arXiv research

On a complex manifold, a co-Higgs bundle is a holomorphic vector bundle with an endomorphism twisted by the tangent bundle. The notion of generalized holomorphic bundle in Hitchin's generalized geometry coincides with that of co-Higgs bundle when the generalized complex manifold is ordinary complex. Schwarzenberger's r…

Identifies images of determinant morphism for specific co-Higgs bundles.

Study finds conditions for minimum of Donaldson's functional on Higgs bundles with non-holomorphic Higgs fields.

The paper extends orthogonal decomposition results to hermitian Higgs bundles.

Generalizes Higgs bundles theory using a vector bundle twist.

Study on harmonic metrics for rank 3 Higgs bundles in Hitchin section.

Study uses Vinberg pairs for Higgs bundles, revealing their role.

In this paper, we consider the gradient flow of the Yang-Mills-Higgs functional for Higgs pairs on a Hermitian vector bundle $(E, H_{0})$ over a compact Kähler manifold $(M, ω)$. We study the asymptotic behavior of the Yang-Mills-Higgs flow for Higgs pairs at infinity, and show that the limiting Higgs sheaf is isomorph…

New Poisson structures found on Higgs bundle moduli spaces.

We consider the gradient flow of the Yang-Mills-Higgs functional of twist Higgs pairs on a Hermitian vector bundle $(E,H_0)$ over a Riemann surface $X$. It is already known the gradient flow with initial data $(A_0,φ_0)$ converges to a critical point $(A_\infty, φ_\infty)$ of this functional. Using a modified Chern-Wei…

Classifies very stable Higgs bundles for complex groups.

Diagonal metrics solve Hermitian-Einstein equations for decomposed Higgs bundles.

Study very stable Higgs bundles on Riemann surfaces, linking to multiplicity and mirror symmetry.

In this note, by using the Yang-Mills-Higgs flow, we show that semistable Higgs bundles with vanishing the first and second Chern numbers over compact Käher manifolds must admit a filtration whose quotients are Hermitian flat Higgs bundles.

We generalize the classical Beauville-Narasimhan-Ramanan correspondence to the case of parabolic Higgs bundles with regular singularities and Higgs $V$-bundles. Using this correspondence along with Bott-Morse theoretic techniques we provide an exact component count for moduli spaces of maximal parabolic $\text{Sp}\left…

Develops Lie-theoretic perspective on Hitchin's equations for cyclic G-Higgs bundles.

Defines and classifies toric co-Higgs bundles on projective toric 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…

We introduce the notion of Hermitian Higgs bundle as a natural generalization of the notion of Hermitian vector bundle and we study some vanishing theorems concerning Hermitian Higgs bundles when the base manifold is a compact complex manifold. We show that a first vanishing result, proved for these objects when the ba…

In this article, we study the Higgs vector bundles $(E,θ)$ over a compact Calabi-Yau manifolds $X$. We use Yang-Mills-Higgs flow to prove that if a semistable Higgs bundle with vanishing Chern classes over a compact connected Calabi-Yau manifold, then the Higgs field $θ$ is trivial. In particular, the vector bundle $E$…

The paper studies the moduli space of Higgs pairs and their geometric properties.

Study Poisson metrics on noncompact Kähler manifolds and their Higgs bundle applications.

We prove a Hitchin-Kobayashi correspondence for extensions of Higgs bundles. The results generalize known results for extensions of holomorphic bundles. Using Simpson's methods, we construct moduli spaces of stable objects. In an appendix we construct Bott-Chern forms for Higgs bundles

Study families of flat connections with nilpotent Higgs fields, showing similar monodromy to regular Higgs bundles.

Study generalizes Hermitian-Einstein equation for cyclic Higgs bundles, proving existence and inequality.

Co-Higgs bundles are Higgs bundles in the sense of Simpson, but with Higgs fields that take values in the tangent bundle instead of the cotangent bundle. Given a vector bundle on P^1, we find necessary and sufficient conditions on its Grothendieck splitting for it to admit a stable Higgs field. We characterize the rank…

A new class of Higgs bundles is introduced in a natural setting. Existence and nonexistence results for Higgs-Hermitian-Yang-Mills metrics are proved.

Fix a $C^\infty$ principal $G$--bundle $E^0_G$ on a compact connected Riemann surface $X$, where $G$ is a connected complex reductive linear algebraic group. We consider the gradient flow of the Yang--Mills--Higgs functional on the cotangent bundle of the space of all smooth connections on $E^0_G$. We prove that this f…

This paper generalizes a topological invariant to cyclic Higgs bundles.

In this paper, we study Higgs bundles on non-compact Hermitian manifolds. Under some assumptions for the underlying Hermitian manifolds which are not necessarily Kähler, we solve the Hermitian-Einstein equation on analytically stable Higgs bundles.

Study shows unique harmonic metrics for certain Higgs bundles over non-compact surfaces.

Computes infinitesimal automorphisms for $L$-valued Higgs bundles, leading to DM stacks.

In this paper, using Donaldson's heat flow, we show that the semi-stability of a Higgs bundle over a compact Kähler manifold implies the existence of approximate Hermitian-Einstein structure on the Higgs bundle.

Study Higgs bundles on smooth projective varieties and their restrictions to curves.

Solves Dirichlet problem for generalized Hitchin's equation on cyclic Higgs bundles.

We study the $2k$-Hitchin equations introduced by Ward \cite{Ward 2} from the geometric viewpoint of Higgs bundles. After an introduction on Higgs bundles and $2k$-Hitchin's equations, we review some elementary facts on complex geometry and Yang-Mills theory. Then we study some properties of holomorphic vector bundles …

Paper solves vortex equations on complex surfaces, linking to Higgs bundle stability.

We define homogeneous principal Higgs and co-Higgs bundles over irreducible Hermitian symmetric spaces of compact type. We provide a classification for each type of object up to isomorphism, which in each case can be interpreted as defining a moduli space.

Study nilpotent Higgs bundles and Calabi-Yau moduli metrics.

Constructs diffeological moduli stacks for Higgs and flat bundles on Kähler manifolds

Defines and analyzes $L^2$ norms on Higgs bundles over $\mathbb{CP}^1$.

The paper studies the locus in the rank 2 Higgs bundle moduli space corresponding to points which are critical for d of the Poisson commuting functions. These correspond to the Higgs field vanishing on a divisor of degree D. The degree D critical locus has an induced integrable system related to K(-D)-twisted Higgs bun…

Study harmonic metrics on Higgs bundles on non-compact Riemann surfaces.

We establish a gluing construction for Higgs bundles over a connected sum of Riemann surfaces in terms of solutions to the $\text{Sp(4}\text{,}\mathbb{R}\text{)}$-Hitchin equations using the linearization of a relevant elliptic operator. The construction can be used to provide model Higgs bundles in all the $2g-3$ exce…

The paper explores Higgs bundles and their moduli spaces on Riemann surfaces.

Given a compact Kähler manifold $X$, there is an equivalence of categories between the completely reducible flat vector bundles on $X$ and the polystable Higgs bundles $(E,\, θ)$ on $X$ with $c_1(E)= 0= c_2(E)$ \cite{SimC}, \cite{Cor}, \cite{UY}, \cite{DonI}. We extend this equivalence of categories to the context of c…

Survey on Higgs bundle moduli spaces and their connected components.

We construct a Petersson-Weil type Kähler form on the moduli spaces of Higgs bundles over a compact Kähler manifold. A fiber integral formula for this form is proved, from which it follows that the Petersson-Weil form is the curvature of a certain determinant line bundle, equipped with a Quillen metric, on the moduli s…