Compact Kahler Manifold Research Articles

Moser-Trudinger inequalities and complex Monge-Ampère equation

Our aim is to give a version of the Moser-Trudinger inequality in the setting of complex geometry. As a very particular case, our result already gives a new Moser-Trudinger inequality for functions in the Sobolev space $W^{1,2}$ of a domain in $R^2$. We also deduce a new necessary condition for the complex Monge-Ampere equation for a given measure on a compact Kahler manifold to admit a Holder continuous solution.

Positivity and the Kodaira embedding theorem

Kodaira embedding theorem provides an effective characterization of projectivity of a Kahler manifold in terms the second cohomology. Recently X. Yang [21] proved that any compact Kahler manifold with positive holomorphic sectional curvature must be projective. This gives a metric criterion of the projectivity in terms of its curvature. In this note, we prove that any compact Kahler manifold with positive 2nd scalar curvature (which is the average of holomorphic sectional curvature over 2-dimensional subspaces of the tangent space) must be projective. In view of generic 2-tori being non-abelian, this new curvature characterization is sharp in certain sense.

Restricted volumes on Kähler manifolds

We study numerical restricted volumes of (1,1) classes on compact Kahler manifolds, as introduced by Boucksom. Inspired by work of Ein-Lazarsfeld-Mustata-Nakamaye-Popa on restricted volumes of line bundles on projective manifolds, we pose a natural conjecture to the effect that irreducible components of the non-Kahler locus of a big class should have vanishing numerical restricted volume. We prove this conjecture when the class has a Zariski decomposition, and give several applications.

Existence of cscK metrics on smooth minimal models

Given a compact Kahler manifold $X$ it is interesting to ask whether it admits a constant scalar curvature Kahler (cscK) metric. In this short note we show that there always exist cscK metrics on compact Kahler manifolds with nef canonical bundle, thus on all smooth minimal models, and also on the blowup of any such manifold. This confirms an expectation of Jian-Shi-Song \cite{JianShiSong} and extends their main result from $K_X$ semi-ample to $K_X$ nef, with a direct proof that does not appeal to the Abundance conjecture. As a byproduct we obtain that the connected component $\mathrm{Aut}_0(X)$ of a compact Kahler manifold with $K_X$ nef is either trivial or a complex torus.

Stability and Hölder regularity of solutions to complex Monge–Ampère equations on compact Hermitian manifolds

Let $(X,\omega)$ be a compact Hermitian manifold. We establish a stability result for solutions to complex Monge-Ampere equations with right-hand side in $L^p$, $p>1$. Using this we prove that the solutions are Holder continuous with the same exponent as in the Kahler case \cite{DDGKPZ14}. Our techniques also apply to the setting of big cohomology classes on compact Kahler manifolds.

The principle of least action in the space of Kähler potentials

Given a compact Kahler manifold, the space $\mathcal H$ of its (relative) Kahler potentials is an infinite dimensional Frechet manifold, on which Mabuchi and Semmes have introduced a natural connection $\nabla$. We study certain Lagrangians on $T\mathcal H$, in particular Finsler metrics, that are parallel with respect to the connection. We show that geodesics of $\nabla$ are paths of least action, and prove a certain convexity property of the least action. This generalizes earlier results of Calabi, Chen, and Darvas.

Berezin–Toeplitz quantization in real polarizations with toric singularities

On a compact Kahler manifold $X$, Toeplitz operators determine a deformation quantization $(\operatorname{C}^\infty(X, \mathbb{C})[[\hbar]], \star)$ with separation of variables [10] with respect to transversal complex polarizations $T^{1, 0}X, T^{0, 1}X$ as $\hbar \to 0^+$ [15]. The analogous statement is proved for compact symplectic manifolds with transversal non-singular real polarizations [13]. In this paper, we establish the analogous result for transversal singular real polarizations on compact toric symplectic manifolds $X$. Due to toric singularities, half-form correction and localization of our Toeplitz operators are essential. Via norm estimations, we show that these Toeplitz operators determine a star product on $X$ as $\hbar \to 0^+$.

Bergman bundles and applications to the geometry of compact complex manifolds

We introduce the concept of Bergman bundle attached to a hermitian manifold X, assuming the manifold X to be compact - although the results are local for a large part. The Bergman bundle is some sort of infinite dimensional very ample Hilbert bundle whose fibers are isomorphic to the standard L² Hardy space on the complex unit ball; however the bundle is locally trivial only in the real analytic category, and its complex structure is strongly twisted. We compute the Chern curvature of the Bergman bundle, and show that it is strictly positive. As a potential application, we investigate a long standing and still unsolved conjecture of Siu on the invariance of plurigenera in the general situation of polarized families of compact Kahler manifolds.

The deformed Hermitian–Yang–Mills equation on the blowup of $\mathbb{P}^n$

We study the deformed Hermitian-Yang-Mills equation on the blowup of complex projective space. Using symmetry, we express the equation as an ODE which can be solved using combinatorial methods if an algebraic stability condition is satisfied. This gives evidence towards a conjecture of the first author, T.C. Collins, and S.-T. Yau on general compact Kahler manifolds.

Holomorphic maps into Grassmann manifolds (harmonic maps into Grassmann manifolds III)

A well-known Calabi’s rigidity theorem on holomorphic isometric immersions into the complex projective space is generalized to the case that the target is the complex Grassmann manifolds. Our strategy is to use the differential geometry of vector bundles and a generalization of do Carmo and Wallach theory developed in Nagatomo (Harmonic maps into Grassmann manifolds. arXiv:mathDG/1408.1504 ). We introduce the associated maps with holomorphic maps to obtain a general rigidity theorem (Theorem 5.6). As applications, several rigidity results on Einstein–Hermitian holomorphic maps are exhibited and we also give an interpretation of the existence of a Kahler structure with an $$S^1$$ -action on the moduli spaces of holomorphic isometric embeddings of a compact Kahler manifold into complex quadrics. Theorem 5.6 also implies classification theorems for equivariant holomorphic maps.

The Modified Cusp Kähler–Ricci Flow and Soliton

In this paper, we study the limiting flow of conical Kahler–Ricci flow modified by a holomorphic vector field on a compact Kahler manifold M which carries an effective $${\mathbb {R}}$$ -ample divisor D with simple normal crossing support. By smooth approximation, we prove the existence and uniqueness of the flow with cusp singularity when the twisted canonical bundle $$K_M+D$$ is ample. At last, we show that the flow converges to a soliton-type metric.

Miyaoka–Yau Inequality for Compact Kähler Manifolds with Semi-positive Canonical Bundles

In this paper, we prove the Miyaoka–Yau inequality for compact Kahler manifolds with semi-positive canonical bundles. The key point of the proof is the estimate for the $$L^2$$ -norm of the scalar curvature along the Kahler–Ricci flow.

A Rigidity Theorem for the deformed Hermitian-Yang-Mills equation

In this paper, we study the deformed Hermitian-Yang-Mills equation on compact Kahler manifold with non-negative orthogonal bisectional curvature. We prove that the curvatures of deformed Hermitian-Yang-Mills metrics are parallel with respect to the background metric if there exists a positive constant C such that $$\begin{aligned} -\frac{1}{C}\omega<\sqrt{-1}F<C\omega . \end{aligned}$$ We also consider the self-shrinker over $$\mathbb {C}^n$$ to the corresponding parabolic flow. We prove that the self-shrinker over $${\mathbb {C}}^n$$ is a quadratic polynomial function. At last, we show a similar rigidity theorem for the J-equations and the self-shrinkers over $${\mathbb {C}}^n$$ to J-flow.

Density currents and relative non‐pluripolar products

We compare the notion of relative non-pluripolar products and that of density currents on compact Kahler manifolds.

Geometry of Twisted Kähler–Einstein Metrics and Collapsing

We prove that the twisted Kahler-Einstein metrics that arise on the base of certain holomorphic fiber space with Calabi-Yau fibers have conical-type singularities along the discriminant locus. These fiber spaces arise naturally when studying the collapsing of Ricci-flat Kahler metrics on Calabi-Yau manifolds, and of the Kahler-Ricci flow on compact Kahler manifolds with semiample canonical bundle and intermediate Kodaira dimension. Our results allow us to understand their collapsed Gromov-Hausdorff limits when the base is smooth and the discriminant has simple normal crossings.

Holomorphic sectional curvature, nefness and Miyaoka–Yau type inequality

On a compact Kahler manifold, we introduce a notion of almost nonpositivity for the holomorphic sectional curvature, which by definition is weaker than the existence of a Kahler metric with semi-negative holomorphic sectional curvature. We prove that a compact Kahler manifold of almost nonpositive holomorphic sectional curvature has a nef canonical line bundle, contains no rational curves and satisfies some Miyaoka-Yau type inequalities. In the course of the discussions, we attach a real value to any fixed Kahler class which, up to a constant factor depending only on the dimension of manifold, turns out to be an upper bound for the nef threshold.

Expanding Kähler–Ricci solitons coming out of Kähler cones

We give necessary and sufficient conditions for a Kahler equivariant resolution of a Kahler cone, with the resolution satisfying one of a number of auxiliary conditions, to admit a unique asymptotically conical (AC) expanding gradient Kahler–Ricci soliton. In particular, it follows that for any $n \in \mathbb{N}_0$ and for any negative line bundle $L$ over a compact Kahler manifold $D$, the total space of the vector bundle $L^{\oplus (n+1)}$ admits a unique AC expanding gradient Kahler–Ricci soliton with soliton vector field a positive multiple of the Euler vector field if and only if $c_1 \Bigl ( K_D \oplus {(L^\ast)}^{\oplus (n+1)} \Bigr ) \gt 0$. This generalizes the examples already known in the literature. We further prove a general uniqueness result and show that the space of certain AC expanding gradient Kahler–Ricci solitons on $\mathbb{C}^n$ with positive curvature operator on $(1, 1)$-forms is path-connected.

Toeplitz Operators with Analytic Symbols

We develop a new semiclassical calculus in analytic regularity, and apply these techniques to the study of Berezin–Toeplitz quantization in real-analytic regularity. We provide asymptotic formulas for the Bergman projector and Berezin–Toeplitz operators on a compact Kahler manifold. These objects depend on an integer N and we study, in the limit $$N\rightarrow +\infty $$ , situations in which one can control them up to an error $$O(e^{-cN})$$ for some $$c>0$$ . We develop a calculus of Toeplitz operators with real-analytic symbols, which applies to Kahler manifolds with real-analytic metrics. In particular, we prove that the Bergman kernel is controlled up to $$O(e^{-cN})$$ on any real-analytic Kahler manifold as $$N\rightarrow +\infty $$ . We also prove that Toeplitz operators with analytic symbols can be composed and inverted up to $$O(e^{-cN})$$ . As an application, we study eigenfunction concentration for Toeplitz operators if both the manifold and the symbol are real-analytic. In this case we prove exponential decay in the classically forbidden region.

Several injectivity theorems on compact Kähler manifolds

In this short note, we use the Bochner technique and the Hodge theory in complex differential geometry to prove several injectivity results for the cohomology of holomorphic vector bundles on compact Kahler manifolds, which generalize Enoki’s original injectivity theorem.

Regularity of weak minimizers of the K-energy and applications to properness and K-stability

Let (X,ω)(X,ω) be a compact Kahler manifold and HH the space of Kahler metrics cohomologous to ωω. If a csck metric exists in HH, we show that all finite energy minimizers of the extended K-energy are smooth csck metrics, partially confirming a conjecture of Y.A. Rubinstein and the second author. As an immediate application, we obtain that the existence of a csck metric in HH implies J-properness of the K-energy, thus confirming one direction of a conjecture of Tian. Exploiting this properness result we prove that an ample line bundle (X,L)(X,L) admitting a csck metric in c1(L)c1(L) is KK-polystable. When the automorphism group is finite, the properness result, combined with a result of Boucksom-Hisamoto-Jonsson, also implies that (X,L)(X,L) is uniformly K-stable.