# 6th Meeting

## AMSS, CAS

Meng Chen (Fudan)

On minimal 3-folds of general type with maximal pluricanonical section index
Let X be a minimal 3-fold of general type. The pluricanonical section index $\delta(X)$ is defined to be the minimal integer m so that $P_m(X) \geq 2$. According to J. A. Chen and M. Chen, one has either $1 \leq \delta(X) \leq 15$ or $\delta(X) = 18$. In this talk, we explain further study on the case $\delta(X) = 18$. A direct corollary is that the 57th canonical map of every minimal 3-fold of general type is stably birational, which improves known results.

Lei Fu (Nankai)

Rigidity of $\ell$-adic sheaves
Let X be a smooth connected algebraic curve over an algebraically closed field k, let S be a finite closed subset in X, and let F be a lisse $\ell$-adic sheaf on X - S. We study the deformation of F. By studying the generic fiber of the universal deformation space, we prove a conjecture of Katz which says that if F is irreducible and physically rigid, then F has rigidity index 2.

Yuchen Liu (Princeton)

The volume of Kähler-Einstein Q-Fano varieties
A complex projective variety is Q-Fano if it has klt singularities and the anti-canonical divisor is Q-Cartier and ample. Starting from dimension 2, the anti-canonical volume of a Q-Fano variety can be arbitrarily large, e.g. weighted projective spaces. Recently, Fujita showed that an n-dimensional Kähler-Einstein Q-Fano variety has volume at most (n + 1)n. In this talk, I will discuss a refinement of Fujita’s volume upper bounds involving invariants of the local singularities. If time permits, I will also talk about an equivalent relation between K-semistability and de Fernex-Ein-Mustata type inequalities. Part of this work is joint with Chi Li.

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