Related papers: The complex geometry of a hypothetical complex str…
We study quaternionic Bott-Chern cohomology on compact hypercomplex manifolds and adapt some results from complex geometry to the quaternionic setting. For instance, we prove a criterion for the existence of HKT metrics on compact…
We consider the complexification of the Henon family of quadratic diffeomorphisms of the plane, and the region of parameter space corresponding to complex horseshoes. We discuss some conjectures about the global topology of the complex…
We discuss the deformed sigma-model that arises when considering four-dimensional N=2 abelian vector multiplets in the presence of an arbitrary chiral background field. In addition, we allow for a class of deformations of special geometry…
We study moduli spaces of certain sextic curves with a singularity of multiplicity 3 from both perspectives of Deligne-Mostow theory and periods of K3 surfaces. In both ways we can describe the moduli spaces via arithmetic quotients of…
In this paper, we study a deformation theory of rigid analytic spaces. We develop a theory of cotangent complexes for rigid geometry which fits in with our deformations. We then use the complexes to give a cohomological description of…
The notions of holomorphic symplectic structures and hypercomplex structures on Courant algebroids are introduced and then proved to be equivalent. These generalize hypercomplex triples and holomorphic symplectic 2-forms on manifolds…
Complex and Hermitian structures on hom-Lie algebras are introduced and some examples of these structures are presented. Also, it is shown that there not exists a proper complex (Hermitian) home-Lie algebra of dimension two. Then using a…
The existence of some complex geometrical structures on a compact manifold such as complex structures, Kaehler (pseudo-Kaehler) structures often impose certain restrictions on its underling topological or differentiable manifold. In this…
We discuss the integrability of orthogonal almost complex structures on Riemannian products of even-dimensional round spheres and give a partial answer to the question raised by E. Calabi concerning the existence of complex structures on a…
We complete the classification of six-dimensional strongly unimodular almost nilpotent Lie algebras admitting complex structures. For several cases we describe the space of complex structures up to isomorphism. As a consequence we determine…
We study the envelopes of meromorphy of neighborhoods of symplectically immersed two-spheres in complex K\"ahler surfaces using the Gromov's theory of pseudoholomorphic curves. The construction of a complete family of holomorphic…
An algebraic deformation theory of module-algebras over a bialgebra is constructed. The cases of module-coalgebras, comodule-algebras, and comodule-coalgebras are also considered.
In this paper we introduce a class of polygonal complexes for which we can define a notion of sectional combinatorial curvature. These complexes can be viewed as generalizations of 2-dimensional Euclidean and hyperbolic buildings. We focus…
The general theory of parabolic geometries is applied to the study of the normal Cartan connections for all hyperbolic and elliptic 6-dimensional CR-manifolds of codimension two. The geometric meaning of the individual components of the…
We discuss our recent results on the existence and classification problem of complex and Kaehler structures on compact solvmanifolds. In particular, we determine in this paper all the complex surfaces which are diffeomorphic to compact…
In this short note, we review the well-known result that there is no orthogonal complex structure on the 6-sphere with respect to the round metric.
Almost paracontact almost paracomplex Riemannian manifolds of the lowest dimension are studied. Such structures are constructed on hyperspheres in 4-dimensional spaces, Euclidean and pseudo-Euclidean, respectively. The obtained manifolds…
There are three complete plane geometries of constant curvature: spherical, Euclidean and hyperbolic geometry. We explain how a closed oriented surface can carry a geometry which locally looks like one of these. Focussing on the hyperbolic…
Let (M,I) be a compact Kaehler manifold admitting a hypercomplex structure. We show that (M, I) admits a natural HKT-metric. This is used to construct a holomorphic symplectic form on (M,I).
We introduce the notion of skew-holomorphic Lie algebroid on a complex manifold, and explore some cohomologies theories that one can associate to it. Examples are given in terms of holomorphic Poisson structures of various sorts.