On the extension problem of pluricanonical forms

We review some recent development on the extension problem of pluricanonical forms from a divisor to the ambient space in [Si], [K5] and [N3] with simplified proofs. A section for a correction is added.Comment: 18 page

Birational geometry and derived categories

This paper is based on a talk at a conference "JDG 2017: Conference on Geometry and Topology". We survey recent progress on the DK hypothesis connecting the birational geometry and the derived categories stating that the K-equivalence of smooth projective varieties should correspond to the equivalence of their derived categories, and the K-inequality to the fully faithful embedding.Comment: 24 page

On effective non-vanishing and base-point-freeness

Let $X$ be a complete normal variety, $B$ an effective $\mathbb{R}$-divisor on $X$, and $D$ a Cartier divisor on $X$. Assume that the pair $(X, B)$ is log terminal. We consider the problem whether $H^0(X, D) \ne 0$ and obtain some results in lower dimensions.Comment: 13 pages, Theorem 5.2 (3) is delete

Flops connect minimal models

A remark on a paper by Birkar-Cascini-Hacon-McKernan.Comment: 5 page

On Fujita's freeness conjecture for 3-folds and 4-folds

The result on 3-fold is now optimal by the result of Ein-Lazarsfeld and Helmke.Comment: AmsTe

Hodge theory on generalized normal crossing varieties

We generalize some results in Hodge theory to generalized normal crossing varieties.Comment: 19 page

Euivalences of derived catgories of sheaves on smooth stacks

We extend Orlov's representability theorem on the equivalence of derived categories of sheaves to the case of smooth stacks associated to normal projective varieties with only quotient singularities.Comment: 29 page

D-equivalence and K-equivalence

Let $X$ and $Y$ be smooth projective varieties over $\mathbb{C}$. They are called {\it $D$-equivalent} if their derived categories of bounded complexes of coherent sheaves are equivalent as triangulated categories, while {\it $K$-equivalent} if they are birationally equivalent and the pull-backs of their canonical divisors to a common resolution coincide. We expect that the two equivalences coincide at least for birationally equivalent varieties. We shall provide a partial answer to the above problem in this paper.Comment: 25 pages, minor chang

Semipositivity theorem for reducible algebraic fiber spaces

We shall prove an extension of the semipositivity theorem for the case of reducible algebraic fiber spaces.Comment: 20 page

General hyperplane sections of nonsingular flops in dimension 3

We give a simple proof of a theorem of Katz and Morrison.Comment: 3 pages, AMS-TE