About

Valentino Tosatti is a Professor of Mathematics and Vice Dean for Faculty Affairs at the Courant Institute, School of Mathematics, Computing, and Data Science at New York University. His research areas encompass complex and differential geometry, geometric analysis, and partial differential equations, with connections to algebraic geometry and dynamical systems. His work includes specific topics such as Kähler geometry, Calabi-Yau manifolds, almost-complex, symplectic, and Hermitian geometry, geometric flows, complex Monge-Ampère equations, transcendental methods in algebraic geometry, and holomorphic dynamics. Tosatti has contributed extensively to the understanding of Calabi-Yau metrics, Kähler-Ricci flows, and degenerations of complex manifolds, among other areas, and has a significant publication record in leading mathematical journals. His expertise combines deep theoretical insights with a focus on the geometric structures underlying complex manifolds, making him a prominent figure in his field.

Research topics

• Mathematics
• Mathematical analysis
• Computer Science
• Pure mathematics
• Physics
• Statistics
• Geometry

Selected publications

• Gromov-Hausdorff limits of immortal Kähler-Ricci flows

  arXiv (Cornell University) · 2026-02-23

  preprintOpen access

  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.

  PublisherDOI

• Nonlinearizable embeddings of elliptic curves in rational surfaces

  arXiv (Cornell University) · 2026-05-05

  preprintOpen accessSenior author

  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.

  PublisherDOI

• Generic regularity of intermediate complex structure limits

  Journal für die reine und angewandte Mathematik (Crelles Journal) · 2026-03-02

  articleOpen accessSenior author

  Abstract We study certain polarized degenerations of Calabi–Yau manifolds near an intermediate complex structure limit, and improve the potential <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>C</m:mi> <m:mn>0</m:mn> </m:msup> </m:math> C^{0} -convergence to a metric convergence result on the generic region for the corresponding collapsing Ricci-flat Kähler metrics.

  PublisherOA PDFDOI

• Nonlinearizable embeddings of elliptic curves in rational surfaces

  ArXiv.org · 2026-05-05

  articleOpen accessSenior author

  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.

  PublisherOA PDF

• Gromov-Hausdorff limits of immortal Kähler-Ricci flows

  ArXiv.org · 2026-01-01

  articleOpen access

  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.

  PublisherOA PDF

• Special issue in memory of Jean-Pierre Demailly

  Pure and Applied Mathematics Quarterly · 2025-01-01

  articleOpen accessSenior author

  shortly before his 65th birthday.He was an outstanding mathematician, who embodied integrity, enthusiasm, and professionalism, coupled with a deep humanity.This special issue of Pure and Applied Mathematics Quarterly pays tribute to his profound contributions to complex analysis and geometry, and his particularly innovative ability to develop powerful analytic tools and apply them to deep problems in algebraic geometry.Jean-Pierre was born in 1957 in Pronne, a small town in the north of France, and was admitted at the Ecole Normale Suprieure in 1975.Under the guidance of Henri Skoda and in contact with Pierre Lelong, his first work addressed a question posed by Jean-Pierre Serre regarding Stein fiber spaces, earning him early recognition.He then turned to the study of closed positive currents, their Lelong numbers, regularization, and the use of Monge-Ampre operators.He established his famous holomorphic Morse inequalities, developed and employed Hrmander's L 2 techniques, Dolbeault cohomology, and particularly refined curvature calculations to obtain vanishing theorems for line and vector bundles over algebraic varieties.His use of analytic methods later allowed for major advancements towards Fujita's conjecture in the

  PublisherOA PDFDOI

• Regularity of the volume function

  Bulletin of the London Mathematical Society · 2025-10-09

  articleSenior authorCorresponding

  Abstract We prove the optimal 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.

  PublisherDOI

• A Cheng-Yau Type Estimate for the Symplectic Calabi-Yau Equation

  2025-01-01

  book-chapter1st authorCorresponding
  PublisherDOI

• Regularity of the volume function

  ArXiv.org · 2025-05-20

  preprintOpen accessSenior author

  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.

  PublisherOA PDFDOI

• The volume of a divisor and cusp excursions of geodesics in hyperbolic manifolds

  Journal of Algebraic Geometry · 2025-07-18

  articleSenior author

  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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -folds. We find that the volume function exhibits a pathological behavior when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N greater-than-or-equal-to 3"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">N\geq 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we obtain examples of a pseudoeffective <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -divisor <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which the volume of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D plus s upper A"> <mml:semantics> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>+</mml:mo> <mml:mi>s</mml:mi> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">D+sA</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s"> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding="application/x-tex">s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> small and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding="application/x-tex">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ample, oscillates between two powers of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s"> <mml:semantics> <mml:mi>s</mml:mi> <mml:annotation encoding="application/x-tex">s</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and we deduce the sharp regularity of this function answering a question of Lazarsfeld. We also show that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h Superscript 0 Baseline left-parenthesis upper X comma left floor m upper D right floor plus upper A right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>h</mml:mi> <mml:mn>0</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>X</mml:mi> <mml:mo>,</mml:mo> <mml:mrow> <mml:mo>⌊</mml:mo> <mml:mi>m</mml:mi> <mml:mi>D</mml:mi> <mml:mo>⌋</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">h^0(X,\left \lfloor mD \right \rfloor +A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> displays a similar oscillatory behavior as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> increases, showing that several notions of numerical dimensions of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D"> <mml:semantics> <mml:mi>D</mml:mi> <mml:annotation encoding="application/x-tex">D</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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.

  PublisherDOI

Recent grants

• Geometric Partial Differential Equations and Complex Geometry

  NSF · $151k · 2019–2022

• Geometry and Analysis on Calabi-Yau and Hermitian Manifolds

  NSF · $191k · 2013–2016

• Geometric Partial Differential Equations and Complex Geometry

  NSF · $167k · 2022–2024

• Partial Differential Equations on Complex and Symplectic Manifolds

  NSF · $30k · 2012–2013

• Geometric Analysis on Complex Manifolds

  NSF · $225k · 2016–2019

Frequent coauthors

• Ben Weinkove

  28 shared

• Yuguang Zhang

  27 shared

• Simion Filip

  University of Chicago

  16 shared

• Mark Gross

  12 shared

• Yang Li

  Massachusetts Institute of Technology

  10 shared

• J. M. Landsberg

  Texas A&M University

  9 shared

• Edward Frenkel

  Texas A&M University

  9 shared

• J.G. Morse

  University of California, Berkeley

  9 shared