Related papers: A global Torelli theorem for hyperkaehler manifold…
We prove that the first order deformations of two smooth projective K3 surfaces are derived equivalent under a Fourier--Mukai transform if and only if there exists a special isometry of the total cohomology groups of the surfaces which…
We prove that given two compact oriented $3$-manifolds $N$ and $M,$ with $M$ satisfying only a mild hypothesis, there is a hyperbolic $3$-manifold $N'$ arbitrarily ``closely related'' to $N,$ and such that $N'$ does not embed in $M.$ For…
We revisit Brunella's proof of the fact that Kato surfaces admit locally conformally K\" ahler metrics, and we show that it holds for a large class of higher dimensional complex manifolds containing a global spherical shell. On the other…
In this paper, we discuss the cycle theory on moduli spaces $\cF_h$ of $h$-polarized hyperk\"ahler manifolds. Firstly, we construct the tautological ring on $\cF_h$ following the work of Marian, Oprea and Pandharipande on the tautological…
The well-known K\"ahler identities naturally extend to the non-integrable setting. This paper deduces several geometric and topological consequences of these extended identities for compact almost K\"ahler manifolds. Among these are…
We investigate the global topology of 3-dimensional Hessian manifolds. We prove that any compact, orientable 3-dimensional Hessian manifold is either a Hantzsche-Wendt manifold or admits the structure of a K\"ahler mapping torus. We analyze…
We study the moduli space of pairs consisting of a smooth cubic surface and a smooth hyperplane section, via a Hodge theoretic period map due to Laza, Pearlstein, and the second named author. The construction associates to such a pair a…
We survey recent advances in the theory of moduli spaces of stable sheaves on hyperk\"ahler manifolds of dimension greater than $2$. We start by recalling the well-known theory in dimension $2$, i.e.~for $K3$ surfaces, emphasizing the…
In previous work, we noted that the known cases of hyper-K\"ahler manifolds satisfy a natural condition on the LLV decomposition of the cohomology; informally, the Verbitsky component is the dominant representation in the LLV decomposition.…
We prove several vanishing theorems for the cohomology of balanced hyperbolic manifolds that we introduced in our previous work and for the $L^2$ harmonic spaces on the universal cover of these manifolds. Other results include a Hard…
This paper investigates the distribution of non-rigid families in a moduli space $\mathcal{M}$ of polarized projective manifolds for which the infinitesimal Torelli theorem holds. Guided by the analogy with unlikely intersection in Shimura…
A Teichm\"uller space $Teich$ is a quotient of the space of all complex structures on a given manifold $M$ by the connected components of the group of diffeomorphisms. The mapping class group $\Gamma$ of $M$ is the group of connected…
We introduce the notion of a Hyper-K\"{a}hler manifold $X$ induced by a Hodge structure of K3-type. We explore this notion for the known deformation types of hyper-K\"{a}hler manifolds studying those that are induced by a K3 or abelian…
Smooth cubic fourfolds are linked to K3 surfaces via their Hodge structures, due to work of Hassett, and via Kuznetsov's K3 category A. The relation between these two viewpoints has recently been elucidated by Addington and Thomas. In this…
The article proves the Infinitesimal Torelli theorem for surfaces subject to the following conditions: 1) the canonical bundle of a surface is ample and generated by its global sections, 2)the geometric genus $p_g \geq 4$, 3) the…
We prove a conjecture of Odaka--Oshima, which says that there is an algebraic description of the Gromov--Hausdorff compactification of all unit-diameter hyperk\"ahler metrics on K3 surfaces. As a corollary, we obtain a classification of the…
We study the global hypoellipticity of the operator $\mathbb{L} = \mathrm{d}_t + \sum_{k=1}^m \omega_k \wedge \partial_{x_k}$, defined on differential forms over product manifolds of the form $M \times \mathbb{T}^m$, where $M$ is a…
This note is a summary of our work [OO] which provides an explicit and global moduli-theoretic framework for the collapsing of Ricci-flat Kahler metrics and we use it to study especially the K3 surfaces case. For instance, it allows us to…
Phenomenological implications of the volume of the Calabi-Yau threefolds on the hidden and observable M-theory boundaries, together with slope stability of their corresponding vector bundles, constrain the set of Kaehler moduli which give…
In the first part we survey some of the known results and conjectures on compact Hyperkaehler (HK) manifolds. In the second part we presents a program which aims to show that HK four-folds whose second cohomology (with 4-tuple cup-product)…