arXiv research

Let X be a compact connected Kaehler manifold such that the holomorphic tangent bundle TX is numerically effective. A theorem of Demailly, Peternell and Schenider says that there is a finite unramified Galois covering M --> X, a complex torus T, and a holomorphic surjective submersion f: M --> T, such that the fibers o…

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.

Given a Kaehlerian holomorphic fiber bundle whose fiber is a compact homogeneous Kaehler manifold, we describe the perturbed Hermitian-Einstein equations relative to certain holomorphic vector bundles. With respect to special metrics on the holomorphic bundles, there is a dimensional reduction procedure which reduces t…

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

Extends Higgs fields theory to complex fiber bundles.

Study Kähler-Ricci flow on manifolds with singularities.

Develops SGH bundles and theories for GC manifolds.

The moduli space of holomorphic fiber bundles ${\cal M}_n(\Si)$ over a compact Riemann surface $\Si$ is considered. A formula for the regularised determinant and an other for the symplectic form at trivial bundle are proposed.

We prove an analogue of the Kobayashi-Hitchin correspondence oncompact connected 3-folds that is fibered on orbifold Riemann surfaces and satisfy an integrability condition, which contains compact connected Sasakian 3-folds. We define mini-holomorphic bundles on such 3-folds and the algebraic Dirac-type singularities o…

The study classifies Kähler threefolds with special fiber bundles.

We study holomorphic extensions of Matsuki orbits in complex Grassmannians.

Study of flows on 7D manifolds with holomorphic properties.

Paper develops a unified framework for Lie algebroid connections on various bundles.

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

We consider a proper flat fibration with real base and complex fibers. First we construct odd characteristic classes for such fibrations by a method that generalizes constructions of Bismut-Lott. Then we consider the direct image of a fiberwise holomorphic vector bundle, which is a flat vector bundle on the base. We gi…

A principal toric bundle $M$ is a complex manifold equipped with a free holomorphic action of a compact complex torus $T$. Such a manifold is fibered over $M/T$, with fiber $T$. We discuss the notion of positivity in fiber bundles and define positive toric bundles. Given an irreducible complex subvariety $X\subset M$ o…

We give a complete proof of a propagation theorem of multiplicity-free property from fibers to spaces of global sections for holomorphic vector bundles. The propagation theorem is formalised in three ways, aiming for producing various multiplicity-free theorems in representation theory for both finite and infinite dime…

The paper explores linear generalised complex structures over vector bundles.

In this paper we show as main results two structure theorems of a compact homogeneous locally conformally Kaehler (or shortly l.c.K.) manifold, a holomorphic structure theorem asserting that it has a structure of holomorphic principal fiber bundle over a flag manifold with fiber a 1-dimensional complex torus, and a met…

Let $\Y$ be a smooth connected manifold, $Σ\subset\C$ an open set and $(σ,y)\to\scrP_y(σ)$ a family of unbounded Fredholm operators $D\subset H_1\to H_2$ of index 0 depending smoothly on $(y,σ)\in \Y\times Σ$ and holomorphically on $σ$. We show how to associate to $\scrP$, under mild hypotheses, a smooth vector bundle …

For each g > 2 and h > 1, we explicitly construct (1) fiber sum indecomposable relatively minimal genus g Lefschetz fibrations over genus h surfaces whose monodromies lie in the Torelli group, (2) fiber sum indecomposable genus g surface bundles over genus h surfaces whose monodromies are in the Torelli group (provided…

Study curvature of direct image bundles in deformations of maps.

Let $π:\mathcal{X}\to M$ be a holomorphic fibration with compact fibers and $L$ a relatively ample line bundle over $\mathcal{X}$. We obtain the asymptotic of the curvature of $L^2$-metric and Qullien metric on the direct image bundle $π_*(L^k\otimes K_{\mathcal{X}/M})$ up to the lower order terms than $k^{n-1}$ for la…

Inspired by a string duality, we construct a deformation family for $G_2$-orbifolds given as total spaces of coassociative fibrations by ADE singularities over a closed and oriented smooth three-manifold $Q$. The deformations are parametrized by sections of a fiber bundle on $Q$ that can be interpreted as spectral/came…

This research constructs hypercomplex structures from twistor spaces.

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

We prove that the twisted Kahler-Einstein metrics that arise on the base of certain holomorphic fiber space with Calabi-Yau fibers have conical-type singularities along the discriminant locus. These fiber spaces arise naturally when studying the collapsing of Ricci-flat Kahler metrics on Calabi-Yau manifolds, and of th…

We prove that a compact smooth 4-manifold admits generalized complex structures of odd type if and only if it has a transversely holomorphic 2-foliation. Consequently, there exist generalized complex structures of odd type on a circle bundle over a closed Seifert fibered 3-manifold.

New curvature assumptions prove Nakano positivity for complex vector bundles.

We extend the equivariant holomorphic Morse inequalities of circle actions to cases with torus and non-Abelian group actions on holomorphic vector bundles over Kahler manifolds and show the necessity of the Kahler condition. For torus actions, there is a set of inequalities for each choice of action chambers specifying…

Let $X\rightarrow {\mathbb P}^1$ be an elliptically fibered $K3$ surface, admitting a sequence $ω_{i}$ of Ricci-flat metrics collapsing the fibers. Let $V$ be a holomorphic $SU(n)$ bundle over $X$, stable with respect to $ω_i$. Given the corresponding sequence $Ξ_i$ of Hermitian-Yang-Mills connections on $V$, we prove …

The Bialynicki-Birula decomposition of the space of lambda-connections restricts to the Morse stratification on the moduli space of Higgs bundles and to the partial oper stratification on the de Rham moduli space of holomorphic connections. For both the Morse and partial oper stratifications, every stratum is a holomor…

Using the recent results of Siebert and Tian about the holomorphicity of genus 2 Lefschetz fibrations with irreducible singular fibers, we show that any genus 2 Lefschetz fibration becomes holomorphic after fiber sum with a holomorphic fibration.

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

In this paper, we construct the moduli space of marked oper structures on a closed, oriented smooth surface of negative Euler characteristic as a holomorphic fiber bundle over Teichmüller space. We prove that the holonomy map from the space of marked oper structures to the moduli space of reductive flat bundles is a ho…

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…

On a compact symplectic manifold $(X,ω)$ with a prequantum line bundle $(L,\nabla,h)$, we consider the one-parameter family of $ω$-compatible complex structures which converges to the real polarization coming from the Lagrangian torus fibration. There are several researches which show that the holomorphic sections of t…

New complex structure on hyperbolic disc within hyperkaehler space.

Let π: V \rightarrow M be a (real or holomorphic) vector bundle whose base has an almost Frobenius structure (\circ_{M},e_{M}, g_{M}) and typical fiber has the structure of a Frobenius algebra (\circ_{V},e_{V},g_{V}). Using a connection D on the bundle V and a morphism α: V \rightarrow TM, we construct an almost Froben…

Introduces Darboux-Lie derivative for fiber bundles.

Investigates conditions for GKM fiber bundles and realizability of fiber bundles of GKM graphs.

Each Morita--Mumford--Miller (MMM) class e_n assigns to each genus g >= 2 surface bundle S_g -> E^{2n+2} -> M^{2n} an integer e_n^#(E -> M) := <e_n,[M]> in Z. We prove that when n is odd the number e_n^#(E -> M) depends only on the diffeomorphism type of E, not on g, M, or the map E -> M. More generally, we prove that …

New examples show not all homology fiber bundles are topological.

The paper studies deformations of Lagrangian fibrations on symplectic manifolds.

Let $π: \mathcal{X}\rightarrow Δ$ be a holomorphic family of compact complex manifolds over an open disk in $\mathbb{C}$. If the fiber $π^{-1}(t)$ for each nonzero $t$ in an uncountable subset $B$ of $Δ$ is Moishezon and the reference fiber $X_0$ satisfies the local deformation invariance for Hodge number of type $(0,1…

The first aim of this paper is to give four types of examples of surface bundles over surfaces with non-zero signature. The first example is with base genus 2, a prescribed signature, a 0-section and the fiber genus greater than a certain number which depends on the signature. This provides a new upper bound on the min…

New exotic 4-manifolds found with fiber bundles.

The paper proves non-existence of positive scalar curvature on certain fiber bundles.