arXiv research

We classify complex compact parallelizable manifolds which admit flat torsion free holomorphic affine connections. We exhibit complex compact manifolds admitting holomorphic affine connections, but no flat torsion free holomorphic affine connections.

Study of affine and projective structures on foliated complex manifolds.

The paper constructs submanifolds with corners in Delzant polytopes from affine subspaces.

We prove that any compact complex manifold with finite fundamental group and algebraic dimension zero admits no holomorphic affine connection.

Flat affine subvarieties found in OT-manifolds.

New complex manifolds found with flat structure.

Study shows complex affine transformations index is at most 2 for Kähler manifolds.

The paper classifies compact affine quaternionic curves and surfaces.

The study explores extensions of Kähler manifolds and their properties.

Affine connections linked to Riccati distributions on compact surfaces.

We develop a theory of stable bundles and affine Hermitian-Einstein metrics for flat vector bundles over a special affine manifold (a manifold admitting an atlas whose gluing maps are all locally constant volume-preserving affine maps). Our paper presents a parallel to Donaldson-Uhlenbeck-Yau's proof of the existence o…

We obtain a characterization of the real Lie algebras admitting abelian complex structures in terms of certain affine Lie algebras $\frak a \frak f \frak f (A)$, where $A$ is a commutative algebra. These affine Lie algebras are natural generalizations of $\frak a \frak f \frak f (\Bbb C)$ and the corresponding Lie grou…

Complex foliation cohomology is shown to be simple.

Affine vector fields on pseudo-Kähler manifolds are symplectic.

Proves algebraic cones for LCK manifolds with potential.

Study intrinsic volume forms on complex hypersurfaces.

Study of flows on complex manifolds with holomorphic properties.

The goal of this paper is to introduce the lifting theory that has an important role in geometry. Therefore, using the lifts of differential geometric structures we show that tangent bundle TM of paracomplex manifold M admits para-complex torsion-free affine connection.

Our aim here is to investigate the holomorphic geometric structures on compact complex manifolds which may not be Kähler. We prove that holomorphic geometric structures of affine type on compact Calabi-Yau manifolds with polystable tangent bundle (with respect to some Gauduchon metric on it) are locally homogeneous. In…

Study on holomorphic structures on complex manifolds with specific properties.

This paper is the first arising from our project announced in math.AG/0211094, "Affine manifolds, log structures, and mirror symmetry." We aim to study mirror symmetry by studying the log structures of Illusie-Fontaine and Kato on degenerations of Calabi-Yau manifolds. The basic idea is that one can associate to certai…

The study limits the cohomological dimension of certain affine manifolds with partially hyperbolic holonomy groups.

It is proved that the moduli space of all connected compact orientable embedded minimal affine Lagrangian submanifolds of a complex equiaffine space constitutes an infinite dimensional Frechet manifold (if it is not the empty set). The moduli space of all connected compact orientable metric Lagrangian embedded surfaces…

Study of polygon degeneration to segments in complex space.

We introduce the notion of a special complex manifold: a complex manifold (M,J) with a flat torsionfree connection \nabla such that (\nabla J) is symmetric. A special symplectic manifold is then defined as a special complex manifold together with a \nabla-parallel symplectic form ω. This generalises Freed's definition …

An affine manifold is a manifold with torsion-free flat affine connection. A geometric topologist's definition of an affine manifold is a manifold with an atlas of charts to the affine space with affine transition functions; a radiant affine manifold is an affine manifold with holonomy consisting of affine transformati…

Complex manifolds with specific geometric structures have infinite fundamental groups.

Leibniz cohomology reveals connections on manifolds.

Study on non-Kähler Calabi-Yau manifolds and their geometric structures.

New iteration method for complex and real Monge-Ampere equations converges under certain conditions.

We construct finite volume hyperbolic manifolds with large symmetry groups. The construction makes use of the presentations of finite Coxeter groups provided by Barot and Marsh and involves mutations of quivers and diagrams defined in the theory of cluster algebras. We generalize our construction by assigning to every …

A general theorem on the existence of natural torsion-free affine connections on a complete family of compact complex submanifolds in a complex manifold is proved. Applications to twistor theory are discussed.

A (flat) affine $3$-manifold is a $3$-manifold with an atlas of charts to an affine space $\mathbb{R}^3$ with transition maps in the affine transformation group $\mathrm{Aff}(\mathbb{R}^3)$. We will show that a connected closed affine $3$-manifold is either an affine Hopf $3$-manifold or decomposes canonically to conca…

The paper studies Anosov holonomy groups in complete affine manifolds.

To every real analytic Riemannian manifold M there is associated a complex structure on a neighborhood of the zero section in the real tangent bundle of M. This structure can be uniquely specified in several ways, and is referred to as a Grauert tube. We say that a Grauert tube is entire if the complex structure can be…

An (flat) affine $3$-manifold is a $3$-manifold with an atlas of charts to an affine space ${\mathbf R}^3$ with transition maps in the affine transformation group $Aff({\mathbf R}^3)$. Equivalently an affine $3$-manifold is a $3$-manifold with a flat torsion-free affine connection. We show that a closed affine $3$-mani…

We study affine maps between affine manifolds. Even when the fibers are compact and diffeomorphic, two of them can inherit different affine structures from the source space. This leads to a fixed linear holonomy deformation theory of the affine structure of an affine manifold. We found various conditions which make the…

This paper describes integral affine structures on compact 3-manifolds.

Characterizes affine vector fields on Finsler manifolds with rigidity results.

Characterizes flat affine connections on manifolds.

Study of symmetries in deformed q-map spaces reveals a complex group structure.

Let $M$ be a compact connected special flat affine manifold without boundary equipped with a Gauduchon metric $g$ and a covariant constant volume form. Let $G$ be either a connected reductive complex linear algebraic group or the real locus of a split real form of a complex reductive group. We prove that a flat princip…

In this paper, we show that a compact affine manifold endowed with an Affine Anosov transformation is finitely covered by a complete affine nilmanifold.

The study compares Kähler and Riemannian normal coordinates on manifolds.

Paper proves certain closed affine manifolds without invariant lines don't exist.

Completed classification of rank 2 affine manifolds in genus 3.

This is an extended example of the study of mirror symmetry via log schemes and the discrete Legendre transform on affine manifolds, introduced by myself and Bernd Siebert in "Mirror Symmetry via Logarithmic Degeneration Data I" (math.AG/0309070). In this paper, I consider the construction as it applies to the Batyrev-…

The paper characterizes flat affine connections on manifolds and Lie groups.