For every smooth complex projective variety W of dimension d and nonnegative Kodaira dimension, we show the existence of a universal constant m depending only on d and two natural invariants of the very general fibres of an Iitaka fibration of W such that the pluricanonical system |mK W | defines an Iitaka fibration. This is a consequence of a more general result on polarized adjoint divisors. In order to prove these results we develop a generalized theory of pairs, singularities, log canonical thresholds, adjunction, etc.

Abstract. We present a complete list of extremal elliptic K3 surfaces (Theorem 1

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Abstract. An endomorphism f of a projective variety X is polarized (resp. quasipolarized) if f * H ∼ qH (linear equivalence) for some ample (resp. nef and big) Cartier divisor H and integer q > 1. First, we use cone analysis to show that a quasi-polarized endomorphism is always polarized, and the polarized property descends via any equivariant dominant rational map. Next, we show that a suitable maximal rationally connected fibration (MRC) can be made f -equivariant using a construction of N. Nakayama, that f descends to a polarized endomorphism of the base Y of this MRC and that this Y is a Q-abelian variety (quasi-étale quotient of an abelian variety). Finally, we show that we can run the minimal model program (MMP) f -equivariantly for mildly singular X and reach either a Q-abelian variety or a Fano variety of Picard number one.As a consequence, the building blocks of polarized endomorphisms are those of Qabelian varieties and those of Fano varieties of Picard number one.Along the way, we show that f always descends to a polarized endomorphism of the Albanese variety Alb(X) of X, and that the pullback of a power of f acts as a scalar multiplication on the Neron-Severi group of X (modulo torsion) when X is smooth and rationally connected.Partial answers about X being of Calabi-Yau type, or Fano type are also given with an extra primitivity assumption on f which seems necessary by an example.

We prove a theorem of Tits type for compact Kähler manifolds, which has been conjectured in the paper [9]. IntroductionWe work over the field C of complex numbers. In this note, we prove first the following result, which also gives an affirmative answer to the conjecture of Tits type for compact Kähler manifolds as formulated in [9].Theorem 1.1. Let X be an n-dimensional (n ≥ 2) compact Kähler manifold and G a subgroup of Aut(X). Then one of the following two assertions holds:(1) G contains a subgroup isomorphic to the non-abelian free group Z * Z, and hence G contains subgroups isomorphic to non-abelian free groups of all countable ranks.(2) There is a finite-index subgroup G 1 of G such that the induced action G 1 |H 1,1 (X)is solvable and Z-connected. Further, the subset N(G 1 ) := {g ∈ G 1 | g is of null entropy} of G 1 is a normal subgroup of G 1 and the quotient group G 1 /N(G 1 ) is a free abelian group of rank r ≤ n − 1 (see 1.2 below for the boundary cases).In Theorems 1.1 and 1.2, the action G|H 1,1 (X) is Z-connected if its Zariski-closure in GL(H 1,1 (X)) is connected with respect to the Zariski topology. X is said to be almost homogeneous (resp. dominated by some closed subgroup H of the identity connected component Aut 0 (X) of Aut(X)) if some Aut 0 (X)-orbit (resp. H-orbit) is dense open in X.A compact Kähler manifold is weak Calabi-Yau if the irregularity q(X) := h 1 (X, O X ) = 0 and the Kodaira dimension κ(X) = 0. A projective manifold is ruled if it is birational to P 1 × (another projective manifold).Theorem 1.2. Let X be an n-dimensional (n ≥ 2) compact Kähler manifold and G a subgroup of Aut(X) such that the induced action G|H 1,1 (X) is solvable and Z-connected.

Research of the first author was supported in part by a NSF grant 6−k i=1 H i ; (m, k) = (0, k) with (1 ≤ k ≤ 6); M 1 , H i are lines through the same point p 1 ; H i = H j is allowed but M 1 = H i . (5) Γ = M 1 + J ℓ=1 g ℓ G ℓ ; (m, k) = (1, 1); g 1 = 1, 2; 2g 1 + J j=2 g j = 5; G 1 is a conic; M 1 andintersect M 1 transversally at the same point p 1 and J − 1 other points; G 1 meets M 1 at two distinct points not in M 1 ∩ G j (j ≥ 2). (9) Γ = M 1 + 4G 1 ; (m, k) = (4, 1); M 1 is a conic; G 1 is a line intersecting M 1 at two distinct points. (10) Y min = P 1 × P 1 ; Γ = M 1 + J ℓ=1 g ℓ G ℓ ; (m, k) = (2, 1); g 1 = 1, 2; r i=2 g i = J j=r+1 g j = 3 − g 1 ; M 1 and G 1 are sections (of both rulings) of self intersection 2 and intersect each other at two distinct points; G i (2 ≤ i ≤ r) and G j (r + 1 ≤ j ≤ J) are distinct fibers of two different rulings such that Sing; H 1 is the unique (−2)-curve on F 2 ; M 1 and G 1 are two sections of self intersection 2 and intersect each other at two distinct points; G j (2 ≤ j ≤ J) are distinct fibers not through M 1 ∩ G 1 ; when h = 0 (resp. h = 3), there is no such G j (resp. no such G 1 ).are distinct fibres of a fixed ruling not through M 1 ∩ G 1 ; G 1 is a section with G 2 1 = −(m − 4); M 1 is a section with M 2 1 = m and meeting G 1 at two distinct points.

Etale endomorphisms of complex projective manifolds are constructed from two building blocks up to isomorphism if the good minimal model conjecture is true. They are the endomorphisms of abelian varieties and the nearlyétale rational endomorphisms of weak Calabi-Yau varieties.Let X be a uniruled nonsingular projective variety. A maximal rationally connected fibration of X in the sense of [10,40] is obtained by a certain rational map X ···→ Chow(X) into the Chow variety Chow(X), which assigns a general point x ∈ X a maximal rationally connected subvariety containing x. Let Y be the normalization of the image of X ···→ Chow(X) and let π : X ···→ Y be the induced rational fibration. Assume that X admits anétale endomorphism f : X → X. Then there is an endomorphism h : Y → Y such that π • f = h • π (cf. Lemma 5.2). Since rationally connected manifolds are simply connected, the endomorphism f is induced from h. In Theorem C below, we shall show that h is nearlyétale.Theorem C. Let X be a projective manifold with anétale endomorphism f . Assume that X is uniruled. Then there exist a projective manifold M with anétale endomorphism(Remark.(1) We have h −1 (Y rat ) = Y rat and the restriction Y rat → Y rat of h isétale for the open subset Y rat ⊂ Y consisting of the smooth points and the points of rational singularity, by Propositions 3.11 and 3.10(2).(2) If Y has the relative canonical model Y can for resolutions of singularities of Y , then, by Lemma 3.8, h lifts to anétale endomorphism of Y can and also to anétale endomorphism of a certain resolution Y of singularities of Y . The recent paper [6] has announced a proof of the existence of minimal models of varieties of general type even in a relative setting. The existence of the relative canonical model Y can follows from the result. Equivariant resolutionsLet V be a normal projective complex variety and let f : V → V be anétale endomorphism. Then there exists a resolution μ : X → V of singularities such that the induced rational mapThis is known as the existence theorem of an equivariant resolution when f is an automorphism. However, the proof is also effective for nonisomorphicétale endomorphisms: a method of resolution of singularities is called to have a functoriality if, for any smooth morphism X → Y , and for the resolutions of singularities X → X and Y → Y given by the method, X is isomorphic to the fiber product X × Y Y . The recent methods by Bierstone-Milman and by Villamayor using the notion of invariant have the functoriality (cf. [5, 14, 15, 39, 55]). Therefore, we call the resolution X → V above also an equivariant resolution even if f is a nonisomorphicétale endomorphism of V . The meaning of our reductionLet X be a nonsingular projective variety with anétale endomorphism f .

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