arXiv research

The paper examines the limit of harmonic flow on flat vector bundles.

Proves a theorem for complex flat vector bundles using differential forms.

New equivalence found for flat vector bundles without extra conditions.

Paper constructs $L^2$ estimates for flat vector bundles and generalizes Prékopa's theorem.

Study projective flat vector bundles over Riemann surfaces using Wronskian line bundles.

This article is devoted to a study of flat orbifold vector bundles. We construct a bijection between the isomorphic classes of proper flat orbifold vector bundles and the equivalence classes of representations of the orbifold fundamental groups of base orbifolds. We establish a Bismut-Zhang like anomaly formula for the…

Characterizes curvature positivity for Riemannian metrics on flat vector bundles.

Equivalence proven between two torsion invariants for flat vector bundles.

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…

We construct a Fourier--Mukai transform for smooth complex vector bundles $E$ over a torus bundle $π:M \to B,$ the vector bundles being endowed with various structures of increasing complexity. At a minimum, we consider vector bundles $E$ with a flat partial unitary connection, that is families or deformations of flat …

We investigate the flat holomorphic vector bundles over compact complex parallelizable manifolds $G / Γ$, where $G$ is a complex connected Lie group and $Γ$ is a cocompact lattice in it. The main result proved here is a structure theorem for flat holomorphic vector bundles $E_ρ$ associated to any irreducible representa…

In this paper, we establish an equality between the analytic torsion introduced by Dar\cite{MR876230} and the orbifold analytic torsion defined by Ma \cite{MR2140438} on a compact manifold with isolated conical singularities which in addition has an orbifold structure. We assume the orbifold flat vector bundle is an ho…

The paper proves conditions for Kähler-Einstein metrics on certain bundles.

The purpose of this paper is to give a proof of the real part of the Riemann-Roch-Grothendieck theorem for complex flat vector bundles at the differential form level in the even dimensional fiber case. The proof is, roughly speaking, an application of the local family index theorem for a perturbed twisted spin Dirac op…

We show that a unipotent vector bundle on a non-Kaehler compact complex manifold does not admit a flat holomorphic connection in general. We also construct examples of topologically trivial stable vector bundle on compact Gauduchon manifold that does not admit any unitary flat connection.

Study Higgs bundles and flat connections on quasi-regular Sasakian manifolds.

Analytic torsion equals dynamical zeta function for certain bundles.

Complex valued analytic torsion and dynamical zeta function studied on locally symmetric spaces.

Study on Gauduchon manifolds finds metrics for projectively flat bundles.

We establish a generic counting formula for the Euler number of a flat vector bundle of rank $2n$ over a $2n$ dimensional closed manifold, in terms of vertices of transversal open coverings of the underlying manifold. We use the Mathai-Quillen formalism to prove our result.

Holomorphic vector bundles on Hopf manifolds admit flat connections.

We present an alternate definition of the mod {\bf Z} component of the Atiyah-Patodi-Singer $η$ invariant associated to (not necessary unitary) flat vector bundles, which identifies explicitly its real and imaginary parts. This is done by combining a deformation of flat connections introduced in a previous paper with t…

The paper proves a section for Anosov vector fields on compact manifolds.

We introduce Dolbeault cohomology valued characteristic classes of Higgs bundles over complex manifolds. Flat vector bundles have characteristic classes lying in odd degree de Rham cohomology and a theorem of Reznikov says that these must vanish in degrees three and higher over compact Kähler manifolds. We provide a si…

We construct explicit complete Ricci-flat metrics on the total spaces of certain vector bundles over flag manifolds of the group $SU(n)$, for all Kähler classes. These metrics are natural generalizations of the metrics of Candelas-de la Ossa on the conifold, Pando Zayas-Tseytlin on the canonical bundle over $\mathbb{CP…

This note describes sharp Milnor--Wood inequalities for the Euler number of flat oriented vector bundles over closed Riemannian manifolds locally isometric to products of hyperbolic planes. One consequence is that such manifolds do not admit an affine structure, confirming Chern--Sullivan's conjecture in this case. The…

In this paper we introduce the notion of almost flatness for (stably) relative bundles on a pair of topological spaces and investigate basic properties of it. First, we show that almost flatness of topological and smooth sense are equivalent. This provides a construction of an almost flat stably relative bundle by usin…

Holomorphic connections on Calabi-Yau manifolds are flat.

Let (E,D,P) be a flat vector bundle with a parabolic structure over a punctured Riemann surface, (M,g). We consider a deformation of the harmonic metric equation which we call the Poisson metric equation. This equation arises naturally as the dimension reduction of the Hermitian-Yang-Mills equation for holomorphic vect…

The paper studies volumes of direct images for high tensor powers of ample bundles.

In this paper, we present two kinds of total Chern forms $c(E,G)$ and $\mathcal{C}(E,G)$ as well as a total Segre form $s(E,G)$ of a holomorphic Finsler vector bundle $π:(E,G)\to M$ expressed by the Finsler metric $G$, which answers a question of J. Faran (\cite{Faran}) to some extent. As some applications, we show tha…

Let M be a simply connected Riemannian symmetric space, with at most one flat direction. We show that every Riemannian (or unitary) vector bundle with parallel curvature over M is an associated vector bundle of a canonical principal bundle, with the connection inherited from the principal bundle. The problem of finding…

The paper proves a quadratic formality for Sasakian manifolds' representation varieties.

We provide notions of numerical effectiveness and numerical flatness for Higgs vector bundles on compact Kähler manifolds in terms of fibre metrics. We prove several properties of bundles satisfying such conditions and in particular we show that numerically flat Higgs bundles have vanishing Chern classes, and that they…

The paper studies special surfaces in 4D space forms with specific geometric properties.

Study Higgs sections and flat sections for nonlinear harmonic bundles.

Researchers calculate the Ray-Singer Torsion for $S^{1}$ bundles.

We characterize all LVMB manifolds X such that the holomorphic tangent bundle TX is spanned at the generic point by a family of global holomorphic vector fields, each of them having non-empty zero locus. We deduce that holomorphic connections on semi-stable holomorphic vector bundles over LVMB manifolds with this previ…

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…

Constructs irreducible flat connections on a Riemann surface.

Study partially hyperbolic flows on flat bundles, proving equivalence for complete affine manifolds.

Considering pseudo-Riemannian $g$-natural metrics on tangent bundles, we prove that the condition of being Ricci soliton is hereditary in the sense that a Ricci soliton structure on the tangent bundle gives rise to a Ricci soliton structure on the base manifold. Restricting ourselves to some class of pseudo-Riemannian …

The paper compares two torsion invariants in complex vector bundles.

A flat complex vector bundle (E,D) on a compact Riemannian manifold (X,g) is stable (resp. polystable) in the sense of Corlette [C] if it has no D-invariant subbundle (resp. if it is the D-invariant direct sum of stable subbundles). It has been shown in [C] that the polystability of (E,D) in this sense is equivalent to…

We summarize the global geometric formulation of Einstein-Scalar-Maxwell theories twisted by flat symplectic vector bundle which encodes the duality structure of the theory. We describe the scalar-electromagnetic symmetry group of such models, which consists of flat unbased symplectic automorphisms of the flat symplect…

The Corlette-Donaldson-Hitchin-Simpson's correspondence states that, on a compact Kähler manifold $(X, ω)$, there is a one-to-one correspondence between the moduli space of semisimple flat complex vector bundles and the moduli space of poly-stable Higgs bundles with vanishing Chern numbers. In this paper, we extend thi…

In this paper, we study numerically flat holomorphic vector bundles over a compact non-Kähler manifold $(X, ω)$ with the Hermitian metric $ω$ satisfying the Gauduchon and Astheno-Kähler conditions. We prove that numerically flatness is equivalent to numerically effectiveness with vanishing first Chern number, semistabl…

In this paper we develop a relative version of T-duality in generalized complex geometry which we propose as a manifestation of mirror symmetry. Let M be an n-dimensional smooth real manifold, V a rank n real vector bundle on M, and nabla a flat connection on V. We define the notion of a nabla-semi-flat generalized com…