. Then I will talk about convexity of multiplicities along geodesics, with applications in valuation theory and K-stability. I will also introduce a metric on the space of filtrations as an analogue of the Darvas metric in complex geometry, and prove the semiconuity of log canonical threshold functions in the spirit of Koll¨¢r-Demailly. This talk is partially based on joint work with Harold Blum and Yuchen Liu.

A del Pezzo surface is a smooth projective surface with ample anticanonical divisor. Over an algebraically closed field, any surface like this is rational. However, without this assumption del Pezzo surfaces exhibit very interesting birational properties. I will survey some old and new results about birational geometry of del Pezzo surfaces over arbitrary fields, mostly focusing on del Pezzo surfaces of degree 4. These can be realized as intersections of two quadrics. It is known that minimal del Pezzo surfaces of degree 4 over a perfect field are never rational. I will provide a detailed description of birational models for such surfaces, and will discuss the properties of their birational automorphism groups. The talk is based on a joint work in progress with A.Trepalin.

Symmetric holomorphic forms on projective manifolds have been studied from various angles in the last ten years : Brotbek-Darondeau and independently Song-Yan Xie have shown that complete intersections of sufficiently high degree and codimension have an ample cotangent bundle, so many symmetric forms. Vice versa Campana and Paun show that if the cotangent bundle is big, then \(X\) is of general type. In this series of lectures I will work under the assumption that the cotangent bundle is pseudoeffective. In general this positivity assumption is too weak to have any useful implications, but we will see that for the cotangent bundle it leads to surprisingly many restrictions on the geometry of the manifold. In particular we expect a nonvanishing property that is analogous to the famous nonvanishing conjecture for the canonical bundle. These talks are based on joint papers with Thomas Peternell and Junyan Cao.

Symmetric holomorphic forms on projective manifolds have been studied from various angles in the last ten years : Brotbek-Darondeau and independently Song-Yan Xie have shown that complete intersections of sufficiently high degree and codimension have an ample cotangent bundle, so many symmetric forms. Vice versa Campana and Paun show that if the cotangent bundle is big, then \(X\) is of general type. In this series of lectures I will work under the assumption that the cotangent bundle is pseudoeffective. In general this positivity assumption is too weak to have any useful implications, but we will see that for the cotangent bundle it leads to surprisingly many restrictions on the geometry of the manifold. In particular we expect a nonvanishing property that is analogous to the famous nonvanishing conjecture for the canonical bundle. These talks are based on joint papers with Thomas Peternell and Junyan Cao.

The perverse filtration captures interesting homological information of algebraic maps. Recent studies of the Hitchin and Beauville-Mukai integrable systems suggest two common features of the perverse filtration for abelian fibrations:
1) the perverse filtration is multiplicative with respect to the cup product;
2) the perversity of tautological classes is governed by the Chern degree.
In this talk, I will explain a unified approach to bothstatements for a natural class of abelian fibrations, namely fibrations incompactified Jacobians. I will discuss several ingredients from coherent derived categories, K-theory/Chow theory, and constructible categories, as well as some applications of our result. Joint work with Davesh Maulik and Junliang Shen.

We continue the discussion of the last talk. I will explain the properties of the \(d_S\)-pseudometric and prove various characterizations of I-good singularities.

In this talk, we will discuss a geometric application of the theory of I-good singularities. I will prove the bijection between geodesic rays and test curves and the bijection between Berman-Boucksom-Jonsson maximal geodesic rays and I-model test curves.

The Green-Griffiths-Lang (GGL) conjecture asserts that any entire curve in a complex projective variety of general type cannot be Zariski dense. This conjecture fascinates many complex geometers, in part due to its arithmetic analogy with the Bombieri-Lang conjecture for rational points in algebraic varieties over number fields. In this talk I will report a recent work with Cadorel and Yamanoi, focusing on the proof of the generalized GGL conjecture for quasi-projective varieties whose topological fundamental groups possess a big and reductive representation into a complex general linear group.

Hua domain, named after Chinese mathematician Loo-Keng Hua, is defined as a domain in \(\mathbb{C}^n\) fibered over an irreducible bounded symmetric domain \(\Omega \subset \mathbb{C}^d\) with the fiber over \(z\in \Omega\) being a \((n-d)\)-dimensional generalized complex ellipsoid \(\Sigma(z)\). In 2015, Tu-Wang obtained the rigidity result that proper holomorphic mappings between two equidimensional Hua domains are biholomorphisms when the sets consisting of boundary points of Hua domains which are not strongly pseudoconvex have complex codimension at least $2$. In this article, we find a counter-example to show that the rigidity result is not true for Hua domains without this condition and obtain the rigidity of proper holomorphic self-mappings of the Hua domains in this case.

This talk mainly concerns the projective, Moishezon and Kahler loci of a family. We first review our recent results on deformation limit and invariance of plurigenera of Moishezon manifolds, based on several joint works with I-Hsun Tsai, Yi Li and Runze Zhang. Then we talk about an in-process joint work with Mu-Lin Li and Mengjiao Wang, on various loci of a family and their applications. Among them are a Chow-type Lemma and a reverse side of one theorem of Zhiwei Wang on the modification of some special complex structure.

Let \(f:X\rightarrow Z\) be a Mori fibre space. McKernan conjectured that the singularities of \(Z\) are bounded in terms of the singularities of \(X\). Shokurov independently proposed a more general conjecture in the setting of pairs. In this talk, we may survey some recent progress on Shokurov¡¯s conjecture on the singularities of the base. Then we reduce various conjectures in birational geometry, including Shokurov¡¯s conjecture on the singularities of the base and boundedness conjecture for rationally connected Calabi-Yau varieties, to a conjecture on multiplicities of fibers of Fano fibrations over curves. This is a joint work with Chuyu Zhou.

I will start by reviewing an old joint work with Davesh Maulik and Yukinobu Toda on relating Gromov-Witten, Gopakumar-Vafa and stable pair invariants on compact Calabi-Yau \(4\)-folds. For non-compact CY\(4\) like local curves, similar invariants can be studied via the perspective of quasimaps to quivers with potentials. In a joint work in progress with Gufang Zhao, we define a virtual count for such quasimaps and prove a gluing formula. Computations of examples will also be discussed.

The framework of derived algebraic geometry (DAG), developed by Toen-Vezzosi, Lurie and many others, allows us to extend Grothendieck's theory of flag schemes of sheaves to the cases of complexes. In the first half of the talk, we will define and study flag schemes of complexes using DAG.
The Borel-Weil-Bott theorem provides a fundamental link between algebraic geometry and representation theory. In the second half of the talk, we will establish a derived generalization of the Borel-Weil-Bott theorem for flag schemes of \([-1,0]\)-perfect complexes.
Based on the second part of the paper arXiv:2212.10488.

We'll start with reviewing Quillen and Lurie's theory of non-abelian derived categories (also known as the "animation" theory). Following that, we will apply this theory to generalize the classical Schur and Weyl functors of \(GL_n\)-representation theory to the cases of connective complexes, and investigate their fundamental properties. If time allows, we will also talk about how to generalize them to non-connective cases using Goodwillie's theory of calculus of functors.
Based on the first part of the paper arXiv:2212.10488.

The Zariski dense orbit conjecture of the arithmetic dynamic is to study the existence of a rational point \(P\) with a Zariski dense orbit of a dominant rational self-map \(f\) of a projective variety \(X\). When there exist such a rational point \(P\), Prof. Kawaguchi and Prof. Silverman conjecture defined the arithmetic degree at \(P\) and claimed that its arithmetic degree is equal to the first dynamical degree of \(f\), i.e., Kawaguchi-Silverman conjecture of the arithmetic dynamic. In this talk, I will explain my recent progress towards on the Zariski dense orbit conjecture and Kawaguchi-Silverman conjecture on automorphisms of projective threefolds from the viewpoint of the explicit birational geometry.

Since the celebrated work by Cartan, distributions with small growth vector \((2,3,5)\) have been studied extensively. In the holomorphic setting, there is a natural correspondence between holomorphic \((2,3,5)\)-distributions and nondegenerate lines on holomorphic contact manifolds of dimension \(5\). We generalize this correspondence to higher dimensions by studying nondegenerate lines on holomorphic contact manifolds and the corresponding class of distributions of small growth vector \((2m, 3m, 3m+2)\) for any positive integer \(m\). This is a joint work with Jun-Muk Hwang.