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

Subadjunction of log canonical divisors II

We extend a subadjunction formula of log canonical divisors as in [K3] to the case when the codimension of the minimal center is arbitrary by using the positivity of the Hodge bundles.Comment: 15 pages, AMS-Te

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