Kodaira Surfaces Research Articles

Stable Bundles on Non-Kähler Elliptic Surfaces

In this paper, we study the moduli spaces of stable rank-2 vector bundles on non-Kähler elliptic surfaces, thus giving a classification of these bundles; in the case of Hopf and Kodaira surfaces, these moduli spaces admit the structure of an algebraically completely integrable Hamiltonian system.

Entropy and collapsing of compact complex surfaces

We study the problem of existence of $\mathcal{F}$ -structures (in the sense of Cheeger and Gromov, but not necessarily polarized) on compact complex surfaces. We give a complete classification of compact complex surfaces of Kahler type admitting $\mathcal{F}$ -structures. In the non-Kahler case we give a complete classification modulo the gap in the classification of surfaces of class VII. We then use these results to study the minimal entropy problem for compact complex surfaces: we prove that, modulo the gap in the classification of surfaces of class VII, all compact complex surfaces of Kodaira dimension less than or equal to 1 have minimal entropy zero and such a surface admits a smooth metric $g$ with zero topological entropy if and only if it is ${\mathbb C} P^2$ , a ruled surface of genus 0 or 1, a Hopf surface, a complex torus, a Kodaira surface, a hyperelliptic surface or a Kodaira surface modulo a finite group. The key result we use to prove this is a new topological obstruction to the existence of metrics with vanishing topological entropy. Finally we show that these results fit perfectly into Wall's study of geometric structures on compact complex surfaces. For instance, we show that the minimal entropy problem can be solved for a minimal compact Kahler surface $S$ of Kodaira dimension $- \infty$ , 0 or 1 if and only if $S$ admits a geometric structure modelled on ${\mathbb C} P^2$ , $S^2 \times S^2$ , $S^2 \times {\mathbb E}^2$ or ${\mathbb E}^4$ .

DEFORMATIONS OF KODAIRA MANIFOLDS

Using compact simple Lie groups and Heisenberg groups, we combine and generalize the constructions of complex structures on Kodaira surfaces and Hopf surfaces. We identify locally complete parameter spaces of deformations of these spaces and analyze the deformation of Kodaira manifolds in details.

Compact moduli spaces of stable sheaves over non-algebraic surfaces

We show that under certain conditions on the topological invariants, the moduli spaces of stable bundles over polarized non-algebraic surfaces may be compactified by allowing at the border isomorphy classes of stable non-necessarily locally-free sheaves. As a consequence, when the base surface is a primary Kodaira surface, we obtain examples of moduli spaces of stable sheaves which are compact holomorphically symplectic manifolds.

Holomorphic vector bundles on primary Kodaira surfaces

It is in general unknown which topological complex vector bundles on a non-algebraic surface admit holomorphic structures. We solve this problem for primary Kodaira surfaces by using results of Kani on curves of genus two with elliptic differentials. Some of the corresponding moduli spaces will be smooth compact and holomorphically symplectic.

A six-dimensional compact symplectic solvmanifold without Kähler structures

Gordon and Benson have proved that if a compact nilmanifold admits a Kahler structure then it is a torus [5] more precisely they proved that the condition (iv) fails for any symplectic structure on a non-toral nilmanifold M. This result was independently proved by Hasegawa [12] by showing that (v) fails for M. For a compact solvmanifold M of dimension 4 it is known that M has a Kahler structure if and only if it is a complex torus or a hyperelliptic surface. In fact, Auslander and Szczarba in [4] proved that if the first Betti number bι(M) of M is 2, M is a fiber bundle over T with fiber T. Then by Ue [19] M has a complex structure only if it is a hyperelliptic surface or a primary Kodaira surface which is a compact nilmanifold. Thus, if M is a Kahler manifold, it must be a hyperelliptic surface. Since !<&ι(M)<4, M can be a Kahler manifold only if it is a complex torus or a hyperelliptic surface. The fact that a hyperelliptic surface is a solvmanifold follows from Auslander [3]. The above result may be generalized as the following conjecture : A compact solvmanifold has a Kahler structure if and only if it is a finite quotient of a complex torus. In contrast to the case of compact nilmanifolds there are compact symplectic

Higher-dimensional Analogues of Inoue and Kodaira Surfaces

Higher-dimensional Analogues of Inoue and Kodaira Surfaces

The Picard group of a primary Kodaira surface

The Picard group of a primary Kodaira surface

Negatively curved metrics on Kodaira surfaces

It has been known that Kodaira surfaces of general type have negative holomorphic tangent bundles (cf. [! 1]). This definition of negativity is algebraic-geometric in nature, i.e. negative in the sense of Grauert [5]. Griffiths [7] discussed the relations between various notions of negativity in algebraic geometry and those in differential geometry, and raised the question of whether a Grauert-negative holomorphic vector bundle also admits a hermitian metric of negative holomorphic bisectional curvature. If it does, we would say that this metric is negatively curved, and the vector bundle is negative in Griffiths' sense. The aim of this work is to show that the tangent bundles of certain type of surfaces considered in [11] by Schneider, including Kodaira surfaces of general type, are negative in Griffiths' sense. We also remark here that Cheung [2] constructed on Kodaira surfaces of general type a "K/ihler metric" of negative holomorphic sectional curvature.

Automorphism groups of Inoue and Kodaira surfaces

Automorphism groups of Inoue and Kodaira surfaces

Hermitian manifolds with semi-positive holomorphic sectional curvature

We prove that a compact Hermitian manifold with semi-positive but not identically zero holomorphic sectional curvature has Kodaira dimension $-\infty$. As applications, we show that Kodaira surfaces and hyperelliptic surfaces can not admit Hermitian metrics with semi-positive holomorphic sectional curvature although they have nef tangent bundles.