The Monge-Ampère equation for non-integrable almost complex structures
We show existence and uniqueness of solutions to the Monge-Ampere equation on compact almost complex manifolds with non-integrable almost complex structure.
math.AP↗
arXiv subjects
Valentino Tosatti
Publications and source records attributed to Valentino Tosatti.
On the $C^{1,1}$ regularity of geodesics in the space of Kähler metrics
We prove that any two Kahler potentials on a compact Kahler manifold can be connected by a geodesic segment of C^{1,1} regularity. This follows from an a priori interior real Hessian bound for solutions of the nondegenerate complex Monge-Ampere equation, which is independent of a positive lower bound for the right hand side.
math.DG↗
Regularity of envelopes in Kähler classes
We establish the C^{1,1} regularity of quasi-psh envelopes in a Kahler class, confirming a conjecture of Berman.
math.CV↗
The Kähler-Ricci flow, Ricci-flat metrics and collapsing limits
We investigate the Kahler-Ricci flow on holomorphic fiber spaces whose generic fiber is a Calabi-Yau manifold. We establish uniform metric convergence to a metric on the base, away from the singular fibers, and show that the rescaled metrics on the fibers converge to Ricci-flat Kahler metrics. This strengthens previous work of Song-Tian and others. We obtain analogous results for degenerations of Ricci-flat Kahler metrics.
math.DG↗
Infinite time singularities of the Kähler-Ricci flow
We study the long-time behavior of the Kahler-Ricci flow on compact Kahler manifolds. We give an almost complete classification of the singularity type of the flow at infinity, depending only on the underlying complex structure. If the manifold is of intermediate Kodaira dimension and has semiample canonical bundle, so that it is fibered by Calabi-Yau varieties, we show that parabolic rescalings around any point on a smooth fiber converge smoothly to a unique limit, which is the product of a Ricci-flat metric on the fiber and of a flat metric on Euclidean space. An analogous result holds for collapsing limits of Ricci-flat Kahler metrics.
math.DG↗
The Monge-Ampère equation for (n-1)-plurisubharmonic functions on a compact Kähler manifold
A C^2 function on C^n is called (n-1)-plurisubharmonic in the sense of Harvey-Lawson if the sum of any n-1 eigenvalues of its complex Hessian is nonnegative. We show that the associated Monge-Ampere equation can be solved on any compact Kahler manifold. As a consequence we prove the existence of solutions to an equation of Fu-Wang-Wu, giving Calabi-Yau theorems for balanced, Gauduchon and strongly Gauduchon metrics on compact Kahler manifolds.
math.DG↗
Finite time collapsing of the Kähler-Ricci flow on threefolds
We show that if on a compact Kahler threefold there is a solution of the Kahler-Ricci flow which encounters a finite time collapsing singularity, then the manifold admits a Fano fibration. Furthermore, if there is finite time extinction then the manifold is Fano and the initial class is a positive multiple of the first Chern class.
math.DG↗
The Calabi-Yau Theorem and Kähler currents
In this note we give an overview of some applications of the Calabi-Yau theorem to the construction of singular positive (1,1) currents on compact complex manifolds. We show how recent developments allow us to give streamlined proofs of existing results, as well as new ones.
math.CV↗
The Aleksandrov-Bakelman-Pucci estimate and the Calabi-Yau equation
We give two applications of the Aleksandrov-Bakelman-Pucci estimate to the Calabi-Yau equation on symplectic four-manifolds. The first is solvability of the equation on the Kodaira-Thurston manifold for certain almost-Kahler structures assuming $S^1$-invariance, extending a result of Buzano-Fino-Vezzoni. The second is to reduce the general case of Donaldson's conjecture to a bound on the measure of a superlevel set of a scalar function.
math.DG↗
Orthogonality of divisorial Zariski decompositions for classes with volume zero
We show that the orthogonality conjecture for divisorial Zariski decompositions on compact Kahler manifolds holds for pseudoeffective (1,1) classes with volume zero.
math.CV↗
Uniqueness of CP^n
We give an exposition of a theorem of Hirzebruch, Kodaira and Yau which proves the uniqueness of the Kahler structure of complex projective space, and of Yau's resolution of the Severi Conjecture.
math.DG↗
An extension of a theorem of Wu-Yau
We show that a compact Kahler manifold with nonpositive holomorphic sectional curvature has nef canonical bundle. If the holomorphic sectional curvature is negative then it follows that the canonical bundle is ample, confirming a conjecture of Yau. The key ingredient is the recent solution of this conjecture in the projective case by Wu-Yau.
math.DG↗
Non-Kähler Calabi-Yau manifolds
We study the class of compact complex manifolds whose first Chern class vanishes in the Bott-Chern cohomology. This class includes all manifolds with torsion canonical bundle, but it is strictly larger. After making some elementary remarks, we show that a manifold in Fujiki's class C with vanishing first Bott-Chern class has torsion canonical bundle. We also give some examples of non-Kahler Calabi-Yau manifolds, and discuss the problem of defining and constructing canonical metrics on them.
math.DG↗
Remarks on the collapsing of torus fibered Calabi-Yau manifolds
One of the main results of the paper arXiv:1108.0967 by Gross-Tosatti-Zhang establishes estimates on the collapsing of Ricci-flat Kahler metrics on holomorphic torus fibrations. We remove a projectivity assumption from these estimates and simplify some of the underlying analysis.
math.DG↗
A singular Demailly-Paun theorem
We give a numerical characterization of the Kahler cone of a possibly singular compact analytic variety which is embedded in a smooth ambient space.
math.CV↗
Kähler currents and null loci
We prove that the non-Kahler locus of a nef and big class on a compact complex manifold bimeromorphic to a Kahler manifold equals its null locus. In particular this gives an analytic proof of a theorem of Nakamaye and Ein-Lazarsfeld-Mustata-Nakamaye-Popa. As an application, we show that finite time non-collapsing singularities of the Kahler-Ricci flow on compact Kahler manifolds always form along analytic subvarieties, thus answering a question of Feldman-Ilmanen-Knopf and Campana. We also extend the second author's results about noncollapsing degenerations of Ricci-flat Kahler metrics on Calabi-Yau manifolds to the nonalgebraic case.
math.CV↗
Inoue surfaces and the Chern-Ricci flow
We investigate the Chern-Ricci flow, an evolution equation of Hermitian metrics, on Inoue surfaces. These are non-Kahler compact complex surfaces of type Class VII. We show that, after an initial conformal change, the flow always collapses the Inoue surface to a circle at infinite time, in the sense of Gromov-Hausdorff.
math.DG↗
Families of Calabi-Yau manifolds and canonical singularities
Given a polarized family of varieties over the unit disc, smooth except over the origin and with smooth fibers Calabi-Yau, we show that the origin lies at finite Weil-Petersson distance if and only if after a finite base change the family is birational to one with central fiber a Calabi-Yau variety with at worst canonical singularities, answering a question of C.-L. Wang. This condition also implies that the Ricci-flat Kahler metrics in the polarization class on the smooth fibers have uniformly bounded diameter, or are uniformly volume non-collapsed.
math.AG↗