arXiv research

In this note we show that if a compact Kahler manifold with trivial canonical bundle is the total space of a holomorphic fibration without singular fibers, then the fibration is a holomorphic fiber bundle. In the algebraic case, the fibration becomes trivial after a finite base change.

Constructs irreducible flat connections on a Riemann surface.

It is proved that on nilmanifolds with abelian complex structure, there exists a canonically constructed non-trivial holomorphic Poisson structure. We identify the necessary and sufficient condition for its associated cohomology to be isomorphic to the cohomology associated to trivial (zero) holomorphic Poisson structu…

Study complex solvmanifolds with trivial canonical bundle and hypercomplex geometry.

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

Logarithmic connections on complex manifolds with trivial tangent bundle.

Holomorphic connections found on Riemann surfaces with Fuchsian monodromy.

Let X be a compact Kahler manifold with a non-trivial holomorphic Poisson structure. Then there exist deformations of non-trivial generalized Kahler structures with one pure spinor on X. We prove that every Poisson submanifold of X is a generalized Kahler submanifold with respect to the deformed generalized Kahler stru…

We introduce the concept of a branched holomorphic Cartan geometry. It generalizes to higher dimension the definition of branched (flat) complex projective structure on a Riemann surface introduced by Mandelbaum. This new framework is much more flexible than that of the usual holomorphic Cartan geometries. We show that…

Vanishing theorems show holomorphic tensor fields on certain Kähler manifolds are trivial.

We determine the 6-dimensional solvmanifolds admitting an invariant complex structure with holomorphically trivial canonical bundle. Such complex structures are classified up to isomorphism, and the existence of strong Kähler with torsion (SKT), generalized Gauduchon, balanced and strongly Gauduchon metrics is studied.…

Investigate pseudoconvexity of locally trivial holomorphic ball bundles over compact Riemann surfaces.

We study the holomorphic vector bundles E over the twistor space Tw(M) of a compact simply connected hyperkähler manifold $M$. We give a characterization of the semistability condition for E in terms of its restrictions to the holomorphic sections of the holomorphic twistor projection π:Tw(M)\rightarrow CP^1. It is sho…

The study classifies holomorphic projective connections on complex threefolds.

We propose a new approach to the value distribution theory of entire holomorphic curves. We define a ``packing density'' of an entire holomorphic curve, and show that it has various non-trivial properties. We prove a ``gap theorem'' for holomorphic maps from elliptic curves to the complex projective space, and study th…

The paper proves a structure theorem for compact Kähler manifolds with semi-positive holomorphic sectional curvature.

The study constructs differential systems on Riemann surfaces and explores their monodromy properties.

This article is based on the methods developed in [AGG]. We construct a complex hyperbolic structure on a trivial disc bundle over a closed orientable surface $Σ$ (of genus 2) thus solving a long standing problem in complex hyperbolic geometry (see [Gol1, p. 583] and [Sch, p. 14]). This example answers also [Eli, Open …

Holomorphic bundles on complex manifolds with boundary are studied, extending results from Donaldson's work.

We investigate the holonomy group of singular Kähler-Einstein metrics on klt varieties with numerically trivial canonical divisor. Finiteness of the number of connected components, a Bochner principle for holomorphic tensors, and a connection between irreducibility of holonomy representations and stability of the tange…

We review some constructions and properties of complex manifolds admitting pluriclosed and balanced metrics. We prove that for a 6-dimensional solvmanifold endowed with an invariant complex structure J having holomorphically trivial canonical bundle the pluriclosed flow has a long time solution for every invariant init…

Decomposes complex manifolds with trivial canonical bundle into homogeneous structures.

Extends noncommutative deformations of holomorphic line bundles on complex tori and their mirror partners.

The Fubini-Study metric minimizes a volume-normalized holomorphic systole in $\mathbb{C}P^n$.

The paper solves Riemann-Hilbert problems using framed holomorphic bundles.

Let $G$ be a connected complex Lie group and $Γ\subset G$ a cocompact lattice. Let $H$ be a complex Lie group. We prove that a holomorphic principal $H$-bundle $E_H$ over $G/Γ$ admits a holomorphic connection if and only if $E_H$ is invariant. If $G$ is simply connected, we show that a holomorphic principal $H$-bundle …

On compact Kähler manifolds, we classify regular holomorphic foliations of codimension 1 whose canonical bundle is numerically trivial.

Classifies 6D homogeneous spaces with holomorphically trivial canonical bundle.

New linking numbers link complex cycles to Calabi-Yau 3-folds.

New dHYM connections found on complex vector bundles.

Variant of Seiberg-Witten equations for multiple-spinors connects to stability of holomorphic bundles.

Under certain integrability and geometric conditions, we prove division theorems for the exact sequences of holomorphic vector bundles and improve the results in the case of Koszul complex. By introducing a singular Hermitian structure on the trivial bundle, our results recover Skoda's division theorem for holomorphic …

There are solved standard problems related to Formal (Holomorphic) Segre preserving Mappings of non-trivial Real-Formal Hypersurfaces in $\mathbb{C}^{2}$.

We define a subset of an almost complex manifold (M,J) to be a holomorphic shadow if it is the image of a J-holomorphic map from a compact complex manifold. Notice that a J-holomorphic curve is a holomorphic shadow, and so is a complex subvariety of a compact complex manifold. We show that under some conditions on an a…

A hypercomplex manifold is a manifold equipped with a triple of complex structures satisfying the quaternionic relations. A holomorphic Lagrangian variety on a hypercomplex manifold with trivial canonical bundle is a holomorphic subvariety which is calibrated by a form associated with the holomorphic volume form; this …

Paper studies Lagrangian submanifolds and their homological monodromy.

The article classifies six-dimensional solvmanifolds with non-invariant trivializing sections of their canonical bundle.

Vanishing theorem for certain tensor fields on compact Hermitian manifolds.

We give examples of symplectic actions of a cyclic group, inducing a trivial action on homology, on four-manifolds that admit Hamiltonian circle actions, and show that they do not extend to Hamiltonian circle actions. Our work applies holomorphic methods to extend combinatorial tools developed for circle actions to stu…

Holomorphic curves found in compact quotients of SL(2,C).

Let $M$ be an irreducible smooth complex projective variety equipped with an action of a compact Lie group $G$, and let $({\mathcal L},h)$ be a $G$-equivariant holomorphic Hermitian line bundle on $M$. Given a compact connected Riemann surface $X$, we construct a $G$-equivariant holomorphic Hermitian line bundle $(L\,,…

We study a smooth analogue of jumping curves of a holomorphic vector bundle, and use Yang-Mills theory over $S ^{2}$ to show that any non-trivial, smooth Hermitian vector bundle $E$ over a smooth simply connected manifold, must have such curves. This is used to give new examples complex manifolds for which a non-tri…

Let $E$ be a closed set in the Riemann sphere $\widehat{\mathbb{C}}$. We consider a holomorphic motion $φ$ of $E$ over a complex manifold $M$, that is, a holomorphic family of injections on $E$ parametrized by $M$. It is known that if $M$ is the unit disk $Δ$ in the complex plane, then any holomorphic motion of $E$ ove…

We provide non trivial examples of solutions to the system of coupled equations introduced by M. García-Fernández for the uniformization problem of a triple $(M,L,E)$ where $E$ is a holomorphic vector bundle over a polarized complex manifold $(M,L)$, generalizing the notions of both constant scalar curvature Kähler met…

Construct Kähler metrics with negative holomorphic bisectional curvature on compact relative Kähler fibrations.

Let $S$ be a compact Sasakian manifold which does not admit non-trivial Hamiltonian holomorphic vector fields. If there exists an Einstein-Sasakian metric on $S$, then it is unique.

Study of flows on 7D manifolds with holomorphic properties.

We develop a non-relativistic twistor theory, in which Newton--Cartan structures of Newtonian gravity correspond to complex three-manifolds with a four-parameter family of rational curves with normal bundle ${\mathcal O}\oplus{\mathcal O}(2)$. We show that the Newton--Cartan space-times are unstable under the general K…