Introduction Chapter O. Notation and preliminaries § 0-1. Kleiman's criterion for ampleness § 0-2. Definitions of terminal, canonical and (weak) log-terminal singularities § 0-3. Canonical varieties § 0-4. The minimal model conjecture Chapter 1. Vanishing theorems § 1-1. Covering Lemma § 1-2. Vanishing theorem of Kawamata and Viehweg § 1-3. Vanishing theorem of Elkik and Fujita Chapter 2. Non-Vanishing Theorem § 2-1. Non-Vanishing Theorem Chapter 3. Base Point Free Theorem § 3-1. Base Point Free Theorem § 3-2. Contractions of extremal faces § 3-3. Canonical rings of varieties of general type Chapter 4. Cone Theorem § 4-1. Rationality Theorem § 4-2. The proof of the Cone Theorem Chapter 5. Flip Conjecture § 5-1. Types of contractions of extremal rays § 5-2. Flips of toric morphisms

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We extend a subadjunction formula of log canonical divisors as in [Kawamata, Contemp. Math. 207 (1997), 79-88] to the case when the codimension of the minimal center is arbitrary by using the positivity of the Hodge bundles.

Let X and Y be smooth projective varieties over C. They are called D-equivalent if their derived categories of bounded complexes of coherent sheaves are equivalent as triangulated categories, while 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 for birationally equivalent varieties. We shall provide a partial answer to the above problem in this paper. for useful discussions or comments and the anonymous referee for suggestions. From D-equivalence to K-equivalenceWe need the concept of Fourier-Mukai transformation:Definition 2.1. Let X and Y be smooth projective varieties, and let p 1 : X ×Y → X and p 2 : X ×Y → Y be projections. For an object e ∈ D(X ×Y ), we define an integral functor Φ e X→Y : D(X) → D(Y ) by Φ e X→Y (a) = p 2 * (p * 1 (a) ⊗ e)

In this paper we shall generalize Ramanujam's form of Kodaira's vanishing theorem [4] in the higher dimensional case. Let X be a non-singular projective algebraic variety defined over the complex number field. A divisor D on X is said to be numerically effective (or semi-positive), if the intersection number (D-C) is non-negative for any curve C on X. The main result in this paper is the following:Theorem 1. Let D be a numerically effective divisor on X. Assume that the highest self-intersection number (D") is positive, where n = dimX. Then Hi( X, •x(-D)) = 0 for 0 < i < n.