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Projet ANR JCJC Paraplui


Parabolic Pluripotential Theory

Introduction

The goal of this project is to develop a parabolic pluripotential theory motivated by the Minimal Model Program (MMP), whose aim is the (birational) classification of projective manifolds. Inspired by the celebrated work of Birkar-Cascini-Hacon-Mckernan which showed the existence of minimal models for a large class of varieties called varieties of general type, Song and Tian have proposed an analytic analogue making use of the Kähler-Ricci flow. As the models involved in this program are necessarily singular, one is lead to develop a theory of weak Monge-Ampere flows. The first steps of a parabolic pluripotential theory have been built by Guedj-Lu-Zeriahi, allowing one to treat Kawamata log terminal singularities. In this project we aim at developing this theory further, extending it to the most general singularities encountered in the MMP, and studying the geometric convergence of the Monge-Ampère flows.

Members

  1. Quang-Tuan Dang, Postdoc ICTP, Trieste.
  2. Alix Deruelle, Professeur, Université Paris-Saclay
  3. Eleonora Di Nezza, Professeure, Sorbonne Université
  4. Vincent Guedj, Professeur, Université de Toulouse
  5. Chinh H. Lu, (PI), Professeur, Université d'Angers


Articles

  1. Q.T. Dang, T.D. Tô, An iterative construction of complete Kähler--Einstein metrics. arXiv:2410.12599.
  2. S. Boucksom, V. Guedj, and C.H. Lu, Volumes of Bott-Chern classes. arXiv:2406.01090.
  3. P. Åhag, R. Czyż, C.H. Lu, and A. Rashkovskii, Kiselman Minimum Principle and Rooftop Envelopes in Complex Hessian Equations. Math. Z. 308, 70 (2024). arXiv:2405.04948.
  4. P. Åhag, R. Czyż, C.H. Lu, and A. Rashkovskii, Geodesic connectivity and rooftop envelopes in the Cegrell classes. Math. Ann. (2024). arXiv:2405.04384.
  5. Q.T. Dang, Hermitian null loci. arXiv:2404.01126.
  6. V. Guedj, A. Trusiani, Kähler-Einstein metrics with positive curvature near an isolated log terminal singularity. To appear in Compositio Math. arXiv:2306.07900.
  7. V. Guedj, H. Guenancia, A. Zeriahi, Diameters of Kähler currents. To appear in J. Reine Angew. Math. arXiv:2310.20482.
  8. V. Guedj, H. Guenancia, A. Zeriahi, Strict positivity of Kähler-Einstein currents. Forum Math. Sigma 12 (2024), Paper No. e68. arXiv:2305.12422.
  9. T. Darvas, E. Di Nezza, C.H. Lu, Relative pluripotential theory on compact Kähler manifolds, arXiv:2303.11584.
  10. V. Guedj, C.H. Lu, Degenerate complex Hessian equations on compact Hermitian manifolds. To appear in Pure and Applied Mathematics Quarterly. arXiv:2302.03354.
  11. D. Angella, V. Guedj, C.H. Lu, Plurisigned hermitian metrics. Trans. Amer. Math. Soc. 376 (2023), 4631--4659. arXiv:2207.04705.
  12. Q.T. Dang, Pluripotential Chern-Ricci flows. Indiana Univ. Math. J. 73 No. 4 (2024), 1401--1441. arXiv:2201.01150.
  13. E. Di Nezza, C.H. Lu, Geodesic distance and Monge-Ampère measures on contact sets. Anal. Math. 48 (2022), no. 2, 451--488, arXiv:2112.09627.
  14. V. Guedj, C.H. Lu, Quasi-plurisubharmonic envelopes 3: Solving Monge-Ampère equations on hermitian manifolds. J. Reine Angew. Math. 800 (2023), 259--298. arXiv:2107.01938.
  15. V. Guedj, C.H. Lu, Quasi-plurisubharmonic envelopes 2: Bounds on Monge-Ampère volumes. Algebr. Geom. 9, No. 6, 688-713 (2022). arXiv:2106.04272.
  16. V. Guedj, C.H. Lu, Quasi-plurisubharmonic envelopes 1: Uniform estimates on Kähler manifolds. To appear in J. Eur. Math. Soc. arXiv:2106.04273.
  17. Q.T. Dang, Pluripotential Monge-Ampère flows in big cohomology classes. J. Funct. Anal. 282 (2022), no. 6, Paper No. 109373, 65 pp. arXiv:2102.05189.
  18. Q.T. Dang, Continuity of Monge-Ampère potentials in big cohomology classes. Int. Math. Res. Not. IMRN 2022, no. 14, 11180--11201. arXiv:2102.02704.

Activities

  1. Non-Kähler geometry, Orsay 17 May 2022, link.
  2. Geometric flows on complex manifolds, Cortona 11-15 September 2023, link.
  3. Workshop in complex geometry, LAREMA 17-19 April 2024, link.
  4. Summer school Calabi-Yau varieties, Cetraro 29 July -- 02 August 2024, link.
  5. Complex Hermitian Geometry in Angers, 19-13 May 2025, link.