Related papers: The LLV decomposition of hyper-Kaehler cohomology

We construct real polarizable Hodge structures on the reduced leafwise cohomology of K\"ahler-Riemann foliations by complex manifolds. As in the classical case one obtains a hard Lefschetz theorem for this cohomology. Serre's K\"ahlerian…

Let M be a hypercomplex Hermitian manifold, (M,I) the same manifold considered as a complex Hermitian with a complex structure I induced by the quaternions. The standard linear-algebraic construction produces a canonical nowhere degenerate…

Let $M$ be a closed symplectic manifold of dimension $2n$ with non-ellipticity. We can define an almost K\"ahler structure on $M$ by using the given symplectic form. Hence, we have a $\G=\pi_1(M)$-invariant almost K\"ahler structure on the…

Let $M^n$ be a compact K$\ddot{a}$hler manifold with almost nonnegative Ricci curvature and nonzero first Betti number. We show that the holomorphic Euler number of $M^n$ vanishes, which gives a new obstruction for compact complex manifolds…

In this paper, we study the degeneration and stability of K\"ahler structures on Calabi--Yau manifolds, namely compact K\"ahler manifolds with trivial canonical bundles, from the viewpoint of deformation theory and Hodge theory. Using the…

We study Kahler manifolds-with-boundary, not necessarily compact, with weakly pseudoconvex boundary, each component of which is compact. If such a manifold $K$ has $l\ge2$ boundary components (possibly $l=\infty$), then it has first betti…

Oeljeklaus-Toma (OT) manifolds are complex non-K\"ahler manifolds whose construction arises from specific number fields. In this note, we compute their de Rham cohomology in terms of invariants associated to the background number field.…

We prove a Kawamata-Viehweg vanishing theorem on a normal compact Kahler space X: if L is a nef line bundle with numerical dimension at least equal to 2, then the q-th cohomology group of K_X+L vanishes for q at least equal to the dimension…

We study the J-invariant and J-anti-invariant cohomological subgroups of the de Rham cohomology of a compact manifold M endowed with an almost-K\"ahler structure (J, \omega, g). In particular, almost-K\"ahler manifolds satisfying a…

We introduce in this paper the notion of Hodge similarities of transcendental lattices of hyperk\"ahler manifolds and investigate the Hodge conjecture for these Hodge morphisms. Studying K3 surfaces with a symplectic automorphism, we prove…

We prove that a standard realization of the direct image complex via the so-called Douady-Barlet morphism associated with a smooth complex analytic surface admits a natural decomposition in the form of an injective quasi-isomorphism of…

We study rank $1$ flat bundles over solvmanifolds whose cohomologies are non-trivial. By using Hodge theoretical properties for all topologically trivial rank $1$ flat bundles, we represent the structure theorem of K\"ahler solvmanifolds as…

It is shown that a compact $n$-dimensional K\"ahler manifold with $\frac{n}{2}$-positive Calabi curvature operator has the rational cohomology of complex projective space. For even $n,$ this is sharp in the sense that the complex quadric…

We propose two conjectures on a moduli theoretic approach to constructing Lagrangian subvarieties of hyperk\"ahler varieties arising from the Kuznetsov components of cubic fourfolds or Gushel--Mukai fourfolds. Then we verify the conjectures…

We prove that there are no unexpected universal integral linear relations and congruences between Hodge, Betti and Chern numbers of compact complex manifolds and determine the linear combinations of such numbers which are bimeromorphic or…

We investigate the Lefschetz standard conjecture for degree $2$ cohomology of hyper-K\"ahler manifolds admitting a covering by Lagrangian subvarieties. In the case of a Lagrangian fibration, we show that the Lefschetz standard conjecture is…

Let $L_f$ be a link of an isolated hypersurface singularity defined by a weighted homogenous polynomial $f.$ In this article, we give ten examples of $2$-connected seven dimensional Sasaki-Einstein manifolds $L_f$ for which $H_{3}(L_f,…

A compact symplectic manifold $(M, \omega)$ is said to satisfy the hard-Lefschetz condition if it is possible to develop an analogue of Hodge theory for $(M, \omega)$. This loosely means that there is a notion of harmonicity of differential…

We discuss a peculiar interplay between the representation theory of the holonomy group of a Riemannian manifold, the Weitzenboeck formula for the Hodge-Laplace operator on forms and the Lichnerowicz formula for twisted Dirac operators. For…

Recently, L.Rozansky and E.Witten (hep-th/9612216) associated to any hyperKaehler manifold X a system of "weights" (numbers, one for each trivalent graph) and used them to construct invariants of topological 3-manifolds. We give a very…