arXiv subjects
Publications and source records attributed to Valentino Tosatti.
We classify compact Kähler surfaces with semipositive anticanonical bundle. In particular, we resolve a question of Demailly--Peternell--Schneider: any blowup of $\mathbb{P}^2$ at 9 points on a smooth cubic has semipositive anticanonical bundle.
We prove the transcendental Morse inequality conjecture of Boucksom-Demailly-Păun-Peternell. Among the corollaries of this result, this proves orthogonality of divisorial Zariski decompositions, shows that the pseudoeffective and movable cones are dual, and establishes the differentiability of the volume function on the big cone, on all compact Kähler manifolds.
We construct a universal affine bundle $E$ over a compact complex manifold $X$ equipped with a probability measure $μ$, which is affine over the vector space $H_{1,1}(X;\mathbb{R})$, and satisfies a number of natural properties. The bundle $E$ carries a natural action of the automorphism group of $(X,μ)$, and provides potentials for all $dd^c$-closed $(1,1)$-forms on $X$. We relate our construction to lifts of the action of the automorphism group of a Calabi--Yau manifold to its universal torsor.
We show that for any smooth cubic in $\mathbb{P}^2$, there exists a dense $G_δ$ set of configurations of 9 distinct points such that blowing up $\mathbb{P}^2$ at these 9 points, the strict transform of the cubic is not linearizable and has nontorsion normal bundle. This answers a problem raised by Ogus in 1975.
We show that the normalized Kähler-Ricci flow on a compact Kähler manifold with semiample canonical bundle converges in the Gromov-Hausdorff topology to the metric completion of the twisted Kähler-Einstein metric on the canonical model, as conjectured by Song-Tian's analytic mimimal model program.
We study certain polarized degenerations of Calabi-Yau manifolds near an intermediate complex structure limit, and improve the potential $C^0$-convergence to a metric convergence result on the generic region for the corresponding collapsing Ricci-flat Kähler metrics.
We study the space of Ricci-flat Kahler metrics on a given Calabi-Yau manifold, pose a number of questions about their possible degenerations, and survey some recent results on these questions.
We prove the optimal $C^{1,1}$ regularity of the volume function on the big cone of a projective manifold, and investigate its regularity when restricted to segments moving in ample directions.
We give a complete description of the behavior of the volume function at the boundary of the pseudoeffective cone of certain Calabi-Yau complete intersections known as Wehler N-folds. We find that the volume function exhibits a pathological behavior when N>=3, we obtain examples of a pseudoeffective R-divisor D for which the volume of D+sA, with s small and A ample, oscillates between two powers of s, and we deduce the sharp regularity of this function answering a question of Lazarsfeld. We also show that h^0(X,[mD]+A) displays a similar oscillatory behavior as m increases, showing that several notions of numerical dimensions of D do not agree and disproving a conjecture of Fujino. We accomplish this by relating the behavior of the volume function along a segment to the visits of a corresponding hyperbolic geodesics to the cusps of a hyperbolic manifold.
We consider the Kähler-Ricci flow on compact Kähler manifolds with semiample canonical bundle and intermediate Kodaira dimension, and show that the flow collapses to a canonical metric on the base of the Iitaka fibration in the locally smooth topology and with bounded Ricci curvature away from the singular fibers. This follows from an asymptotic expansion for the evolving metrics, in the spirit of recent work of the first and third-named authors on collapsing Calabi-Yau metrics, and proves two conjectures of Song and Tian.
In the setting of Donaldson's conjecture on the Calabi-Yau equation on symplectic 4-manifolds, we prove an a priori estimate which in the Kähler case resembles a classical estimate of Cheng-Yau.
We prove that Calabi-Yau metrics on compact Calabi-Yau manifolds whose Kahler classes shrink the fibers of a holomorphic fibration have a priori estimates of all orders away from the singular fibers. To this end we prove an asymptotic expansion of these metrics in terms of powers of the fiber diameter, with k-th order remainders that satisfy uniform C^k-estimates with respect to a collapsing family of background metrics. The constants in these estimates are uniform not only in the sense that they are independent of the fiber diameter, but also in the sense that they only depend on the constant in the estimate for k=0 known from previous work of the second-named author. For k>0 the new estimates are proved by blowup and contradiction, and each additional term of the expansion arises as the obstruction to proving a uniform bound on one additional derivative of the remainder.
We survey some recent developments on solutions of the Kähler-Ricci flow on compact Kähler manifolds which exist for all positive times.
We study the collapsing of Calabi-Yau metrics and of Kahler-Ricci flows on fiber spaces where the base is smooth. We identify the collapsed Gromov-Hausdorff limit of the Kahler-Ricci flow when the divisorial part of the discriminant locus has simple normal crossings. In either setting, we also obtain an explicit bound for the real codimension 2 Hausdorff measure of the Cheeger-Colding singular set, and identify a sufficient condition from birational geometry to understand the metric behavior of the limiting metric on the base.
Given a holomorphic Lagrangian fibration of a compact hyperkahler manifold, we use the differential geometry of the special Kahler metric that exists on the base away from the discriminant locus, and show that the pullback of the tangent bundle of the base to the total space of a family of minimal rational curves admits a parallel splitting. The splitting is nontrivial when the base is not half-dimensional projective space. Combining this with results of Voisin, Hwang and Bakker-Schnell, we deduce that the base must be projective space, a result first proved by Hwang.
We construct examples of canonical closed positive currents on projective K3 surfaces that are not fully supported on the complex points. The currents are the unique positive representatives in their cohomology classes and have vanishing self-intersection. The only previously known such examples were due to McMullen on non-projective K3 surfaces and were constructed using positive entropy automorphisms with a Siegel disk. Our construction is based on a Zassenhaus-type estimate for commutators of automorphisms.
We survey some recent developments on various notions of semipositivity for (1,1)-classes on complex manifolds, and discuss a number of open questions.
We prove that every Gauduchon metric on an Inoue-Bombieri surface admits a strongly leafwise flat form in its $\partial\overline\partial$-class. Using this result, we deduce uniform convergence of the normalized Chern-Ricci flow starting at any Gauduchon metric on all Inoue-Bombieri surfaces. We also show that the convergence is smooth with bounded curvature for initial metrics in the $\partial\overline\partial$-class of the Tricerri/Vaisman metric.