刘   杰

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.