Related papers: Complex hyperk\"ahler structures defined by Donald…

We define and construct the real analytic moduli stack of pluriharmonic bundles on a compact Kaehler manifold X, and show how this is equipped with Hodge and quaternionic structures. This stack maps to the de Rham moduli stack, giving rise…

We study pairs of structures, such as the Poisson-Nijenhuis structures, on the tangent bundle of a manifold or, more generally, on a Lie algebroid or a Courant algebroid. These composite structures are defined by two of the following, a…

We characterize the geometric moduli of non-Kaehler manifolds with torsion. Heterotic supersymmetric flux compactifications require that the six-dimensional internal manifold be balanced, the gauge bundle be hermitian Yang-Mills, and also…

The standard formulation of Jacobi manifolds in terms of differential operators on line bundles, although effective at capturing most of the relevant geometric features, lacks a clear algebraic interpretation similar to how Poisson algebras…

A trisymplectic structure on a complex 2n-manifold is a triple of holomorphic symplectic forms such that any linear combination of these forms has constant rank 2n, n or 0, and degenerate forms in $\Omega$ belong to a non-degenerate quadric…

We give a characterisation of Atiyah's and Hitchin's transverse Hilbert schemes of points on a symplectic surface in terms of bi-Poisson structures. Furthermore, we describe the geometry of hyperk\"ahler manifolds arising from the…

An associated Nijenhuis tensor of endomorphisms in the tangent bundle is introduced. Special attention is paid to such tensors for an almost hypercomplex structure and the metric of Hermitian-Norden type. There are studied relations between…

The relation between nilmanifolds with left-invariant complex structure and iterated principal holomorphic torus bundles is clarified and we give criteria under which deformations in the large are again of such type. As an application we…

We answer a question posed independently by Fels-Huckleberry-Wolf and Looijenga concerning the geometric meaning of small deformations of twistor cycles in the K3 period domain. These are shown to induce complex-hyperk\"ahler metrics on…

This paper concerns the cohomological aspects of Donaldson-Thomas theory for Jacobi algebras and the associated cohomological Hall algebra, introduced by Kontsevich and Soibelman. We prove the Hodge-theoretic categorification of the…

We introduce the notion of infinitesimal variations of mixed Hodge structures and invariants associated to them. We describe these invariants in the case of a pair $(X,Y)$ with $X$ a Fano 3-fold and $Y$ a smooth anticanonical K3 surface and…

We study a number of local and global classification problems in generalized complex geometry. In the first topic, we characterize the local structure of generalized complex manifolds by proving that a generalized complex structure near a…

We propose a generalization of the Hodge $dd_c$-lemma to the case of hyperk\"ahler manifolds. As an application of this result we derive the global construction of the fourth order transgression of the Chern character forms of…

A system of functional equations relating the Euler characteristics of moduli spaces of stable representations of quivers and the Euler characteristics of (Hilbert scheme-type) framed versions of quiver moduli is derived. This is applied to…

In this paper we will introduce a new notion of geometric structures defined by systems of closed differential forms in term of the Clifford algebra of the direct sum of the tangent bundle and the cotangent bundle on a manifold. We develop…

We study HKT structures on nilpotent Lie groups and on associated nilmanifolds. We exhibit three weak HKT structures on $\R^8$ which are homogeneous with respect to extensions of Heisenberg type Lie groups. The corresponding hypercomplex…

We investigate the conditions under which astheno-K\"ahler structures can be identified on the product of two compact trans-Sasakian manifolds of dimensions greater than 2.

We give an introduction to Joyce's construction of the motivic Hall algebra of coherent sheaves on a variety M. When M is a Calabi-Yau threefold we define a semi-classical integration map from a Poisson subalgebra of this Hall algebra to…

We study almost complex structures with lower bounds on the rank of the Nijenhuis tensor. Namely, we show that they satisfy an $h$-principle. As a consequence, all parallelizable manifolds and all manifolds of dimension $2n\geq 10$…

In this expository paper we review some twistor techniques and recall the problem of finding compact differentiable manifolds that can carry both K\"ahler and non-K\"ahler complex structures. Such examples were constructed independently by…