arXiv research

Study complex solvmanifolds with trivial canonical bundle and hypercomplex geometry.

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

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.

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

Decomposes complex manifolds with trivial canonical bundle into homogeneous structures.

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.…

The study classifies holomorphic projective connections on complex threefolds.

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…

Let (M,I,J,K) be a compact hypercomplex manifold admitting an HKT-metric. Assume that the canonical bundle of (M,I) is trivial as a holomorphic line bundle. We show that the holonomy of Obata connection on M is contained in SL(n,H). In Appendix we apply these arguments to compact nilmanifolds equipped with abelian hype…

Classifies 6D homogeneous spaces with holomorphically trivial canonical bundle.

Smooth complex surfaces with triple intersections using differential geometry.

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

Holomorphic vector fields and anti-canonical divisors on complex manifolds are studied.

We use hyperbolic geometry to construct simply-connected symplectic or complex manifolds with trivial canonical bundle and with no compatible Kahler structure. We start with the desingularisations of the quadric cone in C^4: the smoothing is a natural S^3-bundle over H^3, its holomorphic geometry is determined by the h…

The classical Beauville-Bogomolov Decomposition Theorem asserts that any compact Kähler manifold with numerically trivial canonical bundle admits an étale cover that decomposes into a product of a torus, and irreducible, simply-connected Calabi-Yau-- and holomorphic-symplectic manifolds. The decomposition of the simply…

We use Chern-Weil theory for Hermitian holomorphic vector bundles with canonical connections for explicit computation of the Chern forms of trivial bundles with special non-diagonal Hermitian metrics. We prove that every del-dellbar exact real form of the type (k,k) on an n-dimensional complex manifold X arises as a di…

Study numerically flat foliations on Kähler manifolds, proving new splitting theorems.

New complex non-Kähler manifolds with specific properties are constructed.

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

The article describes canonical metrics on holomorphic fibre bundles.

For each holomorphic vector bundle we construct a holomorphic bundle 2-gerbe that geometrically represents its second Beilinson-Chern class. Applied to the cotangent bundle, this may be regarded as a higher analogue of the canonical line bundle in complex geometry. Moreover, we exhibit the precise relationship between …

Defines complex structure for families of Hilbert spaces with reasonable curvature.

We show that the deformation space of complex parallelisable nilmanifolds can be described by polynomial equations but is almost never smooth. This is remarkable since these manifolds have trivial canonical bundle and are holomorphic symplectic in even dimension. We describe the Kuranishi space in detail in several exa…

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 …

In this note we show that if a projective manifold admits a Kähler metric with negative holomorphic sectional curvature then the canonical bundle of the manifold is ample. This confirms a conjecture of the second author.

We show that if a compact complex manifold admits a Kähler metric whose holomorphic sectional curvature is everywhere non positive and strictly negative in at least one point, then its canonical bundle is positive.

We prove that a smooth complex projective threefold with a Kähler metric of negative holomorphic sectional curvature has ample canonical line bundle. In dimensions greater than three, we prove that, under equal assumptions, the nef dimension of the canonical line bundle is maximal. With certain additional assumptions, …

We describe a mathematically rigorous differential model for B-type open-closed topological Landau-Ginzburg theories defined by a pair $(X,W)$, where $X$ is a non-compact Kählerian manifold with holomorphically trivial canonical line bundle and $W$ is a complex-valued holomorphic function defined on $X$ and whose criti…

On a compact oriented four-manifold with an orientation preserving involution c, we count solutions of Seiberg-Witten equations, which are moreover symmetrical in relation to c, to construct "real" Seiberg-Witten invariants. Using Taubes' results, we prove that on a symplectic almost complex manifold with an antisymple…

Logarithmic connections on complex manifolds with trivial tangent bundle.

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

The paper constructs a canonical connection on bundles over Riemann surfaces and relates it to the theta divisor.

Constructs irreducible flat connections on a Riemann surface.

The paper extends a theorem about Kähler manifolds with quasi-negative curvature to almost quasi-negative curvature.

We show that on an HKT manifold the holonomy of the Obata connection is contained in SL(n,H) if and only if the Lee form is an exact one form. As an application, we show compact HKT manifolds with holomorphically trivial canonical bundle which are not balanced. A simple criterion for non-existence of HKT metric on hype…

We study a problem of the geometric quantization for the quaternion projective space. First we explain a Kaehler structure on the punctured cotangent bundle of the quaternion projective space, whose Kaehler form coincides with the natural symplectic form on the cotangent bundle and show that the canonical line bundle o…

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…

Study on conical Laplacian operators on Riemann surfaces, focusing on determinants and moduli spaces.

We show that a compact Kahler manifold with nonpositive holomorphic sectional curvature has nef canonical bundle. If the holomorphic sectional curvature is negative then it follows that the canonical bundle is ample, confirming a conjecture of Yau. The key ingredient is the recent solution of this conjecture in the pro…

The $\bar{\partial}_{_{J}}$ operator over an almost complex manifold induces canonical connections of type $(0,1)$ over the bundles of $(p,0)$-forms. If the almost complex structure is integrable then the previous connections induce the canonical holomorphic structures of the bundles of $(p,0)$-forms. For $p=1$ we can …

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…

For smooth families of projective algebraic curves, we extend the notion of intersection pairing of metrized line bundles to a pairing on line bundles with flat relative connections. In this setting, we prove the existence of a canonical and functorial "intersection" connection on the Deligne pairing. A relationship is…

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…

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 …

New dHYM connections found on complex vector bundles.

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

Study of Hermitian structures on toric suspensions of balanced manifolds.

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