Jie Liu

We give an overview of the problem. In particular, the notion of rigidity and superrigidity will be introduced and the main theorem will be explained.

In this talk, we firstly review the definition of canonical singularities and we also give some basic examples. Then we prove the so-called Noether-Fano criterion: If a Fano manifold \(X\) of Picard number one is not weakly superrigid, then there exists a positive integer \(m\) and a movable linear system \(|M|\subset|-mK_X|\) such that \(\left(X,\frac{1}{m}|M|\right)\) is not canonical.

The goal of this this talk is aimed to prove Corti's criterion of (log)-canonical singularities via multiplicties (10.3 of [10]): Let \(M\) be a movable linear system on a smooth variety \(X\). (a) If \((X,\frac{1}{m}\vert M\vert)\) is not canonical at \(x\in X\), then \(mult_x \vert M\vert>m\). (b) If \((X,\frac{1}{m}\vert M\vert)\) is not log canonical at \(x\in X\), then \(mult_x (M\cdot M)>4m^2\).

Various generalizations of Kodaira's vanishing theorem will be discussed in this talk, including Kawamata-Viehweg vanishing theorem and Nadel vanishing theorem (see Chapter II of [7] and Section 2 of [8]).

In this talk we focus on the change of singularities as we restrict a linear system to a hyperplane section (so called "adjunction"). We will see that, contrast with multiplicty, the singularity is made worse by restriction to a general hypersurface through a non-canonical center. The aim is to proved 10.4 in [10] and we follow Section 5 of [10] (compare it with the classical adjunction in [7] and [8]).

The aim of this talk is to prove that a subvariety of a smooth hypersurface can not be unexpectedly singular along a large dimensional subset (Section 3 of [10]). One may note that a generalization to complete intersections was given in [15].

This talk is aimed to present several results on the singularities of monomial ideals. In particular, we explain the relation between the singularities and the associted Newton polytope. One can follow Section 9.3 of [11] (see also Section 8 of [10]).

This talk is aimed to understand the relation between the singularities and the global sections of adjoint bundle (10.6 of [10]). The most general version of the statement is given in [16] and a simplified version is 10.6 of [10]. One can follow the discussion in Section 6 of [10], but the general case should also be explained.

We will discuss more general results in this talk and try to summerize some open problems in this direction. Especially, we shall try to give an overview on the relation of rigidity and K-stability of Fano varieties ([12], [16] and [17] etc.).