Related papers: In\'egalit\'es de Morse et vari\'et\'es de Moishez…

We prove that Demailly's holomorphic Morse inequalities hold true for complex orbifolds by using a heat kernel method. Then we introduce the class of Moishezon orbifolds and as an application of our inequalties, we give a geometric…

Let $X$ be a compact complex manifold such that its canonical bundle $K_X$ is numerically trivial. Assume additionally that $X$ is Moishezon or $X$ is Fujiki with dimension at most four. Using the MMP and classical results in foliation…

We study the existence of $L^2$ holomorphic sections of invariant line bundles over Galois coverings of Zariski open sets in Moishezon manilolds. We show that the von Neuman dimension of the space of $L^2$ holomorphic sections is bounded…

Based on our recent adaptation of the adiabatic limit construction to the case of complex structures, we prove the fact that the deformation limiting manifold of any holomorphic family of Moishezon manifolds is Moishezon. Two new…

The present paper mainly presents, for example, explicit classifications of compact smooth manifolds having non-empty boundaries and simple structures where the dimensions are general. Studies of this type is fundamental and important. They…

Given a (smooth) complex analytic family of compact complex manifolds, we prove that the central fibre must be Moishezon if the other fibres are Moishezon. Using a "strongly Gauduchon metric" on the central fibre whose existence was proved…

The goal of this paper is to generalize Demailly's asymptotic holomorphic Morse inequalities to the case of a covering manifold of a compact manifold. We shall obtain estimates which involve Atiyah's ``normalized dimension'' of the square…

Let X be a compact Moishezon manifold which becomes projective after blowing up a smooth subvariety $Y \subset X$. We assume also that there exists a proper map $\rho :X \to X'$ onto a projective variety X' with $\rho(Y)$ a point, such that…

Let X be a compact manifold with boundary and let L^k be a high power of a hermitian holomorphic line bundle over X. When X has no boundary, Demailly's holomorphic Morse inequalities give asymptotic bounds on the dimensions on the Dolbeault…

Consider an action of a connected compact Lie group on a compact complex manifold $M$, and two equivariant vector bundles $L$ and $E$ on $M$, with $L$ of rank 1. The purpose of this paper is to establish holomorphic Morse inequalities \`{a}…

We determine the 6-dimensional solvmanifolds admitting an invariant complex structure with holomorphically trivial canonical bundle. Such complex structures are classified up to isomorphism, and the existence of strong K\"ahler with torsion…

It is known that the Picard group of a complex manifold can be expressed as a Deligne cohomology group. One may wonder if the same holds for the Picard group of a smooth algebraic variety and Deligne-Beilinson cohomology but this is not…

Let $X$ be a compact complex, not necessarily K\"ahler, manifold of dimension $n$. We characterise the volume of any holomorphic line bundle $L\to X$ as the supremum of the Monge-Amp\`ere masses $\int_X T_{ac}^n$ over all closed positive…

For a compact complex manifold, we introduce holomorphic foliations associated with certain abelian subgroups of the automorphism group. Such foliations are generalizations of holomorphic principal torus bundles. If there exists a…

We construct Morse homology groups associated with any regular function on a smooth complex algebraic variety, allowing singular and non-compact critical loci. These groups are generated by critical points of a certain large pertubation of…

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…

The main goal of this work is to present a detailed study of the foundations of Complex Geometry, highlighting its geometrical, topological and analytical aspects. Beginning with a preliminary material, such as the basic results on…

Let $(X,\omega)$ be a Hermitian manifold and let $(E,h^E)$, $(F,h^F)$ be two Hermitian holomorphic line bundle over $X$. Suppose that the maximal rank of the Chern curvature $c(E)$ of $E$ is $r$, and the kernel of $c(E)$ is foliated, i.e.…

Using a result of Fujita on approximate Zariski decompositions and the singular version of Demailly's holomorphic Morse inequalities as obtained by Bonavero, we express the volume of a line bundle in terms of the absolutely continuous parts…

Let $\pi: \mathcal{X}\rightarrow \Delta$ be a holomorphic family of compact complex manifolds over an open disk in $\mathbb{C}$. If the fiber $\pi^{-1}(t)$ for each nonzero $t$ in an uncountable subset $B$ of $\Delta$ is Moishezon and the…