Related papers: On the homotopy types of compact kaehler and compl…

We recently constructed examples of compact Kaeler manifolds which do not have the homotopy type of a projective complex manifold. They were however obtained by blowing-up certain complex tori, which are themselves deformation equivalent to…

Voisin constructed a series of examples concerning simply connected compact K\"ahler manifolds of even dimensions, which do not have the rational homotopy type of a complex projective manifold starting from dimension six. In this note, we…

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 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…

We first notice in this article that if a compact K\"{a}hler manifold has the same integral cohomology ring and Pontrjagin classes as the complex projective space $\mathbb{C}P^n$, then it is biholomorphic to $\mathbb{C}P^n$ provided $n$ is…

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…

We give four constructions of non-$\partial\bar\partial$ (hence non-K\"ahler) manifolds: (1) A simply connected page-$1$-$\partial\bar\partial$-manifold (2) A simply connected $dd^c+3$-manifold (3) For any $r\geq 2$, a simply connected…

We show the existence of a compact K\"ahler manifold which does not fit in a proper flat family over an irreducible base with one projective (possibly singular) fiber. We also give a topological version of this statement. This strengthens…

The aim of this paper is to continue the study of Kodaira dimension for almost complex manifolds, focusing on the case of compact $4$-dimensional solvmanifolds without any integrable almost complex structure. According to the classification…

In this expository paper (Bourbaki talk) we survey results of Claire Voisin showing that there exist compact Kaehler manifolds which are not homeomorphic to any projective manifold.

We show some results of compact Kahler manifolds with elliptic homotopy type. In complex dimension 4 we list the Hodge diamonds of compact Kahler manifolds with elliptic homotopy type. In general dimension we obtain a partial…

We propose that under certain conditions heterotic string compactifications on half-flat and nearly-Kahler manifolds are equivalent. Based on this correspondence we argue that the moduli space of the nearly-Kahler manifolds under discussion…

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…

We observed in our previous paper that all the complex structures on four-dimensional compact solvmanifolds, including tori, are left-invariant. In this paper we will give an example of a six-dimensional compact solvmanifold which admits a…

We exhibit a family of complex manifolds, which has a member at each odd complex dimension and which has the same cohomology groups as the complex projective space at that dimension, but not homotopy equivalent to it. We also analyze the…

The paper studies the deformation theory of a holomorphic surjective map from a normal compact complex space to a compact Kaehler manifold and describes the component of the space of holomorphic maps, generalizing results in the projective…

In this note we begin a systematic study of compact conformal manifolds of SCFTs in four dimensions (our notion of compactness is with respect to the topology induced by the Zamolodchikov metric). Supersymmetry guarantees that such…

In this paper we study smooth complex projective $4$-folds which are topologically equivalent. First we show that Fano fourfolds are never oriented homeomorphic to Ricci-flat projective fourfolds and that Calabi-Yau manifolds and…

We show that closed, connected 4-manifolds up to connected sum with copies of the complex projective plane are classified in terms of the fundamental group, the orientation character and an extension class involving the second homotopy…

For compact complex surfaces (M^4, J) of Kaehler type, it was previously shown that the sign of the Yamabe invariant Y(M) only depends on the Kodaira dimension Kod (M, J). In this paper, we prove that this pattern in fact extends to all…