Big Cohomology Classes Research Articles

Twisted Kähler–Einstein metrics in big classes

AbstractWe prove existence of twisted Kähler–Einstein metrics in big cohomology classes, using a divisorial stability condition. In particular, when is big, we obtain a uniform Yau–Tian–Donaldson (YTD) existence theorem for Kähler–Einstein (KE) metrics. To achieve this, we build up from scratch the theory of Fujita–Odaka type delta invariants in the transcendental big setting, using pluripotential theory. We do not use the K‐energy in our arguments, and our techniques provide a simple roadmap to prove YTD existence theorems for KE type metrics, that only needs convexity of the appropriate Ding energy. As an application, we give a simplified proof of Li–Tian–Wang's existence theorem in the log Fano setting.

Mabuchi geometry of big cohomology classes

Abstract Let X be a compact Kähler manifold. Fix a big ( 1 , 1 ) {(1,1)} -cohomology class α with smooth representative θ. We study the spaces ℰ p ⁢ ( X , θ ) {\mathcal{E}^{p}(X,\theta)} of finite energy Kähler potentials for each p ≥ 1 {p\geq 1} . We define a metric d p {d_{p}} without using the Finsler geometry nor solving Monge–Ampère-type equations. This construction generalizes the usual d p {d_{p}} -metric defined for an ample class.

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.

Pluripotential Monge-Ampère flows in big cohomology classes

We study pluripotential complex Monge-Ampère flows in big cohomology classes on compact Kähler manifolds. We use the Perron method, considering pluripotential subsolutions to the Cauchy problem. We prove that, under natural assumptions on the data, the upper envelope of all subsolutions is continuous in space and semi-concave in time, and provides a unique pluripotential solution with such regularity. We apply this theory to study pluripotential Kähler-Ricci flows on compact Kähler manifolds of general type as well as on stable varieties

Continuity of Monge–Ampère Potentials in Big Cohomology Classes

Abstract Extending Di Nezza–Lu’s approach [16] to the setting of big cohomology classes, we prove that solutions of degenerate complex Monge–Ampère equations on compact Kähler manifolds are continuous on a Zariski open set. This allows us to show that singular Kähler–Einstein metrics on log canonical varieties of general type have continuous potentials on the ample locus outside of the non-klt part.

C1,1 regularity of geodesics of singular Kähler metrics

We show the optimal $C^{1,1}$ regularity of geodesics in nef and big cohomology class on K\"ahler manifolds away from the non-K\"ahler locus, assuming sufficiently regular initial data. As a special case, we prove the $C^{1,1}$ regularity of geodesics of K\"ahler metrics on compact K\"ahler varieties away from the singular locus. Our main novelty is an improved boundary estimate for the complex Monge-Amp\`ere equation that does not require strict positivity of the reference form near the boundary. We also discuss the case of some special geodesic rays.

The metric geometry of singularity types

Abstract Let X be a compact Kähler manifold. Given a big cohomology class { θ } {\{\theta\}} , there is a natural equivalence relation on the space of θ-psh functions giving rise to 𝒮 ⁢ ( X , θ ) {\mathcal{S}(X,\theta)} , the space of singularity types of potentials. We introduce a natural pseudo-metric d 𝒮 {d_{\mathcal{S}}} on 𝒮 ⁢ ( X , θ ) {\mathcal{S}(X,\theta)} that is non-degenerate on the space of model singularity types and whose atoms are exactly the relative full mass classes. In the presence of positive mass we show that this metric space is complete. As applications, we show that solutions to a family of complex Monge–Ampère equations with varying singularity type converge as governed by the d 𝒮 {d_{\mathcal{S}}} -topology, and we obtain a semicontinuity result for multiplier ideal sheaves associated to singularity types, extending the scope of previous results from the local context.

Log-concavity of volume and complex Monge–Ampère equations with prescribed singularity

Let $$(X,\omega )$$ be a compact Kahler manifold. We prove the existence and uniqueness of solutions to complex Monge–Ampere equations with prescribed singularity type. Compared to previous work, the assumption of small unbounded locus is dropped, and we work with general model type singularities. We state and prove our theorems in the context of big cohomology classes, however our results are new in the Kahler case as well. As an application we confirm a conjecture by Boucksom–Eyssidieux–Guedj–Zeriahi concerning log-concavity of the volume of closed positive (1, 1)-currents. Finally, we show that log-concavity of the volume in complex geometry corresponds to the Brunn–Minkowski inequality in convex geometry, pointing out a dictionary between our relative pluripotential theory and P-relative convex geometry. Applications related to stability and existence of csck metrics are treated elsewhere.

Envelopes with Prescribed Singularities

We prove that quasi-plurisubharmonic envelopes with prescribed analytic singularities in suitable big cohomology classes on compact K\"ahler manifolds have the optimal $C^{1,1}$ regularity on a Zariski open set. This also proves regularity of certain pluricomplex Green's functions on K\"ahler manifolds. We then go on to prove the same regularity for envelopes when the manifold is assumed to have boundary. As an application, we answer affirmatively a question of Ross--Witt-Nystr\"om concerning the Hele-Shaw flow on an arbitrary Reimann surface.

From Monge\u2013Amp\xe8re equations to envelopes and geodesic rays in the zero temperature limit

Let (X,theta ) be a compact complex manifold X equipped with a smooth (but not necessarily positive) closed (1, 1)-form theta . By a well-known envelope construction this data determines, in the case when the cohomology class [theta ] is pseudoeffective, a canonical theta -psh function u_{theta }. When the class [theta ] is Kähler we introduce a family u_{beta } of regularizations of u_{theta }, parametrized by a large positive number beta , where u_{beta } is defined as the unique smooth solution of a complex Monge–Ampère equation of Aubin–Yau type. It is shown that, as beta rightarrow infty , the functions u_{beta } converge to the envelope u_{theta } uniformly on X in the Hölder space C^{1,alpha }(X) for any alpha in ]0,1[ (which is optimal in terms of Hölder exponents). A generalization of this result to the case of a nef and big cohomology class is also obtained and a weaker version of the result is obtained for big cohomology classes. The proofs of the convergence results do not assume any a priori regularity of u_{theta }. Applications to the regularization of omega -psh functions and geodesic rays in the closure of the space of Kähler metrics are given. As briefly explained there is a statistical mechanical motivation for this regularization procedure, where beta appears as the inverse temperature. This point of view also leads to an interpretation of u_{beta } as a “transcendental” Bergman metric.

$L^1$ metric geometry of big cohomology classes

Suppose $(X,\omega)$ is a compact Kahler manifold of dimension $n$, and $\theta$ is closed $(1,1)$-form representing a big cohomology class. We introduce a metric $d_1$ on the finite energy space $\mathcal{E}^1(X,\theta)$, making it a complete geodesic metric space. This construction is potentially more rigid compared to its analog from the Kahler case, as it only relies on pluripotential theory, with no reference to infinite dimensional $L^1$ Finsler geometry. Lastly, by adapting the results of Ross and Witt Nystrom to the big case, we show that one can construct geodesic rays in this space in a flexible manner.

On the singularity type of full mass currents in big cohomology classes

Let$X$be a compact Kähler manifold and$\{\unicode[STIX]{x1D703}\}$be a big cohomology class. We prove several results about the singularity type of full mass currents, answering a number of open questions in the field. First, we show that the Lelong numbers and multiplier ideal sheaves of$\unicode[STIX]{x1D703}$-plurisubharmonic functions with full mass are the same as those of a current with minimal singularities. Second, given another big and nef class$\{\unicode[STIX]{x1D702}\}$, we show the inclusion${\mathcal{E}}(X,\unicode[STIX]{x1D702})\cap \operatorname{PSH}(X,\unicode[STIX]{x1D703})\subset {\mathcal{E}}(X,\unicode[STIX]{x1D703})$. Third, we characterize big classes whose full mass currents are ‘additive’. Our techniques make use of a characterization of full mass currents in terms of the envelope of their singularity type. As an essential ingredient we also develop the theory of weak geodesics in big cohomology classes. Numerous applications of our results to complex geometry are also given.

Corrigendum: Viscosity Solutions to Complex Monge‐Ampère Equations

AbstractThe proof of the comparison principle in the article cited above is not complete. We provide here an alternative proof, valid in the ample locus of any big cohomology class, and discuss the resulting modifications.© 2017 Wiley Periodicals, Inc.

Rational curves on compact Kähler manifolds

We present an inductive strategy to show the existence of rational curves on compact Kaehler manifolds which are not minimal models but have a pseudoeffective canonical bundle. The tool for this inductive strategy is a weak subadjunction formula for lc centers associated to certain big cohomology classes. This subadjunction formula is based, as in the projective case, on positivity arguments for relative adjoint classes.

Stability of Monge–Ampère Energy Classes

We show that the non-pluripolar product of positive currents is a bimeromorphic invariant. Under some natural assumptions, we show that the (weighted) energy associated with big cohomology classes are also bimeromorphic invariants. We compare the weighted energy functionals of currents with respect to different cohomology classes and establish quantitative estimates between big capacities.

Hölder continuous solutions to Monge–Ampère equations

Let (X,\omega) be a compact Kähler manifold. We obtain uniform Hölder regularity for solutions to the complex Monge-Ampère equation on X with L^p right hand side, p>1 . The same regularity is furthermore proved on the ample locus in any big cohomology class. We also study the range \operatorname{MAH}(X,\omega) of the complex Monge-Ampère operator acting on \omega -pluri-subharmonic Hölder continuous functions. We show that this set is convex, by sharpening\break Ko\l odziej's result that measures with L^p -density belong to \operatorname{MAH}(X,\omega) and proving that \operatorname{MAH}X,\omega) has the “ L^p -property”, p>1 . We also describe accurately the symmetric measures it contains.

A variational approach to complex Monge-Ampère equations

We show that degenerate complex Monge-Ampere equations in a big cohomology class of a compact Kahler manifold can be solved using a variational method, without relying on Yau’s theorem. Our formulation yields in particular a natural pluricomplex analogue of the classical logarithmic energy of a measure. We also investigate Kahler-Einstein equations on Fano manifolds. Using continuous geodesics in the closure of the space of Kahler metrics and Berndtsson’s positivity of direct images, we extend Ding-Tian’s variational characterization and Bando-Mabuchi’s uniqueness result to singular Kahler-Einstein metrics. Finally, using our variational characterization we prove the existence, uniqueness and convergence as k→∞ of k-balanced metrics in the sense of Donaldson both in the (anti)canonical case and with respect to a measure of finite pluricomplex energy.

Stability of solutions to complex Monge–Ampère equations in big cohomology classes

Stability of solutions to complex Monge–Ampère equations in big cohomology classes

On the coniveau of certain sub-Hodge structures

We study the generalized Hodge conjecture for certain sub-Hodge structure defined as the kernel of the cup product map with a big cohomology class, which is of Hodge coniveau at least 1. As predicted by the generalized Hodge conjecture, we prove that the kernel is supported on a divisor, assuming the Lefschetz standard conjecture.

Complex Monge-Ampère equations

This is a survey of some of the recent developments in the theory of complex Monge-Ampere equations. The topics discussed include refinements and simplifications of classical a priori estimates, methods from pluripotential theory, variational methods for big cohomology classes, semiclassical constructions of solutions of homogeneous equations, and envelopes.