Quot-scheme limit of Fubini-Study metrics and Donaldson's functional for vector bundles

For a holomorphic vector bundle $E$ over a polarised K\"ahler 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 Donaldson's functional is coercive on the set of Fubini-Study metrics if $E$ is slope stable, and give a new proof of Hermitian-Einstein metrics implying slope stability.

\'Etale triviality of finite equivariant vector bundles

Let H be a complex Lie group acting holomorphically on a complex analytic space X such that the restriction to X_{\mathrm{red}} of every H-invariant regular function on X is constant. We prove that an H-equivariant holomorphic vector bundle E over X is $H$-finite, meaning f_1(E)= f_2(E) as H-equivariant bundles for two distinct polynomials f_1 and f_2 whose coefficients are nonnegative integers, if and only if the pullback of E along some H-equivariant finite \'etale covering of X is trivial as an H-equivariant bundle.

A remark on characterizations of Griffiths positivity through asymptotic conditions

In this paper, we study characterizations of Griffiths semi-positivity through [Formula: see text]-estimates of the [Formula: see text]-equation and [Formula: see text]-extension theorems for symmetric powers of a holomorphic vector bundle. We also investigate several versions of the converse of the Demailly–Skoda theorem.

Uniqueness of optimal symplectic connections

Consider a holomorphic submersion between compact Kähler manifolds, such that each fibre admits a constantscalar curvature Kähler metric. When the fibres admit continuous automorphisms, a choice of fibrewise constant scalarcurvature Kähler metric is not unique. An optimal symplectic connection is a choice of fibrewise constant scalar curvature Kähler metric satisfying a geometric partial differential equation. The condition generalises the Hermite-Einstein condition for a holomorphic vector bundle through the induced fibrewise Fubini-Study metric on the associated projectivisation. We prove various foundational analytic results concerning optimal symplectic connections. Our main result proves that optimal symplectic connections are unique, up to the action of the automorphism group of the submersion, when they exist. Thus optimal symplectic connections are canonical relatively Kähler metrics when they exist. In addition, we show that the existence of an optimal symplectic connection forces the automorphism group of the submersion to be reductive and that an optimal symplectic connection is automatically invariant under a maximal compact subgroup of this automorphism group. We also prove that when a submersion admits an optimal symplectic connection, it achieves the absolute minimum of a natural log norm functional, which we define.

Kobayashi—Hitchin corrispondence for twisted vector bundles

We prove the Kobayashi—Hitchin correspondence and the approximate Kobayashi—Hitchin correspondence for twisted holomorphic vector bundles on compact Kähler manifolds. More precisely, if X is a compact manifold and g is a Gauduchon metric on X, a twisted holomorphic vector bundle on X is g−polystable if and only if it is g−Hermite-Einstein, and if X is a compact Kähler manifold and g is a Kähler metric on X, then a twisted holomorphic vector bundle on X is g−semistable if and only if it is approximate g−Hermite-Einstein.

Branched Holomorphic Cartan Geometries and Calabi–Yau Manifolds

We introduce the concept of a branched holomorphic Cartan geometry. It generalizes to higher dimension the definition of branched (flat) complex projective structure on a Riemann surface introduced by Mandelbaum [25]. This new framework is much more flexible than that of the usual holomorphic Cartan geometries. We show that all compact complex projective manifolds admit a branched flat holomorphic projective structure. We also give an example of a non-flat branched holomorphic normal projective structure on a compact complex surface. It is known that no compact complex surface admits such a structure with empty branching locus. We prove that non-projective compact simply connected Kähler Calabi–Yau manifolds do not admit any branched holomorphic projective structure. The key ingredient of its proof is the following result of independent interest: if E is a holomorphic vector bundle over a compact simply connected Kähler Calabi–Yau manifold and E admits a holomorphic connection, then E is a trivial holomorphic vector bundle and any holomorphic connection on E is trivial.

Metrik Finsler Pseudo-Konveks Kuat pada Bundel Vektor Holomorfik

Rizza-negativity of holomorphic vector bundle is a sufficient condition for the negativity of . In the present paper, we shall discuss that as a special case, using the Rizza metric which is derived from a Hermitian metric also implies the negativity of . Further we showed that for the negative holomorphic vector bundle there is a pseudo-convex Finsler metric with negative curvature. Keywords: Hermitian metric, Rizza metric, Rizza-negativity.

Goresky–Pardon lifts of Chern classes and associated Tate extensions

Let $X$ be an irreducible complex-analytic variety, ${\mathcal{S}}$ a stratification of $X$ and ${\mathcal{F}}$ a holomorphic vector bundle on the open stratum ${X\unicode[STIX]{x0030A}}$. We give geometric conditions on ${\mathcal{S}}$ and ${\mathcal{F}}$ that produce a natural lift of the Chern class $\operatorname{c}_{k}({\mathcal{F}})\in H^{2k}({X\unicode[STIX]{x0030A}};\mathbb{C})$ to $H^{2k}(X;\mathbb{C})$, which, in the algebraic setting, is of Hodge level ${\geqslant}k$. When applied to the Baily–Borel compactification $X$ of a locally symmetric variety ${X\unicode[STIX]{x0030A}}$ and an automorphic vector bundle ${\mathcal{F}}$ on ${X\unicode[STIX]{x0030A}}$, this refines a theorem of Goresky–Pardon. In passing we define a class of simplicial resolutions of the Baily–Borel compactification that can be used to define its mixed Hodge structure. We use this to show that the stable cohomology of the Satake ($=$ Baily–Borel) compactification of ${\mathcal{A}}_{g}$ contains nontrivial Tate extensions.

Positivity of metrics on conic neighborhoods of 1-convex submanifolds

Let [Formula: see text] be a holomorphic submersion from a complex manifold [Formula: see text] onto a 1-convex manifold [Formula: see text] with exceptional set [Formula: see text] and [Formula: see text] a holomorphic section. Let [Formula: see text] be a plurisubharmonic exhaustion function which is strictly plurisubharmonic on [Formula: see text] with [Formula: see text] For every holomorphic vector bundle [Formula: see text] there exists a neighborhood [Formula: see text] of [Formula: see text] for [Formula: see text] conic along [Formula: see text] such that [Formula: see text] can be endowed with Nakano strictly positive Hermitian metric. Let [Formula: see text] [Formula: see text] be a given holomorphic function. There exist finitely many bounded holomorphic vector fields defined on a Stein neighborhood [Formula: see text] of [Formula: see text] conic along [Formula: see text] with zeroes of arbitrary high order on [Formula: see text] and such that they generate [Formula: see text] Moreover, there exists a smaller neighborhood [Formula: see text] such that their flows remain in [Formula: see text] for sufficiently small times thus generating a local dominating spray.

Chern forms of holomorphic Finsler vector bundles and some applications

In this paper, we present two kinds of total Chern forms [Formula: see text] and [Formula: see text] as well as a total Segre form [Formula: see text] of a holomorphic Finsler vector bundle [Formula: see text] expressed by the Finsler metric [Formula: see text], which answers a question of Faran [The equivalence problem for complex Finsler Hamiltonians, in Finsler Geometry, Contemporary Mathematics, Vol. 196 (American Mathematical Society, Providence, RI, 1996), pp. 133–144] to some extent. As some applications, we show that the signed Segre forms [Formula: see text] are positive [Formula: see text]-forms on [Formula: see text] when [Formula: see text] is of positive Kobayashi curvature; we prove, under an extra assumption, that a Finsler–Einstein vector bundle in the sense of Kobayashi is semi-stable; we introduce a new definition of a flat Finsler metric, which is weaker than Aikou’s one [Finsler geometry on complex vector bundles, in A Sampler of Riemann–Finsler Geometry, MSRI Publications, Vol. 50 (Cambridge University Press, 2004), pp. 83–105] and prove that a holomorphic vector bundle is Finsler flat in our sense if and only if it is Hermitian flat.