Compact Hermitian Research Articles

The complex Monge–Ampère equation with a gradient term

We consider the complex Monge-Amp\`ere equation with an additional linear gradient term inside the determinant. We prove existence and uniqueness of solutions to this equation on compact Hermitian manifolds.

Second order estimates for complex Hessian equations on Hermitian manifolds

<p style='text-indent:20px;'>We derive second order estimates for <inline-formula><tex-math id="M1">\begin{document}$ \chi $\end{document}</tex-math></inline-formula>-plurisubharmonic solutions of complex Hessian equations with right hand side depending on the gradient on compact Hermitian manifolds.

Compact Hermitian Symmetric Spaces, Coadjoint Orbits, and the Dynamical Stability of the Ricci Flow

Using a stability criterion due to Kröncke, we show, providing $${n\ne 2k}$$ , the Kähler–Einstein metric on the Grassmannian $$Gr_{k}(\mathbb {C}^{n})$$ of complex k-planes in an n-dimensional complex vector space is dynamically unstable as a fixed point of the Ricci flow. This generalises the recent results of Kröncke and Knopf–Sesum on the instability of the Fubini–Study metric on $$\mathbb {CP}^{n}$$ for $$n>1$$ . The key to the proof is using the description of Grassmannians as certain coadjoint orbits of SU(n). We are also able to prove that Kröncke’s method will not work on any of the other compact, irreducible, Hermitian symmetric spaces.

Second order estimates for complex Hessian equations with gradient terms on both sides

In this paper, we derive the a priori second order estimate for admissible solutions in Γk+1 cone of complex Hessian equations with both sides of the equation depending on the gradient on compact Hermitian manifolds.

Dolbeault cohomology of compact complex manifolds with an action of a complex Lie group

Let G be a complex Lie group acting on a compact complex Hermitian manifold M by holomorphic isometries. We prove that the induced action on the Dolbeault cohomology and on the Bott–Chern cohomology is trivial. We also apply this result to compute the Dolbeault cohomology of Vaisman manifolds.

On the structure of Hermitian manifolds with semipositive Griffiths curvature

In this paper we establish partial structure results on the geometry of compact Hermitian manifolds of semipositive Griffiths curvature. We show that after appropriate arbitrary small deformation of the initial metric, the null spaces of the Chern–Ricci two-form generate a holomorphic, integrable distribution. This distribution induces an isometric, holomorphic, almost free action of a complex Lie group on the universal cover of the manifold. Our proof combines the strong maximum principle for the Hermitian Curvature Flow (HCF), new results on the interplay of the HCF and the torsion-twisted connection, and observations on the geometry of the torsion-twisted connection on a general Hermitian manifold.

On Bismut flat manifolds

In this paper, we give a classification of all compact Hermitian manifolds with flat Bismut connection. We show that the torsion tensor of such a manifold must be parallel, thus the universal cover of such a manifold is a Lie group equipped with a bi-invariant metric and a compatible left invariant complex structure. In particular, isosceles Hopf surfaces are the only Bismut flat compact non-Kähler surfaces, while central Calabi-Eckmann threefolds are the only simply-connected compact Bismut flat threefolds.

The complex Hessian equations with gradient terms on Hermitian manifolds

In this paper, we consider the complex Hessian equation involving additional gradient terms on both left and right hand sides. We derive the second order estimates for χ˜-plurisubharmonic solutions to this equation on compact Hermitian manifolds. Such an estimate may be helpful to study the existence and regularity for the solution of the equation.

A note on compact 𝜅-solutions of Kähler-Ricci flow

In this paper, we give a complete classification of κ \kappa -solutions of Kähler-Ricci flow on compact complex manifolds. Namely, they must be quotients of products of irreducible compact Hermitian symmetric manifolds.

Canonical Metrics on Holomorphic Filtrations over Compact Hermitian Manifolds

The purpose of this paper is twofold. We first solve the Dirichlet problem for $$\tau $$-Hermitian–Einstein equations on holomorphic filtrations over compact Hermitian manifolds. Secondly, by using Uhlenbeck–Yau’s continuity method, we prove the existence of approximate $$\tau $$-Hermitian–Einstein structure on holomorphic filtrations over closed Gauduchon manifolds.

A Dolbeault-Dirac Spectral Triple for Quantum Projective Space

The notion of a Kahler structure for a differential calculus was recently introduced by the second author as a framework in which to study the noncommutative geometry of the quantum flag manifolds. It was subsequently shown that any covariant positive definite Kahler structure has a canonically associated triple satisfying, up to the compact resolvent condition, Connes' axioms for a spectral triple. In this paper we begin the development of a robust framework in which to investigate the compact resolvent condition, and moreover, the general spectral behaviour of covariant Kahler structures. This framework is then applied to quantum projective space endowed with its Heckenberger-Kolb differential calculus. An even spectral triple with non-trivial associated K-homology class is produced, directly q-deforming the Dirac-Dolbeault operator of complex projective space. Finally, the extension of this approach to a certain canonical larger class of compact quantum Hermitian symmetric spaces is discussed in detail.

A Dolbeault–Hilbert complex for a variety with isolated singular points

Given a compact Hermitian complex space with isolated singular points, we construct a Dolbeault-type Hilbert complex whose cohomology is isomorphic to the cohomology of the structure sheaf. We show that the corresponding K-homology class coincides with the one constructed by Baum-Fulton-MacPherson.

Hermitian-Einstein Metrics For Higgs Bundles Over Complete Hermitian Manifolds

In this paper, we solve the Dirichlet problem for the Hermitian-Einstein equations on Higgs bundles over compact Hermitian manifolds. Then we prove the existence of the Hermitian-Einstein metrics on Higgs bundles over a class of complete Hermitian manifolds.

On Analytic Todd Classes of Singular Varieties

AbstractLet $(X,h)$ be a compact and irreducible Hermitian complex space. This paper is devoted to various questions concerning the analytic K-homology of $(X,h)$. In the 1st part, assuming either $\dim (\operatorname{sing}(X))=0$ or $\dim (X)=2$, we show that the rolled-up operator of the minimal $L^2$-$\overline{\partial }$ complex, denoted here $\overline{\eth }_{\textrm{rel}}$, induces a class in $K_0 (X)\equiv KK_0(C(X),\mathbb{C})$. A similar result, assuming $\dim (\operatorname{sing}(X))=0$, is proved also for $\overline{\eth }_{\textrm{abs}}$, the rolled-up operator of the maximal $L^2$-$\overline{\partial }$ complex. We then show that when $\dim (\operatorname{sing}(X))=0$ we have $[\overline{\eth }_{\textrm{rel}}]=\pi _*[\overline{\eth }_M]$ with $\pi :M\rightarrow X$ an arbitrary resolution and with $[\overline{\eth }_M]\in K_0 (M)$ the analytic K-homology class induced by $\overline{\partial }+\overline{\partial }^t$ on $M$. In the 2nd part of the paper we focus on complex projective varieties $(V,h)$ endowed with the Fubini–Study metric. First, assuming $\dim (V)\leq 2$, we compare the Baum–Fulton–MacPherson K-homology class of $V$ with the class defined analytically through the rolled-up operator of any $L^2$-$\overline{\partial }$ complex. We show that there is no $L^2$-$\overline{\partial }$ complex on $(\operatorname{reg}(V),h)$ whose rolled-up operator induces a K-homology class that equals the Baum–Fulton–MacPherson class. Finally in the last part of the paper we prove that under suitable assumptions on $V$ the push-forward of $[\overline{\eth }_{\textrm{rel}}]$ in the K-homology of the classifying space of the fundamental group of $V$ is a birational invariant.

RC-POSITIVITY, VANISHING THEOREMS AND RIGIDITY OF HOLOMORPHIC MAPS

AbstractLet $M$ and $N$ be two compact complex manifolds. We show that if the tautological line bundle ${\mathcal{O}}_{T_{M}^{\ast }}(1)$ is not pseudo-effective and ${\mathcal{O}}_{T_{N}^{\ast }}(1)$ is nef, then there is no non-constant holomorphic map from $M$ to $N$. In particular, we prove that any holomorphic map from a compact complex manifold $M$ with RC-positive tangent bundle to a compact complex manifold $N$ with nef cotangent bundle must be a constant map. As an application, we obtain that there is no non-constant holomorphic map from a compact Hermitian manifold with positive holomorphic sectional curvature to a Hermitian manifold with non-positive holomorphic bisectional curvature.

On the Laplace–Beltrami operator on compact complex spaces

Let ( X , h ) (X,h) be a compact and irreducible Hermitian complex space of complex dimension v &gt; 1 v&gt;1 . In this paper we show that the Friedrichs extension of both the Laplace–Beltrami operator and the Hodge–Kodaira Laplacian acting on functions has discrete spectrum. Moreover, we provide some estimates for the growth of the corresponding eigenvalues, and we use these estimates to deduce that the associated heat operators are trace class. Finally we give various applications to the Hodge–Dolbeault operator and to the Hodge–Kodaira Laplacian in the setting of Hermitian complex spaces of complex dimension 2 2 .

Bott–Chern harmonic forms on complete Hermitian manifolds

Let [Formula: see text] be a Hermitian manifold of complex dimension [Formula: see text]. Assume that the torsion of the Chern connection [Formula: see text] is bounded, and that there exists a [Formula: see text]exhausting function [Formula: see text] such that [Formula: see text] are bounded. We characterize [Formula: see text] Bott–Chern harmonic forms, extending the usual result that holds on compact Hermitian manifolds. Finally, if [Formula: see text] is Kähler complete, [Formula: see text], with [Formula: see text] bounded, and the sectional curvature is bounded, then we get a vanishing theorem for [Formula: see text] Bott–Chern harmonic [Formula: see text]-forms, if [Formula: see text].

On Grauert—Riemenschneider Type Criteria

Let (X, ω) be a compact Hermitian manifold of complex dimension n. In this article, we first survey recent progress towards Grauert-Riemenschneider type criteria. Secondly, we give a simplified proof of Boucksom’s conjecture given by the author under the assumption that the Hermitian metric ω satisfies \(\partial \overline \partial {\omega ^l} = {\rm{for}}\;{\rm{all}}\;l\), i.e., if T is a closed positive current on X such that ∫X\(T_{ac}^n>0\), then the class {T} is big and X is Kahler. Finally, as an easy observation, we point out that Nguyen’s result can be generalized as follows: if \(\partial \overline \partial \omega = 0\), and T is a closed positive current with analytic singularities, such that ∫X\(T_{ac}^n>0\), then the class {T} is big and X is Kahler.

The Fu–Yau equation on compact astheno-Kähler manifolds

In this paper, we study the Fu–Yau equation on compact Hermitian manifolds and prove the existence of solutions of equation on astheno-Kähler manifolds. We also prove the uniqueness of solutions of Fu–Yau equation when the slope parameter α is negative.

Stability and regularity of solutions of the Monge–Ampère equation on Hermitian manifolds

We prove stability of solutions of the complex Monge–Ampère equation on compact Hermitian manifolds, when the right hand side varies in a bounded set in Lp, p>1 and it is bounded away from zero. Such solutions are shown to be Hölder continuous. As an application we extend a recent result of Székelyhidi and Tosatti on Kähler–Einstein equation from Kähler to Hermitian manifolds.