Jie Liu

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.