arXiv research

Study on stable vector bundles over Gauduchon manifolds.

The purpose of this paper is to investigate canonical metrics on a semi-stable vector bundle E over a compact Kahler manifold X. It is shown that, if E is semi-stable, then Donaldson's functional is bounded from below. This implies that E admits an approximate Hermitian-Einstein structure, generalizing a classic result…

By the work of Hong and Tian it is known that given a holomorphic vector bundle E over a compact Kahler manifold X, the Yang-Mills flow converges away from an analytic singular set. If E is semi-stable, then the limiting metric is Hermitian-Einstein and will decompose the limiting bundle into a direct sum of stable bun…

Study on hermitian Yang-Mills connections on blown-up manifolds.

Let $(\mathcal{E}, φ)$ be a rank two co-Higgs vector bundles on a Kähler compact surface $X$ with $φ\in H^0(X,End(\mathcal{E})\otimes T_X)$ nilpotent. If $(\mathcal{E}, φ)$ is semi-stable, then one of the following holds up to finite \' etale cover: $i)$ $X$ is uniruled. $ii)$ $X$ is a torus and $(\mathcal{E}, φ)$ is s…

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.

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…

The study characterizes wobbly rank-2 bundles on Riemann surfaces using spectral curves.

Constructs geometric models for moduli spaces of Higgs bundles over Riemann sphere.

Let E be a Real or Quaternionic Hermitian vector bundle over a Klein surface M. We study the action of the gauge group of E on the space of Galois-invariant unitary connections and we show that the closure of a semi-stable orbit contains a unique unitary orbit of projectively flat, Galois-invariant connections. We then…

The paper studies singularities in a complex flow related to mean curvature.

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…

The Chow-Mumford (CM) line bundle is a functorial line bundle on the base of any family of klt Fano varieties. It is conjectured that it yields a polarization on the moduli space of K-poly-stable klt Fano varieties. Proving ampleness of the CM line bundle boils down to showing semi-positivity/positivity statements abou…

Moduli spaces of real bundles over a real curve arise naturally as Lagrangian submanifolds of the moduli space of semi-stable bundles over a complex curve. In this paper, we adapt the methods of Atiyah-Bott's "Yang-Mills over a Riemann Surface" to compute Z/2-Betti numbers of these spaces, proving formulas recently obt…

Moduli spaces of semi-stable real and quaternionic vector bundles of a fixed topological type admit a presentation as Lagrangian quotients, and can be embedded into the symplectic quotient corresponding to the moduli variety of semi-stable holomorphic vector bundles of fixed rank and degree on a smooth complex projecti…

The paper studies connections on stable bundles and their continuity under metric variations.

Toric hyperk{ä}hler manifolds are quaternion analog of toric varieties. Bielawski pointed out that they can be glued by cotangent bundles of toric varieties. Following his idea, viewing both toric varieties and toric hyperk{ä}her manifolds as GIT quotients, we first establish geometrical criteria for the semi-stable po…

In this paper, we study semistable Higgs sheaves over compact Kähler manifolds, we prove that there is an approximate admissible Hermitian-Einstein structure on a semi-stable reflexive Higgs sheaf and consequently, the Bogomolove type inequality holds on a semi-stable reflexive Higgs sheaf.

Study real semi-stable degenerations and describe real loci via blow-ups.

In this paper we look at two naturally occurring situations where the following question arises. When one can find a metric so that a Chern-Weil form can be represented by a given form ? The first setting is semi-stable Hartshorne-ample vector bundles on complex surfaces where we provide evidence for a conjecture of Gr…

Paper proves convex domains have one maximum for semi-stable solutions.

Study hermitian Yang-Mills connections on pullback bundles for holomorphic submersions.

The paper studies HYM connections on stable vector bundles over Kähler manifolds.

Fix a simple complex Lie group G and a principal sl(2,C) subalgebra of Lie(G). Then the moduli space of semi-stable, topologically trivial G-Higgs bundles on a hyperbolic, spin Riemann surface acquires a marked point. This is the unique C*-fixed point on the Hitchin section. We describe a universal analytic family of d…

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…

New stability criteria for vector bundles linked to Hermite-Einstein geometry.

The Ricci flow on the 2-sphere with marked points is shown to converge in all three stable, semi-stable, and unstable cases. In the stable case, the flow was known to converge without any reparametrization, and a new proof of this fact is given. The semi-stable and unstable cases are new, and it is shown that the flow …

Extends classical stability results to new geometric settings.

The notion of flat $λ$-connections as the interpolation of usual flat connections and Higgs fields was suggested by Deligne and further studied by Simpson. Mochizuki established the Kobayashi--Hitchin-type theorem for $λ$-flat bundles ($λ\neq 0$), which is called the Mochizuki correspondence. In this paper, on the one …

Let $\mathcal{M}_{n,d}$ be the moduli space of semi-stable rank $n$, trace-free Higgs bundles with fixed determinant of degree $d$ on a Riemann surface of genus at least $3$. We determine the following automorphism groups of $\mathcal{M}_{n,d}$: (i) the group of automorphisms as a complex analytic variety, (ii) the gro…

Analytic torsion behavior studied for degenerating manifolds with equivariant bundles.

We examine a moduli problem for real and quaternionic vector bundles on a smooth complex projective curve with a fixed real structure, and we give a gauge-theoretic construction of moduli spaces for semi-stable such bundles with fixed topological type. These spaces embed onto connected subsets of real points inside a c…

We study the moduli space M(G,A) of flat G-bundles on an Abelian surface A, where G is a compact, simple, simply connected, connected Lie group. Equivalently, M(G,A) is the (coarse) moduli space of s-equivalence classes of holomorphic semi-stable G_C-bundles with trivial Chern classes where G_C is the complexified grou…

We prove that any flat family $(\mathcal{ F}_u)_{u\in U}$ of rank 2 torsion-free sheaves on a Gauduchon surface defines a continuous map on the semi-stable locus $U^{\mathrm {ss}}:=\{u\in U \ |\ \mathcal{ F}_u\hbox{ is slope semi-stable}\}$ with values in the Donaldson-Uhlenbeck compactification of the corresponding in…

New algorithm learns mappings between metric spaces, achieving strong consistency.

In this paper, we will provide a review of the geometric construction, proposed by Witten, of the SU(n) quantum representations of the mapping class groups which are part of the Reshetikhin-Turaev TQFT for the quantum group U_q(sl(n, C)). In particular, we recall the differential geometric construction of Hitchin's pro…

Paper solves Hermitian-Einstein equations on noncompact manifolds.

Over the moduli space of rank $n$ semi-stable lattices is a universal family of tori. Along the fibers, there are natural differential operators and differential equations, particularly, the heat equations and the Fokker-Planck equations in statistical mechanics. In this paper, we explain why, by taking averages over t…

Let $Y$ be a CW-complex with a single 0-cell, $K$ its Kan group, a model for the loop space of $Y$, and let $G$ be a compact, connected Lie group. We give an explicit finite dimensional construction of generators of the equivariant cohomology of the geometric realization of the cosimplicial manifold $\roman{Hom}(K,G)$ …

Quaternion-Kähler manifolds' stability and rigidity of scalar curvature studied.

We algebraically prove K-stability of polarized Calabi-Yau varieties and canonically polarized varieties with mild singularities. In particular, the} "stable varieties" introduced by Kollar-Shepherd-Barron and Alexeev, which form compact moduli space, are proven to be K-stable although it is well known that they are \t…

A group is properly 3-realizable if it is the fundamental group of a compact polyhedron whose universal covering is proper homotopically equivalent to some 3-manifold. We prove that when such a group is also quasi-simply filtered then it has {\em pro-(finitely generated free) fundamental group at infinity} and {\em sem…

I study Gromov-Hausdorff limits of complex curves endowed with singular flat metrics of constant diameter. I formulate a criterion that the limit is collapsed in terms of a certain piecewise affine weight function on the dual intersection complex of a semi-stable model of the degeneration introduced by Kontsevich and S…

New findings on Mabuchi energy and stability of manifolds.

The paper proves stability for Einstein metrics with special twisted spinors.

In their papers published in 1993 and 1994, by expressing certain physical quantity in two distinct ways, Bershadsky-Cecotti-Ooguri-Vafa discovered a remarkable equivalence between Ray-Singer analytic torsion and elliptic instanton numbers for Calabi-Yau threefolds. After their discovery, in a paper published in 2008, …

In our previous work [PSSW], we showed that the Ricci flow on S^2 whose initial metric has conical singularities \sum_{j=1}^k β_j[p_j] converges to a constant curvature metric with conic singularities (in the stable and semi-stable cases) or to a gradient shrinking soliton with conical singularities (in the unstable ca…

We revisit Atiyah and Bott's study of Morse theory for the Yang-Mills functional over a Riemann surface, and establish new formulas for the minimum codimension of a (non-semi-stable) stratum. These results yield the exact connectivity of the natural map (C_{min} E)//G(E) --> Map^E (M, BU(n)) from the homotopy orbits of…