Valentino Tosatti

Professor of Mathematics
Courant Institute of Mathematical Sciences
New York University
251 Mercer St, New York, NY 10012
Office: Warren Weaver Hall 1001
Email: tosatti@cims.nyu.edu

I do research in complex and differential geometry, geometric analysis and PDEs, and their connections to algebraic geometry and dynamical systems.
Some specific topics are: Kähler geometry, Calabi-Yau manifolds, almost-complex, symplectic and Hermitian geometry, geometric flows, complex Monge-Ampère equations, transcendental methods in algebraic geometry, dynamics on K3 surfaces.

My CV is available here.

Teaching

Fall 2023 - Analysis
Spring 2023 - Analysis
Fall 2022 - Differential Geometry I

Publications and Preprints

(with J. Lesieutre and S. Filip) The volume of a divisor and cusp excursions of geodesics in hyperbolic manifolds
[arXiv] preprint 2023
(with H.-J. Hein) Smooth asymptotics for collapsing Calabi-Yau metrics
[arXiv] preprint 2021
(with Y. Li) Special Kähler geometry and holomorphic Lagrangian fibrations
[arXiv] to appear in C. R. Math. Acad. Sci. Paris
Semipositive line bundles and (1,1)-classes
[arXiv] to appear in Acta Math. Sin. (Engl. Ser.)
(with Y. Li) Diameter bounds for degenerating Calabi-Yau metrics
[arXiv] to appear in J. Differential Geom.
(with S. Filip) Gaps in the support of canonical currents on projective K3 surfaces
[arXiv] [journal] J. Geom. Anal. 34 (2024), no. 3, Paper No. 76, 14 pp.
(with S. Filip) Canonical currents and heights for K3 surfaces
[arXiv] [journal] Camb. J. Math. 11 (2023), no.3, 699-794.
(with Y. Li) On the collapsing of Calabi-Yau manifolds and Kähler-Ricci flows
[arXiv] [journal] J. Reine Angew. Math. 800 (2023), 155-192.
(with D. Angella) Leafwise flat forms on Inoue-Bombieri surfaces
[arXiv] [journal] J. Funct. Anal. 285 (2023), no.5, Paper No. 110015
(with B. Weinkove) The Chern-Ricci flow
[arXiv] [journal] Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 33 (2022), no.1, 73-107.
(with T. Collins) Restricted volumes on Kähler manifolds
[arxiv] [journal] Ann. Fac. Sci. Toulouse Math. 31 (2022), no.3, 907-947.
(with S. Filip) Kummer rigidity for K3 surface automorphisms via Ricci-flat metrics
[arxiv] [journal] Amer. J. Math. 143 (2021), no.5, 1431-1462.
(with S. Kołodziej) Morse-type integrals on non-Kähler manifolds
[arXiv] [journal] Pure Appl. Math. Q. 17 (2021), no.3, 991-1004.
(with B. Weinkove) The complex Monge-Ampère equation with a gradient term
[arXiv] [journal] Pure Appl. Math. Q. 17 (2021), no.3, 1005-1024.
Ricci-flat metrics and dynamics on K3 surfaces
[arXiv] [journal] Boll. Unione Mat. Ital. 14 (2021), no.1, 191-209.
(with H.-J. Hein) Higher-order estimates for collapsing Calabi-Yau metrics
[arxiv] [journal] Camb. J. Math. 8 (2020), no.4, 683-773.
(with M. Gross and Y. Zhang) Geometry of twisted Kähler-Einstein metrics and collapsing
[arXiv] [journal] Comm. Math. Phys. 380 (2020), no.3, 1401-1438.
(with Y. Zhang) Collapsing hyperkähler manifolds
[arxiv] [journal] Ann. Sci. Éc. Norm. Supér. 53 (2020), no.3, 751-786.
Collapsing Calabi-Yau manifolds
[arXiv] [journal] Surveys in Differential Geometry 23 (2018), 305-337, International Press, 2020.
(with B. Weinkove) Hermitian metrics, (n-1,n-1) forms and Monge-Ampère equations
[arXiv] [journal] J. Reine Angew. Math. 755 (2019), 67-101.
(with N. McCleerey) Pluricomplex Green's functions and Fano manifolds
[arxiv] [journal] Épijournal Geom. Algébrique 3 (2019), Art. 9, 15pp.
(with J. Chu and B. Weinkove) The Monge-Ampère equation for non-integrable almost complex structures
[arxiv] [journal] J. Eur. Math. Soc. (JEMS) 21 (2019), no.7, 1949-1984.
Orthogonality of divisorial Zariski decompositions for classes with volume zero
[arxiv] [journal] Tohoku Math. J. 71 (2019), no.1, 1-8.
(with S. Filip) Smooth and rough positive currents
[arxiv] [journal] Ann. Inst. Fourier (Grenoble) 68 (2018), no.7, 2981-2999.
KAWA lecture notes on the Kähler-Ricci flow
[arxiv] [journal] Ann. Fac. Sci. Toulouse Math. 27 (2018), no.2, 285-376.
(with B. Weinkove) The Aleksandrov-Bakelman-Pucci estimate and the Calabi-Yau equation
[arxiv] [book] in Nonlinear Analysis in Geometry and Applied Mathematics, Part 2, 147-158, Harvard CMSA Ser. Math. 2, International Press, 2018.
Regularity of envelopes in Kähler classes
[arxiv] [journal] Math. Res. Lett. 25 (2018), no.1, 281-289.
Nakamaye's Theorem on complex manifolds
[arxiv] [book] in Algebraic Geometry: Salt Lake City 2015. Part 1, 633-655, Proc. Sympos. Pure Math. 97.1, American Mathematical Society, 2018.
(with B. Weinkove and X. Yang) The Kähler-Ricci flow, Ricci-flat metrics and collapsing limits
[arxiv] [journal] Amer. J. Math. 140 (2018), no.3, 653-698.
(with Y. Zhang) Finite time collapsing of the Kähler-Ricci flow on threefolds
[arxiv] [journal] Ann. Sc. Norm. Super. Pisa Cl. Sci. 18 (2018), no.1, 105-118.
(with J. Chu and B. Weinkove) C^1,1 regularity for degenerate complex Monge-Ampère equations and geodesics rays
[arxiv] [journal] Comm. Partial Differential Equations 43 (2018), no.2, 292-312.
(with G. Székelyhidi and B. Weinkove) Gauduchon metrics with prescribed volume form
[arxiv] [journal] Acta Math. 219 (2017), no.1, 181-211.
(with X. Yang) An extension of a theorem of Wu-Yau
[arXiv] [journal] J. Differential Geom. 107 (2017), no.3, 573-579.
(with J. Chu and B. Weinkove) On the C^1,1 regularity of geodesics in the space of Kähler metrics
[arxiv] [journal] Ann. PDE 3 (2017), no.2, 3:15.
Uniqueness of CPⁿ
[arxiv] [journal] Expo. Math. 35 (2017), no.1, 1-12.
(with B. Weinkove) The Monge-Ampère equation for (n-1)-plurisubharmonic functions on a compact Kähler manifold
[arXiv] [journal] J. Amer. Math. Soc. 30 (2017), no.2, 311-346.
(with S. Fang, B. Weinkove and T. Zheng) Inoue surfaces and the Chern-Ricci flow
[arxiv] [journal] J. Funct. Anal. 271 (2016), no.11, 3162-3185.
The Calabi-Yau Theorem and Kähler currents
[arxiv] [journal] Adv. Theor. Math. Phys. 20 (2016), no.2, 381-404.
(with M. Gross and Y. Zhang) Gromov-Hausdorff collapsing of Calabi-Yau manifolds
[arXiv] [journal] Comm. Anal. Geom. 24 (2016), no.1, 93-113.
(with T. Collins) A singular Demailly-Păun theorem
[arxiv] [journal] C. R. Math. Acad. Sci. Paris 354 (2016), no.1, 91-95.
(with H.-J. Hein) Remarks on the collapsing of torus fibered Calabi-Yau manifolds
[arxiv] [journal] Bull. Lond. Math. Soc. 47 (2015), no.6, 1021-1027.
(with T. Collins) Kähler currents and null loci
[arXiv] [journal] Invent. Math. 202 (2015), no.3, 1167-1198.
(with Y. Zhang) Infinite time singularities of the Kähler-Ricci flow
[arxiv] [journal] Geom. Topol. 19 (2015), no.5, 2925-2948.
Families of Calabi-Yau manifolds and canonical singularities
[arxiv] [journal] Int. Math. Res. Not. IMRN (2015) no.20, 10586-10594.
(with Y. Wang, B. Weinkove and X. Yang) C^2,α estimates for nonlinear elliptic equations in complex and almost complex geometry
[arxiv] [journal] Calc. Var. Partial Differential Equations 54 (2015), no.1, 431-453.
Non-Kähler Calabi-Yau manifolds
[arxiv] [book] Contemp. Math. 644 (2015), 261-277.
(with B. Weinkove and X. Yang) Collapsing of the Chern-Ricci flow on elliptic surfaces
[arXiv] [journal] Math. Ann. 362 (2015), no.3-4, 1223-1271.
(with B. Weinkove) On the evolution of a Hermitian metric by its Chern-Ricci form
[arXiv] [journal] J. Differential Geom. 99 (2015), no.1, 125-163.
(with Y. Zhang) Triviality of fibered Calabi-Yau manifolds without singular fibers
[arXiv] [journal] Math. Res. Lett. 21 (2014), no.4, 905-918.
(with T. Collins) An extension theorem for Kähler currents with analytic singularities
[arXiv] [journal] Ann. Fac. Sci. Toulouse Math. 23 (2014), no.4, 893-905.
(with B. Weinkove) The Chern-Ricci flow on complex surfaces
[arXiv] [journal] Compos. Math. 149 (2013), no.12, 2101-2138.
(with M. Gross and Y. Zhang) Collapsing of abelian fibred Calabi-Yau manifolds
[arXiv] [journal] Duke Math J. 162 (2013), no.3, 517-551.
Blowup behavior of the Kähler-Ricci flow on Fano manifolds
[arXiv] [journal] Univ. Iagel. Acta Math. 50 (2012), 117-126.
Calabi-Yau manifolds and their degenerations
[arXiv] [journal] Ann. N.Y. Acad. Sci. 1260 (2012), 8-13.
(with B. Weinkove) Plurisubharmonic functions and nef classes on complex manifolds
[arXiv] [journal] Proc. Amer. Math. Soc. 140 (2012), no.11, 4003-4010.
The K-energy on small deformations of constant scalar curvature Kähler manifolds
[arXiv] [book] in Advances in Geometric Analysis, 139-150, Advanced Lectures in Math. 21, International Press, 2012.
Kähler-Einstein metrics on Fano surfaces
[arXiv] [journal] Expo. Math. 30 (2012), no.1, 11-31.
(with G. Székelyhidi) Regularity of weak solutions of a complex Monge-Ampère equation
[arXiv] [journal] Anal. PDE 4 (2011), no.3, 369-378.
Degenerations of Calabi-Yau metrics
[arXiv] [journal] in Geometry and Physics in Cracow, Acta Phys. Polon. B Proc. Suppl. 4 (2011), no.3, 495-505.
(with B. Weinkove) The Calabi-Yau equation on the Kodaira-Thurston manifold
[arXiv] [journal] J. Inst. Math. Jussieu 10 (2011), no.2, 437-447.
(with B. Weinkove)The Calabi-Yau equation, symplectic forms and almost complex structures
[arXiv] [book] in Geometry and Analysis, No. 1, 475-493, Advanced Lectures in Math. 17, International Press, 2011.
(with B. Weinkove) The complex Monge-Ampère equation on compact Hermitian manifolds
[arXiv] [journal] J. Amer. Math. Soc. 23 (2010), no.4, 1187-1195.
(with B. Weinkove) Estimates for the complex Monge-Ampère equation on Hermitian and balanced manifolds
[arXiv] [journal] Asian J. Math. 14 (2010), no.1, 19-40.
Adiabatic limits of Ricci-flat Kähler metrics
[arXiv] [journal] J. Differential Geom. 84 (2010), no.2, 427-453.
Kähler-Ricci flow on stable Fano manifolds
[arXiv] [journal] J. Reine Angew. Math. 640 (2010), 67-84.
Limits of Calabi-Yau metrics when the Kähler class degenerates
[arXiv] [journal] J. Eur. Math. Soc. (JEMS) 11 (2009), no.4, 755-776.
(with B. Weinkove and S.-T. Yau) Taming symplectic forms and the Calabi-Yau equation
[arXiv] [journal] Proc. London Math. Soc. 97 (2008), no.2, 401-424.
A general Schwarz Lemma for almost-Hermitian manifolds
[arXiv] [journal] Comm. Anal. Geom. 15 (2007), no.5, 1063-1086.
(with B. Weinkove) The Calabi Flow with small initial energy
[arXiv] [journal] Math. Res. Lett. 14 (2007), no.6, 1033-1039.
On the Critical Points of the E_k Functionals in Kähler Geometry
[arXiv] [journal] Proc. Amer. Math. Soc. 135 (2007), no.12, 3985-3988.
(with L. Pietronero, E. Tosatti and A. Vespignani) Explaining the uneven distribution of numbers in nature
[arXiv] [journal] Phys. A, 293 (2001), 297-304.

Thesis, notes and other material

Geometry of complex Monge-Ampère equations
[PDF] 4 June 2009, PhD Thesis, Harvard University, Advisor: Prof. Shing-Tung Yau.
Extremal Sobolev inequalities and applications
[PDF] 12 March 2005, Minor Thesis, Harvard University, Advisor: Prof. Ben Weinkove.
Index theorems for morphisms of vector bundles and foliations
[PDF] (in Italian) 10 June 2004, Laurea Thesis, University of Pisa, Advisor: Prof. Marco Abate.
A finiteness theorem for polarized manifolds
(with Y. Zhang) [PDF] [book] 28 July 2015, a short note, appendix to this paper, published in Uniformization, Riemann-Hilbert Correspondence, Calabi-Yau Manifolds & Picard-Fuchs Equations, 551-592, Advanced Lectures in Math. 42, International Press, 2018.
Metric limits of Calabi-Yau manifolds
[journal] Notices Amer. Math. Soc. 65 (2018), no.4, 413-414.
Extension of semi-flat forms across singular fibers
[PDF] [journal] 27 March 2018, a short note, appendix to this paper, published in J. Eur. Math. Soc. (JEMS) 25 (2023), no. 2, 633-679.
Jean-Pierre Demailly (1957-2022)
[journal] Notices Amer. Math. Soc. 70 (2023), no.5, 791-792.

Editorial boards

Links

Other events that I recently co-organized

My Erdős number is 3.