Related papers: The LLV decomposition of hyper-Kaehler cohomology

We provide new families of compact complex manifolds with no K\"ahler structure carrying symplectic structures satisfying the \textit{Hard Lefschetz Condition}. These examples are obtained as compact quotients of the solvable Lie group…

We prove that for every compact K\"ahler manifold $X$ there exists an $L$-infinity morphism, lifting the usual cup product in cohomology, from the Kodaira-Spencer differential graded Lie algebra to the suspension of the space of linear…

A locally conformally Kahler (LCK) manifold is a complex manifold admitting a Kahler covering, with the monodromy acting on this covering by homotheties. We define three cohomology invariants, the Lee class, the Morse-Novikov class, and the…

For a compact Lie group G we define a regularized version of the Dolbeault cohomology of a G-equivariant holomorphic vector bundles over non-compact Kahler manifolds. The new cohomology is infinite-dimensional, but as a representation of G…

In complex dimensions $\geq 3$, we provide a geometric existence for generalized ALG complete non-compact Ricci flat K\"ahler manifolds with Schwartz decay i.e. metric decay in any polynomial rate to an ALG model $\mathbb{C}\times Y$ modulo…

Let M be a compact locally conformal hyperkaehler manifold. We prove a version of Kodaira-Nakano vanishing theorem for M. This is used to show that M admits no holomorphic differential forms, and the cohomology of the structure sheaf…

We generalize to nearly K\"ahler manifolds of arbitrary dimensions most of the Hodge-theoretic results for nearly K\"ahler $6$-manifolds that were established by Verbitsky. In particular, for a compact nearly K\"ahler manifold of any…

We study the Lefschetz standard conjecture on a smooth complex projective variety X. In degree 2, we reduce it to a local statement concerning deformations of vector bundles on X. When X is hyperk\"ahler, we show that the existence of…

Let $(X,J,\omega,g)$ be a complete $n$-dimensional K\"ahler manifold. A Theorem by Gromov \cite{G} states that the if the K\"ahler form is $d$-bounded, then the space of harmonic $L_2$ forms of degree $k$ is trivial, unless $k=\frac{n}{2}$.…

We prove that a hyper-K\"ahler fourfold satisfying a mild topological assumption is of K3$^{[2]}$ deformation type. This proves in particular a conjecture of O'Grady stating that hyper-K\"ahler fourfolds of K3$^{[2]}$ numerical type are of…

We investigate when the Chevalley-Eilenberg differential of a complex Lie algebroid on a manifold with boundary admits a Hodge decomposition. We introduce the concepts of Cauchy-Riemann structures, elliptic and non-elliptic boundary points…

We derive sufficient conditions for the vanishing of plurigenera, $p_m(J), m>0$, on compact (l|k)-strong, $\omega^l\wedge \partial\bar\partial \omega^k=0$, Kaehler manifolds with torsion. In particular, we show that the plurigenera of…

The weight $\theta$-sheaf $\underline{\mathbb{R}}_{X,\theta}$ helps us to reinterpret Morse-Novikov cohomologies via sheaf theory. We give several theorems of K\"{u}nneth and Leray-Hirsch types. As applications, we prove that the…

We study the behavior of connections and curvature under the HK/QK correspondence, proving simple formulae expressing the Levi-Civita connection and Riemann curvature tensor on the quaternionic K\"ahler side in terms of the initial…

In this paper, we prove a $\partial\bar{\partial}$-type lemma on compact K\"ahler manifolds for logarithmic differential forms valued in the dual of a certain pseudo-effective line bundle, thereby confirming a conjecture proposed by X. Wan.…

In this thesis we use the Beauville-Bogomolov decomposition to compute the LLV algebra of smooth projective complex varieties admitting a holomorphic symplectic form, generalizing known results from hyperk\"ahler and abelian varieties.…

For the Kirillov-Poisson structure on the vector space $\g^*$, where $\g$ is a finite-dimensional Lie algebra, it is known at least two canonical deformations quantization of this structure: they are the M. Kontsevich universal formula [K],…

A locally conformally Kahler (LCK) manifold is a complex manifold with a Kahler structure on its covering and the deck transform group acting on it by holomorphic homotheties. One could think of an LCK manifold as of a complex manifold with…

We conjecture that certain curvature invariants of compact hyperk\"ahler manifolds are positive/negative. We prove the conjecture in complex dimension four, give an "experimental proof" in higher dimensions, and verify it for all known…

In this article, we discuss the spaces of harmonic forms $\mathcal{H}^{\bullet}_{d}$ over a closed almost K\"{a}hler manifold $(X, J,\omega)$. We show that if the almost complex structure $J$ on the almost K\"{a}hler manifold $X$ is not too…