Related papers: Geometry of compact complex homogeneous spaces wit…

We discuss aspects of the algebraic geometry of compact non-commutative Calabi-Yau manifolds. In this setting, it is appropriate to consider local holomorphic algebras which can be glued together into a compact Calabi-Yau algebra. We…

The study of the geometry of Calabi-Yau fourfolds is relevant for compactifications of string theory, M-theory, and F-theory to various dimensions. In the first part of this thesis, we study the action of mirror symmetry on two-dimensional…

It is argued that black hole condensation can occur at conifold singularities in the moduli space of type II Calabi--Yau string vacua. The condensate signals a smooth transition to a new Calabi--Yau space with different Euler characteristic…

We prove that a Stein manifold admits a closed holomorphic 1-form without zeros in every class of the first cohomology group. We also prove an approximation result for closed holomorphic 1-forms without zeros defined in a neighborhood of a…

We prove that any region $\Gamma$ in a homogeneous $n$-dimensional and locally compact separable metric space $X$, where $n\geq 2$, cannot be irreducibly separated by a closed $(n-1)$-dimensional subset $C$ with the following property: $C$…

We prove that compact Calabi--Yau varieties with certain isolated singularities are projective. In dimension 3 we do this by analysis, supposing given conifold metrics. In higher dimensions it follows more readily from Ohsawa's degenerate…

Kobayashi [Duke Math. J. (1992)] gave a necessary condition for the existence of compact Clifford-Klein forms in terms of Cartan projections and non-compact dimensions of reductive subgroups. We extend his method to non-reductive subgroups,…

Dimer models have been used in string theory to construct path algebras with relations that are 3-dimensional Calabi-Yau Algebras. These constructions result in algebras that share some specific properties: they are finitely generated…

The moduli space $\mathcal M(r,n)$ of framed torsion free sheaves on the projective plane with rank $r$ and second Chern class equal to $n$ has the natural action of the $(r+2)$-dimensional torus. In this paper, we look at the fixed point…

n this paper we define an invariant of a pair of 6 dimensional symplectic %optional manifold with vanishing 1st Chern class and its Lagrangian submanifold with vanishing Maslov index. This invariant is a function on the set of the path…

As was recently observed by M. Xu and J. Wolf, there is a gap in Berard Bergery's classification of odd dimensional positively curved homogeneous spaces. Since this classification has been used in other papers as well, we give a modern,…

Earlier we introduced and studied the concept of holomorphic {\it branched Cartan geometry}. We define here a foliated version of this notion; this is done in terms of Atiyah bundle. We show that any complex compact manifold of algebraic…

We study low-degree curves on one-parameter Calabi-Yau hypersurfaces, and their contribution to the space-time superpotential in a superstring compactification with D-branes. We identify all lines that are invariant under at least one…

An orbifold version of Bogomolov decomposition theorem is established for compact K\"ahler spaces with quotient singularities and first Chern class zero.The proof is a direct adaptation of the classical smooth case, using Ricci-flat…

All spaces are assumed to be separable and metrizable. We show that, assuming the Axiom of Determinacy, every zero-dimensional homogeneous space is strongly homogeneous (that is, all its non-empty clopen subspaces are homeomorphic), with…

We study rank 2 torus-equivariant torsion-free sheaves on the complex projective space. For reflexive sheaves we derive a simple formula for the Chern polynomial, and in the general torsion-free case we introduce an iterative construction…

We study new compactifications of the SO(32) heterotic string theory on compact complex non-Kahler manifolds. These manifolds have many interesting features like fewer moduli, torsional constraints, vanishing Euler character and vanishing…

Below, by space we mean a separable metrizable zero-dimensional space. It is studied when the space can be embedded in a Cantor set while maintaining the algebraic structure. Main results of the work: every space is an open retract of a…

We discuss the use of gauge fields to stabilize complex structure moduli in Calabi-Yau three-fold compactifications of heterotic string and M-theory. The requirement that the gauge fields in such models preserve supersymmetry leads to a…

We prove an analog of the Tian-Todorov theorem for twisted generalized Calabi-Yau manifolds; namely, we show that the moduli space of generalized complex structures on a compact twisted generalized Calabi-Yau manifold is unobstructed and…