Deformation types of real and complex manifolds

Main topic of the paper is the determination, for a compact complex manifold $M$, of the class of manifolds $X$ which are deformation equivalent to it. If $M$ is a complex torus, then also $X$ is so. After describing the structure of principal holomorphic torus bundles over curves, a similar result (stability by deformations in the large) is obtained also for the latter class of manifolds. A section of the paper is devoted to the structure of principal holomorphic torus bundl...

In this article we study compact K\"ahler manifolds $X$ admitting non-singular holomorphic vector fields with the aim of extending to this setting the classical birational classification of projective varieties with tangent vector fields. We prove that any such a K\"ahler manifold $X$ admits an arbitrarily small deformation of a particular type which is a suspension over a torus; that is, a quotient of $F\times \mbb C^s$ fibering over a torus $T=\mbb C^s/\Lambda$. We derive s...

Here, we resume and broaden the results concerned which appeared in math.AG/0101098 and math.AG/0104021. We start from summing up our example of a complex algebraic surface which is not deformation equivalent to its complex conjugate and which, moreover, has no homeomorphisms reversing the canonical class. Then, we construct several series of higher dimensional compact complex manifolds having the same property. We end with discussing applications to the Dif=Def problems, to ...

It is proved that the number of deformation types of complex structures on a fixed oriented smooth four-manifold can be arbitrarily large. The considered examples are locally simple abelian covers of rational surfaces.

We introduce a new notion of deformation of complex structure, which we use as an adaptation of Kodaira's theory of deformations, but that is better suited to the study of noncompact manifolds. We present several families of deformations illustrating this new approach. Our examples include toric Calabi--Yau threefolds, cotangent bundles of flag manifolds, and semisimple adjoint orbits, and we describe their Hodge theoretical invariants, depicting Hodge diamonds and KKP diamon...

We describe the family of real structures $\sigma$ on principal holomorphic torus bundles $X$ over tori, and prove its connectedness when the complex dimension is at most three. From this and previous results of the authors follows that the differentiable type (more precisely, the orbifold fundamental group) determines the deformation type of the pair $(X, \sigma)$ provided we have complex dimension at most three, fibre dimension one, and a certain 'reality' condition on the ...

In this article, we consider Cayley deformations of a compact complex surface in a Calabi--Yau four-fold. We will study complex deformations of compact complex submanifolds of Calabi--Yau manifolds with a view to explaining why complex and Cayley deformations of a compact complex surface are the same. We in fact prove that the moduli space of complex deformations of any compact complex embedded submanifold of a Calabi--Yau manifold is a smooth manifold.

This paper concerns with deformations of noncompact complex hyperbolic manifolds (with locally Bergman metric), varieties of discrete representations of their fundamental groups into $PU(n,1)$ and the problem of (quasiconformal) stability of deformations of such groups and manifolds in the sense of L.Bers and D.Sullivan. Despite Goldman-Millson-Yue rigidity results for such complex manifolds of infinite volume, we present different classes of such manifolds that allow non-t...

Let $X$ be a compact K\"ahler manifold with vanishing Riemann curvature. We prove that there exists a manifold $X'$, deformation equivalent to $X$, which is not an analytification of any projective variety, if and only if $H^0(X, \Omega^2) \neq 0$. Using this, we recover a recent theorem of Catanese and Demleitner, which states that a rigid smooth quotient of a complex torus is always projective. We also produce many examples of non-algebraic flat K\"ahler manifolds with vani...

Deformation theory of complex manifolds is a classical subject with recent new advances in the noncompact case using both algebraic and analytic methods. In this note, we recall some concepts of the existing theory and introduce new notions of deformations for manifolds with boundary, for compactifiable manifolds, and for $q$-concave spaces. We highlight some of the possible applications and give a list of open questions which we intend as a guide for further research in th...