Related papers: Compact K\" ahler threefolds of $\pi_1$-general ty…

Let $(M^n, g)$ be a compact K\"ahler manifold with nonpositive bisectional curvature. We show that a finite cover is biholomorphic and isometric to a flat torus bundle over a compact K\"ahler manifold $N^k$ with $c_1 < 0$. This confirms a…

Let $X$ be a compact K{\"a}hler manifold of dimension three. We prove that there exists a projective manifold $Y$ such that $\pi\_1(X)\simeq \pi\_1(Y)$. We also prove the bimeromorphic existence of algebraic approximations for compact…

We classify all smooth compact connected K\"ahler threefolds that admit the structure of a $C^\infty$-fiber bundle over the circle. This generalizes the work of Hao and Schreieder in the projective case. In contrast to the projective case,…

Let X be a compact Kaehler threefold such that the base of the MRC-fibration has dimension two. We prove that X is bimeromorphic to a Mori fibre space. Together with our earlier result arXiv:1304.4013 this completes the MMP for compact…

We provide infinitely many examples of pairs of diffeomorphic, non simply connected K\" ahler manifolds of complex dimension three with different Kodaira dimensions. Also, in any possible Kodaira dimension we find infinitely many pairs of…

Let X be a compact K\"ahler manifold such that the universal cover admits a compactification. We conjecture that the fundamental group is almost abelian and reduce it to a classical conjecture of Iitaka.

We construct classes of K\"ahler groups that do not have finite classifying spaces and are not commensurable to subdirect products of surface groups. Each of these groups is the fundamental group of the generic fibre of a holomorphic map…

We give a bimeromorphic classification of compact K\"ahler manifolds of Kodaira codimension one that admit a holomorphic one form without zeros.

In this paper we describe the construction of a new class of non-K\"ahler compact complex manifolds. They can be seen as a generalization of Sankaran, OT and LVMB manifolds. Moreover, we give properties of these new spaces. Their Kodaira…

We study holomorphic 2-forms on projective (or compact Kaehler) threefolds not of general type and prove that in almost all cases the 2-form is created by some standard process. This means roughly that every 2-form is induced by a…

This is a sequel to [Ca01]=math.AG/0110051. We define the bimeromorphic {\it category} of geometric orbifolds. These interpolate between (compact K\" ahler) manifolds and such manifolds with logarithmic structure. These geometric orbifolds…

We prove the Jordan property for groups of bimeromorphic selfmaps of three-dimensional compact K\"ahler varieties of non-negative Kodaira dimension and positive irregularity.

Let $M$ be a compact Riemannian manifold, $\pi:\widetilde{M}\rightarrow M$ be the universal covering and $\omega$ be a smooth $2$-form on $M$ with $\pi^*\omega$ cohomologous to zero. Suppose the fundamental group $\pi_1(M)$ satisfies…

This paper proves that the universal covering of a compact K\"{a}hler manifold with small positive sectional curvature in a certain sense is contractible.

We show that in every dimension greater than or equal to 4, there exist compact Kaehler manifolds which do not have the homotopy type of projective complex manifolds. Thus they a fortiori are not deformation equivalent to a projective…

We prove that every compact K\"ahler threefold has arbitrarily small deformations to some projective manifolds, thereby solving the Kodaira problem in dimension 3.

We construct a compact 6-dimensional solvmanifold endowed with a non-trivial invariant generalized K\"ahler structure and which does not admit any K\"ahler metric. This is in contrast with the case of nilmanifolds which cannot admit any…

We show that the bigraded quasi-isomorphism type of the bigraded, bidifferential algebra of forms on a compact K\"ahler manifold generally contains more information than the de Rham cohomology algebra with its real Hodge structure. More…

We classify non-algebraic compact K\"ahler threefolds admitting an endomorphism $f: X \to X$ of degree at least two.

A new class of compact K\"ahler manifolds, called special, is defined, which are the ones having no surjective meromorphic map to an orbifold of general type. The special manifolds are in many respect higher-dimensional generalisations of…