A normal projective complex surface is called a rational homology projective plane if it has the same Betti numbers with the complex projective plane CP 2 . It is known that a rational homology projective plane with quotient singularities has at most 5 singular points. So far all known examples have at most 4 singular points. In this paper, we prove that a rational homology projective plane S with quotient singularities such that K S is nef has at most 4 singular points except one case. The exceptional case comes from Enriques surfaces with a configuration of 9 smooth rational curves whose Dynkin diagram is of type 3A 1 ⊕ 2A 3 .We also obtain a similar result in the differentiable case and in the symplectic case under certain assumptions which all hold in the algebraic case.

Abstract. We find generators of the group of birational automorphisms of the Hessian surface of a general cubic surface. Its nonsingular minimal model is a K3 surface with the Picard lattice of rank 16 which embeds naturally in the even unimodular lattice II 1,25 of rank 26 and signature (1, 25). The generators are related to reflections with respect to some Leech roots. A similar observation was made first in the case of quartic Kummer surfaces in the work of Kondō. We shall explain how our generators are related to the generators of the group of birational automorphisms of a general quartic Kummer surface which is birationally isomorphic to a special Hessian surface. IntroductionLet S : F (x 0 , x 1 , x 2 , x 3 ) = 0 be a nonsingular cubic surface in P 3 over C. Its Hessian surface is a quartic surface defined by the determinant of the matrix of second-order partial derivatives of the polynomial F . When F is general enough, the quartic H is irreducible and has 10 nodes. It contains also 10 lines which are the intersection lines of five planes in general linear position. The union of these five planes is classically known as the Sylvester pentahedron of S. The equation of S can be written as the sum of cubes of some linear forms defining the five planes. A nonsingular model of H is a K3-surfaceH. Its Picard number ρ satisfies the inequality ρ ≥ 16. In this paper we give an explicit description of the group Bir(H) of birational isomorphisms of H when S is general enough so that ρ = 16. Although H, in general, does not have any non-trivial automorphisms (because S does not), the group Bir(H) ∼ = Aut(H) is infinite. We show that it is generated by the automorphisms defined by projections from the nodes of H, a birational involution which interchanges the nodes and the lines, and the inversion automorphisms of some elliptic pencils onH. This can be compared with the known structure of the group of automorphisms of the Kummer surface associated to the Jacobian abelian surface of a general curve of genus 2 (see [Ke2], [Ko]). The latter surface is birationally isomorphic to the Hessian H of a cubic surface ([Hu1]), but the Picard number ofH is equal to 17 instead of 16. We use the method for computing Bir(H) employed by Kondō in [Ko]. We show that the Picard lattice S H of the K3-surfacẽ H can be primitively embedded into the unimodular lattice L = Λ ⊥ U of signature

We show that Mukai's classification of finite groups which may act symplectically on a complex K3 surface extends to positive characteristic p under assumptions that (i) the order of the group is coprime to p and (ii) either the surface or its quotient is not birationally isomorphic to a supersingular K3 surface with Artin invariant 1. In the case without assumption (ii) we classify all possible new groups which may appear. We prove that assumption (i) on the order of the group is always satisfied if p > 11. For p = 2, 3, 5, 11, we give examples of K3 surfaces with finite symplectic automorphism groups of order divisible by p which are not contained in Mukai's list.

A Kummer surface is the minimal desingularization of the surface T/i, where T is a complex torus of dimension 2 and i the involution automorphism on T. T is an abelian surface if and only if its associated Kummer surface is algebraic. Kummer surfaces are among classical examples of K3-surfaces (which are simply-connected smooth surfaces with a nowhere-vanishing holomorphic 2-form), and play a crucial role in the theory of K3-surfaces. In a sense, all Kummer surfaces (resp. algebraic Kummer surfaces) form a 4 (resp. 3)-dimensional subset in the 20 (resp. 19)-dimensional family of K3-surfaces (resp. algebraic K3 surfaces).

First, we shall formulate and prove Theorem of Lie-Kolchin type for a cone and derive some algebro-geometric consequences. Next, inspired by a recent result of Dinh and Sibony we pose a conjecture of Tits type for a group of automorphisms of a complex variety and verify its weaker version. Finally, applying Theorem of Lie-Kolchin type for a cone, we shall confirm the conjecture of Tits type for complex tori, hyperkähler manifolds, surfaces, and minimal threefolds.Here, connected means that its Zariski closure in GL (NS(X) C ) is connected. This property already provides a useful tool in studying groups of automorphisms of projective varieties (see [Og2], [Zh1] for some concrete applications). Another important feature of G * is that it preserves the ample cone Amp (X) and the nef cone Amp (X), the closure of the ample cone, both of which encode much information on geometry of X.Keeping these two features in mind, we shall prove first the following Theorem of Lie-Kolchin type (see also Theorem 2.1 and Corollary 2.3), and derive some direct consequences for groups of automorphisms of projective varieties or compact Kähler manifolds (Theorems 3.1-3.5):Theorem 1.1. (Theorem of Lie-Kolchin type for a cone) Let V be a finite dimensional real vector space, and let C = {0} be a strictly convex closed cone of V . Let G be a connected (in the sense above) solvable subgroup of GL(V ) such that G(C) ⊆ C. Then G has a common eigenvector in the cone C.This theorem is purely algebraic and a priori has nothing to do with algebraic geometry or complex geometry. In our proof, we use Birkhoff-Perron-Frobenius' Theorem [Bir] and the proof of Dinh-Sibony [DS] for the case where G is abelian.