arXiv research

Article establishes criteria for multiplier Hermitian-Einstein metrics on KSM-manifolds.

New Kähler metrics generalize Calabi's and relate to Fano manifolds.

This paper describes how, in the case of algebraic surfaces, the well-known theorem of Donaldson-Uhlenbeck-Yau can be proved in a framework of generalized 'multiplier ideal sheaves', following the ideas of Siu. The key concept is that the destabilizing sheaf satisfies a differential inclusion relation. This relation is…

Hermitian-Einstein metrics linked to stability of bundles on orbifolds.

Paper defines new stability and metrics for complex spaces.

New insights prevent certain types of metrics on compact spaces.

We describe and construct here pseudo-Hermitian structures $θ$ without torsion (i.e. with transversal symmetry) whose Webster-Ricci curvature tensor is a constant multiple of the exterior differential $dθ$. We call these structures pseudo-Hermitian Einstein and our result states that they all can be derived locally fro…

The Horrocks-Mumford bundle $E$ is a famous stable complex vector bundle of rank 2 on 4-dimensional complex projective space. By construction, $E$ has a natural Hermitian metric $h_1$. On the other hand, stability implies the existence of a Hermitian-Einstein metric in $E$ which is unique up to a positive scalar. Now t…

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

We develop a theory of stable bundles and affine Hermitian-Einstein metrics for flat vector bundles over a special affine manifold (a manifold admitting an atlas whose gluing maps are all locally constant volume-preserving affine maps). Our paper presents a parallel to Donaldson-Uhlenbeck-Yau's proof of the existence o…

Iterative method finds Hermitian-Einstein metrics on stable bundles.

New metrics found on non-Kähler complex manifolds.

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

Study proves correspondence for special bundles on complex surfaces.

Existence of twisted Hermitian-Einstein metrics on unstable vector bundles

We adapt the notions of stability of holomorphic vector bundles in the sense of Mumford-Takemoto and Hermitian-Einstein metrics in holomorphic vector bundles for canonically polarized framed manifolds, i.e. compact complex manifolds X together with a smooth divisor D such that K_X \otimes [D] is ample. It turns out tha…

This paper produces explicit strongly Hermitian Einstein-Maxwell solutions on the smooth compact $4$-manifolds that are $S^2$-bundles over compact Riemann surfaces of any genus. This generalizes the existence results by C. LeBrun in arXiv:1411.3992 and arXiv:1504.06669. Moreover, by calculating the (normalized) Einstei…

Paper solves Hermitian-Einstein equations on noncompact manifolds.

We prove the existence of a Hermitian-Einstein metric on holomorphic vector bundles with a Hermitian metric satisfying the analytic stability condition, under some assumption for the underlying Kähler manifolds. We also study the curvature decay of the Hermitian-Einstein metrics. It is useful for the study of the class…

Constructs stable Hilbert bundles on curves using Diophantine approximation.

Given a smooth complex projective variety X and a smooth divisor D on X, we prove the existence of Hermitian-Einstein connections, with respect to a Poincaré-type metric on X - D, on polystable parabolic principal Higgs bundles with parabolic structure over D, satisfying certain conditions on its restriction to D.

A Hermitian Einstein-Weyl manifold is a complex manifold admitting a Ricci-flat Kaehler covering W, with the deck transform acting on W by homotheties. If compact, it admits a canonical Vaisman metric, due to Gauduchon. We show that a Hermitian Einstein-Weyl structure on a compact complex manifold is determined by its …

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…

In this paper, we introduce notions of nonlinear stabilities for a relative ample line bundle over a holomorphic fibration and define the notion of a geodesic-Einstein metric on this line bundle, which generalize the classical stabilities and Hermitian-Einstein metrics of holomorphic vector bundles. We introduce a Dona…

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…

We consider a notion of balanced metrics for triples (X,L,E) which depend on a parameter α, where X is smooth complex manifold with an ample line bundle L and E is a holomorphic vector bundle over X. For generic choice of α, we prove that the limit of a convergent sequence of balanced metrics leads to a Hermitian-Einst…

Construct Hermitian-Einstein metrics on stable holomorphic vector bundles using dynamical methods.

Let $X$ be a compact Gauduchon manifold, and let $E$ and $V_0$ be holomorphic vector bundles over $X$. Suppose that $E$ is stable when considering all subsheaves preserved by a Higgs field $θ\in H^0($End$(E)\otimes V_0)$. Then a modified version of the Donaldson heat flow converges along a subsequence of times to a sol…

Study stability of non-Kähler Calabi-Yau metrics using critical points of generalized Einstein Hilbert action.

Unique solution found for Demailly's equation on stable bundles.

We prove that a simpy connected Hermitian Einstein 4-manifold with non-negative sectional curvature is isometric to complex projective space $\mathbb{C}\mathbb{P}^{2}$ with the Fubini-Study metric or isometric to the product $\mathbb{S}^{2}\times \mathbb{S}^{2}$ with the canonical metric.

In this paper, we compute the index form of the multiply twisted products. We study the Killing vector fields on the multiply twisted product manifolds and determine the Killing vector fields in some cases. We compute the curvature of the multiply twisted products with a semi-symmetric metric connection and show that t…

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…

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

We provide non trivial examples of solutions to the system of coupled equations introduced by M. García-Fernández for the uniformization problem of a triple $(M,L,E)$ where $E$ is a holomorphic vector bundle over a polarized complex manifold $(M,L)$, generalizing the notions of both constant scalar curvature Kähler met…

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

In this paper, we define a semi-symmetric metric Killing vector field, then study semi-symmetric metric Killing vector fields on warped and multiply warped products with a semi-symmetric metric connection. We also study Killing and 2-Killing vector fields on multiply warped products.

In this paper, we study the Einstein multiply warped products with a semi-symmetric non-metric connection and the multiply warped products with a semi-symmetric non-metric connection with constant scalar curvature, we apply our results to generalized Robertson-Walker spacetimes with a semi-symmetric non-metric connecti…

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.

For a holomorphic vector bundle $E$ over a polarised Kähler manifold, we establish a direct link between the slope stability of $E$ and the asymptotic behaviour of Donaldson's functional, by defining the Quot-scheme limit of Fubini-Study metrics. In particular, we provide an explicit estimate which proves that Donaldso…

Let E_G be a principal G-bundle over a compact connected Kähler manifold, where G is a connected reductive complex linear algebraic group. We show that E_G is semistable if and only if it admits approximate Hermitian-Einstein structures.

New obstruction found for Hull-Strominger system solutions.

Let X be a smooth complex projective curve and S a finite subset of X. We show that an orthogonal or symplectic parabolic Higgs bundle on X with parabolic structure over S admits a Hermitian-Einstein connection if and only if it is polystable.

We note that the Bogomolny equation for abelian vortices is precisely the condition for invariance of the Hermitian-Einstein equation under a degenerate conformal transformation. This leads to a natural interpretation of vortices as degenerate hermitian metrics that satisfy a certain curvature equation. Using this view…

Solves Lempert's question on Nakano semi-positivity preservation.

Study on four-dimensional Ricci solitons and multiply warped Ricci flow solutions.

Proves a conjecture about Riemann surfaces using PDEs.

Over a compact Kähler manifold, we provide a Fredholm alternative result for the Lichnerowicz operator associated to a Kähler metric with conic singularities along a divisor. We deduce several existence results of constant scalar curvature Kähler metrics with conic singularities: existence result under small deformatio…