Jie Liu

The Hassett-Keel program relates different birational compactifications of moduli of curves as variation of log canonical models of \(\overline{M}_g\) which are connected by elementary birational transformations.
In this talk, we will discuss the higher dimensional generalization, with a particular focus on the case of K3 surfaces with anti-symplectic involutions. Specifically, we will introduce the Hassett-Keel-Looijenga program and show how the wall crossing for K-moduli of log del Pezzo pairs comes into play. I will also survey some recent progress on wall-crossing of moduli spaces of K3 surfaces.

Given a local system on a complex algebraic variety, what are the subvarieties on which the monodromy drops? The talk will discuss these monodromy special loci, a natural generalisation of (the positive period dimension components of) the Hodge loci.

A classical result by Hassett and Tschinkel shows that there exist infinitely many inequivalent equivariant compactifications of vector groups into projective spaces of dimension at least 6. In this talk£¬we will give a non-commutative analog of this result for the Heisenberg groups. We prove that there exist infinitely many inequivalent equivariant compactifications of Heisenberg groups into odd dimensional projective spaces. This is a joint work with Zhijun Luo.

Lazarsfeld raised the problem that any surjective morphism \(f\) from a rational homogeneous space \(S\) of Picard number 1 to a projective manifold \(X\) must satisfy that either \(X\) is a projective space or \(f\) is an isomorphism. This problem was completely resolved in the affirmative by Hwang and Mok. In this talk, we will provide a partial extension of Lazarsfeld's problem to Fano manifolds of Picard number 1 that have big tangent bundles. As an application, we study the bigness of the tangent bundles of Fano manifolds with Picard number 1. This talk is based on my joint work with Guolei Zhong.

In this talk, I will show that the K-moduli space of Fano threefolds obtained by blowing up \(\mathbb{P}^3\) along \((2,3)\)-complete intersection curves is isomorphic to a VGIT moduli space studied by Casalaina-Martin-Jensen-Laza. In particular, it is a two-step birational modification of the GIT moduli space of \((3,3)\)-curves on \(\mathbb{P}^1×\mathbb{P}^1\). Our strategy is the moduli continuity method with moduli of lattice-polarized K3 surfaces, general elephants and Sarkisov links as new ingredients. Based on joint work with Yuchen Liu.

I will report the recent progress of Mukai type conjecture which gives a character-ization of product of projective spaces by using a variant of Shokurov's complexity. This is based on working with Joshua Enwright, Stefano Filipazzi, Joaquin Moraga, Roberto Svaldi, Chengxi Wang, Kiwamu Watanabe.

According to the minimal model program, Fano varieties are fundamental research objects in birational geometry. Motivated by the classification of canonical Fano 3-folds, we are interested in the boundedness of important invariants of canonical Fano 3-folds such as anticanonical degree and Fano index.
In this short course, we will explain recent progress on the study of these two invariants, and introduce basic tools including foliations, orbifold Chern classes, Riemmann--Roch formula for canonical 3-folds, and partial classification of canonical singularities.

Fogarty's theorem tells us that Hilbert scheme of points of a smooth algebraic surface is a smooth variety, making Hilbert-Chow morphism in this case a very well-studied crepant resolution. Being a semi-small map, the Hilbert-Chow morphism often serves as an example for computation involving the celebrated decomposition theorem for perverse sheaves. However, if we allow the local systems to have (wild) positive characteristic, the decomposition theorem fails; but we can still decompose the pushforward of constant sheaf manually using Deligne's construction. This talk will focus on this manual decomposition and its interesting, close relation with the decomposition matrix of modular representation of permutation groups.

Hassett showed that there are natural reduction morphisms between moduli spaces of weighted pointed stable curves when we reduce weights. I will discuss some joint work with Ziquan Zhuang which constructs similar morphisms between moduli of stable pairs in higher dimension.

For the classification of canonical Fano threefolds, it is crucial to understand the behavior of their anti-canonical degrees. It is conjectured that the upper bound of degrees should be 72. In this talk, I will present that this conjecture holds when the Fano threefold is \(\mathbb{Q}\)-factorial and of Picard number one. This is a joint work with Chen Jiang and Jie Liu.

The Kawamata-Morrison cone conjecture predicts the geometry of the nef cone and the movable cone of a variety with trivial canonical class. In this talk, we will discuss families of varieties with trivial canonical class and vanishing irregularity. We will study the relative nef cone and the relative movable cone of such families, using machinery from the Minimal Model Program. As application, we will show the relative cone conjecture for families whose very general fibre is a projective hyperkähler manifold of one of the known deformation types. This is joint work with Andreas Höring and Gianluca Pacienza.

An automorphism of a cKM \(X\) is said to be numerically trivial (in \(\text{Aut}_{\mathbb{Q}}(X))\) if it acts trivially on rational cohomology, and cohomologically trivial (in \(\text{Aut}_{\mathbb{Z}}(X))\) if it acts trivially on integral cohomology.
The interest for these notions stems from the theory of period maps and from Teichmueller theory.
The quotients of these groups by \(\text{Aut}^0(S)\) were shown to be finite in the 70's by Fujiki, and there has been ever since much work of several authors in the case of algebraic surfaces S, dedicated mostly to the group \(\text{Aut}_{\mathbb{Q}}(X))\) of numerically trivial automorphisms.
After recalling the general status of the problematic of automorphisms with some topological triviality, and the earlier contributions which I gave with Wenfei Liu according to the classification of surfaces, I will concentrate on the difficult open case of properly elliptic surfaces, that is, minimal surfaces \(S\) of Kodaira dimension 1.
Here I shall report on joint results with Matthias Schuett and Wenfei Liu: for \(\chi >0\) we show that all 2-generated abelian groups appear as \(\text{Aut}_{\mathbb{Q}}(X))\), and that there are upper bounds for \(N := |\text{Aut}_{\mathbb{Q}}(X))|\), depending on the bigenus \(P_2(S)\) and on the irregularity \(q(S)\). Our results are sharp in the isotrivial case.
As Noether said, curves were created by God, and surfaces by the devil, since it is very easy to make mistakes on subtle issues: several authors claimed that in this situation there are no numerically trivial automorphisms if \(\chi , p_g >0\)...
In the case where \(\chi=0\), \(S\) is isogenous to an elliptic product, and if if \(\text{Aut}^0(S)\) is infinite (i.e., S is pseudo-elliptic), we showed that the index of \(\text{Aut}^0(S)\) inside \(\text{Aut}_{\mathbb{Z}}(S)\) is at most 2, and classify exactly the cases where the index is 2.
If \(S\) is not pseudo elliptic, but with \(\chi=0\), we show that \(\text{Aut}_{\mathbb{Z}}(S)\) can be only \(\mathbb{Z}/2, \mathbb{Z}/3, (\mathbb{Z}/2)^2\), and that the first two cases do effectively occur. This is done in joint work also with Davide Frapporti and Christian Gleissner.
Also for \(\chi >0\), we do not have examples where \(\text{Aut}_{\mathbb{Z}}(S)\) has strictly more than 3 elements, but conjecturing the upper bound 3 would be too bold.
Together with Frapporti we found recently the record winning value 192 for \(N\) in the case where \(S\) is of general type (then \(N\) is bounded by a universal constant).

A question that turns out to be extremely entertaining and instructive concern the graded ring over an orbifold curve \(C\) of genus 2 marked with distinct points \(P + Q\) in \(|K_C|\), and polarized by the fractional divisor \(\frac{1}{2}P + \frac{3}{5}Q\). Theorem 1 says that \(C\) is the weighted plane curve \(C(11)\) in \(\mathbb{P}(1,2,5)\). The calculation provides an opportunity to explain many points around orbifold RR, that are elementary but deserve to be better known.
Theorem 1 forms base camp for an ascent to one interesting case of a K3 surface and a Fano 3-fold of codimension 5. GRDB list more than 50,000 candidate Hilbert series of Fano 3-folds, of which only around 1,000 have been seriously studied and partly understood. Current joint work with SUZUKI Kaori successfully constructs a case of #41058, and corrects the GRDB entry for #41245.

The concept of K-stability arises from the existence of Kähler-Einstein metrics on Fano manifolds in complex geometry. Recently, algebraic geometers have used tools from K-stability theory to construct compact moduli spaces of Fano varieties; however, the study of specific examples remains poorly understood. Even for certain surface pairs, the wall-crossing phenomenon remains unclear. In this talk, I will first give a brief review of these theories. We will then apply the wall-crossing theory of K-moduli spaces to study the K-moduli spaces of two classes of log del Pezzo surface pairs.
The main results consist of two parts. The first is the study of the K-moduli space of degree 8 del Pezzo surface pairs \((\mathbb{F}_1, C)\) where \(C \in |−2K_{\mathbb{F}_1}|\). This moduli space is closely related to the moduli space of certain K3 surfaces with anti-symplectic involutions. We use tools involving normalized volumes and T-singularity theory to determine all possible degenerations on the boundary. We then employ the theory of complexity-one torus actions to calculate the positions of critical walls and classify critical surface pairs. Finally, we describe the wall-crossing behavior of this K-moduli space via the local VGIT presentation. Notably, our moduli space provides the first example of a K-moduli space that can be associated with non-reductive GIT.
The second part concerns the study of the moduli space of degree 9 del Pezzo surfaces with multiple boundaries \((\mathbb{P}^2; aQ + bL)\), where \(Q\) is a plane quintic and \(L\) is a line. This provides the first example of higher-dimensional wall-crossing in K-moduli spaces. This K-moduli space has a natural connection with the VGIT moduli space of degree 5 pairs investigated by Laza. We prove that our K-moduli space is isomorphic to Laza¡¯s moduli space when the coefficients of the surface pairs satisfy specific constraints.

In this talk I will first explain the well-known Hamiltonian reduction construction of the closure of the minimal nilpotent orbit \(O_{\text{min}}\) in \(\mathfrak{so}(2n,\mathbb{C})\). Then I will present Kostant's theorem on highest weight varieties and show that in the above special case, there is an interpretation of Kostant's theorem via Hamiltonian reduction. Next I will prove the closure of \(O_{\text{min}}\) is isomorphic to the affinization of \(T^*(\text{SL}(2n+2)/[P,P])\) where \(P\) is certain parabolic subgroup of \(\text{SL}(2n+2)\). If time permits I will also explain conjectures on generalizations of the above results.

Let \(C\) be a curve of genus \(g\) , and \(J\) its Jacobian; let \(G\) be a finite group acting on \(C\), hence also on \(J\). A result of Kollár-Larsen implies that the quotient \(J/G\) either has Kodaira dimension zero, or is uniruled. I will show that the latter case can occur only for \(g \le 5\) , and give examples in this range. The question is motivated by a classical problem about the algebraic cycle \([C]-[(-1)^*C]\) on \(J\).

The minimal log discrepancy is an invariant of singularities related to termination of flips. The ACC for minimal log discrepancies is still unknown in dimension three, and it is one of the most important remaining problems in the minimal model theory of threefolds. In the talk, I will introduce a proof of the ACC for minimal log discrepancies on smooth threefolds. More details will be explained in my seminar talk at Peking University on the following day, 16 Oct.

We present an adjunction formula for foliations on varieties and we consider applications of the adjunction formula to the cone theorem for rank one foliations and the study of foliation singularities. Joint work with C. Spicer.

The previous talk is about Higgs fields and flat connections on vector bundles. In the first half of this talk, we consider how the picture changes when we add log poles. It turns out that the de Rham moduli space is, up to a twist and over an open of the Hitchin base, a Galois cover of the Dolbeault moduli space. This happens due to a curious appearance of the Artin-Schreier map on the residues. In the second half of this talk, we consider the case with G-bundles and no poles. We show that the analogous results on semistability and cohomology also hold. This relies on a detailed study of Ngô's regular centralizers, and the theory of Theta-semistability. Based on joint works with Mark de Cataldo and Andres Fernandez Herrero.

The Non Abelian Hodge Theory for a curve in characteristic p relates the moduli of flat connections on the curve and the moduli of Higgs bundles on the Frobenius twist of the curve. Previously, it is known that both moduli stacks sit over a Hitchin base, and that they differ by a twist of a torsor under a Picard stack over the Hitchin base. Now we know that, by shrinking the Picard stack and its torsor simultaneously, we can make the twist respect semistability. Consequently, the semistable Dolbeault and de Rham moduli spaces differ by a twist of a torsor under a connected group scheme over the Hitchin base. We then deduce some results on the cohomologies of the moduli spaces, some of which entail new results in characteristic zero. Based on joint works with Mark de Cataldo and Michael Groechenig.

We give an overview on recent progress on the numerical characterization of the hard Lefschetz classes. During the lectures, the following topics will be discussed: positivity in algebraic geometry, the extremals of geometric inequalities, Lorentzian polynomials and combinatorial poset inequalities. Based on joint works with J. Hu and S. Shang.

The wild Riemann-Hilbert correspondence relates irregular singular connections with Stokes local systems. As a direct generalization, the correspondence also holds for reductive groups G. Based on the wild Riemann-Hilbert correspondence, I will discuss Stokes G-local systems (or Stokes G-representations) together with its moduli space in the viewpoint of nonabelian Hodge correspondence. This is joint work with Pengfei Huang.

The celebrated Simpson¡¯s tame nonabelian Hodge correspondence shows that the correct objects from the Betti side are the so-called filtered local systems. However, such nonabelian Hodge correspondence was only known at the level of categories until recently, due to the lack of a construction of the moduli spaces of filtered local systems. In this talk, we will introduce filtered local systems and demonstrate the construction of their moduli spaces. This method is applicable to general reductive groups. Based on joint work with Hao Sun.

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.