Related papers: A note on compact solvmanifolds with Kaehler struc…

Each choice of a K\"ahler class on a compact complex manifold defines an action of the Lie algebra $\slt$ on its total complex cohomology. If a nonempty set of such K\"ahler classes is given, then we prove that the corresponding…

Given a compact hyperkaehler manifold $M$ and a holomorphic bundle B over $M$, we consider a Hermitian connection $\nabla$ on B which is compatible with all complex structures on $M$ induced by the hyperkaehler structure. Such a connection…

Suppose $G$ is a connected complex Lie group and $H$ is a closed complex subgroup such that $X := G/H$ is Kaehler and the codimension of the top non-vanishing homology group of $X$ with coefficients in $\mathbb Z_2$ is less than or equal to…

It has been conjectured by Fino and Vezzoni that a compact complex manifold admitting both a compatible SKT and a compatible balanced metric also admits a compatible K\"ahler metric. Using the shear construction and classification results…

We give a complex polarized variation of Hodge structure over a compact K"ahler manifold $M$ which controls all finite-dimensional complex polarized variations of Hodge structure over $M$ and their tensor relations. As a corollary, we…

We prove that a compact contact threefold which is bimeromorphically equivalent to a Kaehler manifold and not rationally connected is the projectivised tangent bundle of a Kaehler surface.

We call complex quasifold of dimension k a space that is locally isomorphic to the quotient of an open subset of the space C^k by the holomorphic action of a discrete group; the analogue of a complex torus in this setting is called a…

It is known that there exist complex solvmanifolds $(\Gamma\backslash G,J)$ whose canonical bundle is trivialized by a holomorphic section which is not invariant under the action of $G$. The main goal of this article is to classify the…

An old open question in non-K\"ahler geometry predicts that any compact Hermitian manifold with constant holomorphic sectional curvature must be K\"ahler or Chern flat. The conjecture is known to be true in dimension $2$ due to the work by…

Using examples of compact complex 3-manifolds which arise as twistor spaces, we show that the class of compact complex manifolds bimeromorphic to K\"ahler manifolds is not stable under small deformations of complex structure.

Let $X$ be a connected compact complex manifold admitting a finite surjective map $A \to X$ from a complex torus $A.$ We prove that up to finite \'etale cover, $X$ is a product of projective spaces and a torus.

A strong KT (SKT) manifold consists of a Hermitian structure whose torsion three-form is closed. We classify the invariant SKT structures on four-dimensional solvable Lie groups. The classification includes solutions on groups that do not…

We prove that any invariant strong Kahler structure with torsion (SKT structure) on a flag manifold M=G/K of a semisimple compact Lie group G is Kahler. As an application we describe invariant generalized Kahler structures on M.

Let $X$ be a Calabi-Yau threefold. A conifold transition first contracts $X$ along disjoint rational curves with normal bundles of type $(-1,-1)$, and then smooth the resulting singular complex space $\bar{X}$ to a new compact complex…

In this paper we provide a positive answer to a conjecture due to A. J. Di Scala, A. Loi, H. Hishi (see [3, Conjecture 1]) claiming that a simply-connected homogeneous K\"ahler manifold M endowed with an integral K\"ahler form $\mu\omega$,…

In this short note, we consider a normal compact K\"ahler klt space $X$ whose canonical divisor $K_X$ is pseudo-effective, and give a dynamical criterion for $X$ to be a $Q$-complex torus. We show that, if such $X$ admits an int-amplified…

A vector bundle E on a projective variety X is called finite if it satisfies a nontrivial polynomial equation with integral coefficients. A theorem of Nori implies that E is finite if and only if the pullback of E to some finite etale…

Let X be a compact Kaehler manifold of complex dimension n. Let G be a connected solvable subgroup of the automorphism group Aut(X), and let N(G) be the normal subgroup of G of elements of null entropy. One of the goals of this paper is to…

The smooth action of a compact Lie group on a compact manifold can be resolved to an iterated space, as made explicit by Pierre Albin and the second author. On the resolution the lifted action has fixed isotropy type, in an iterated sense,…

Let $M$ be a singular hyperkaehler variety, obtained as a moduli space of stable holomorphic bundles on a compact hyperkaehler manifold (alg-geom/9307008). Consider $M$ as a complex variety in one of the complex structures induced by the…