Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 039, 19 pages      arXiv:2310.10152      https://doi.org/10.3842/SIGMA.2024.039
Contribution to the Special Issue on Differential Geometry Inspired by Mathematical Physics in honor of Jean-Pierre Bourguignon for his 75th birthday

Entropy for Monge-Ampère Measures in the Prescribed Singularities Setting

Eleonora Di Nezza a, Stefano Trapani b and Antonio Trusiani c
a) IMJ-PRG, Sorbonne Université & DMA, École Normale Supérieure, Université PSL, CNRS, 4 place Jussieu & 45 Rue d'Ulm, 75005 Paris, France
b) Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy
c) Chalmers University of Technology, Chalmers tvärgata 3, 41296 Göteborg, Sweden

Received October 16, 2023, in final form May 04, 2024; Published online May 08, 2024

Abstract
In this note, we generalize the notion of entropy for potentials in a relative full Monge-Ampère mass $\mathcal{E}(X, \theta, \phi)$, for a model potential $\phi$. We then investigate stability properties of this condition with respect to blow-ups and perturbation of the cohomology class. We also prove a Moser-Trudinger type inequality with general weight and we show that functions with finite entropy lie in a relative energy class $\mathcal{E}^{\frac{n}{n-1}}(X, \theta, \phi)$ (provided $n>1$), while they have the same singularities of $\phi$ when $n=1$.

Key words: Kähler manifolds; Monge-Ampère energy; entropy; big classes.

pdf (461 kb)   tex (28 kb)  

References

  1. Bedford E., Taylor B.A., Fine topology, Šilov boundary, and $(dd^c)^n$, J. Funct. Anal. 72 (1987), 225-251.
  2. Berman R.J., Berndtsson B., Moser-Trudinger type inequalities for complex Monge-Ampère operators and Aubin's ''hypothèse fondamentale'', Ann. Fac. Sci. Toulouse Math. (6) 31 (2022), 595-645, arXiv:1109.1263.
  3. Berman R.J., Boucksom S., Eyssidieux P., Guedj V., Zeriahi A., Kähler-Einstein metrics and the Kähler-Ricci flow on log Fano varieties, J. Reine Angew. Math. 751 (2019), 27-89, arXiv:1111.7158.
  4. Berman R.J., Boucksom S., Guedj V., Zeriahi A., A variational approach to complex Monge-Ampère equations, Publ. Math. Inst. Hautes Études Sci. 117 (2013), 179-245, arXiv:0907.4490.
  5. Berman R.J., Boucksom S., Jonsson M., A variational approach to the Yau-Tian-Donaldson conjecture, J. Amer. Math. Soc. 34 (2021), 605-652, arXiv:1509.04561.
  6. Berman R.J., Darvas T., Lu C.H., Convexity of the extended K-energy and the large time behavior of the weak Calabi flow, Geom. Topol. 21 (2017), 2945-2988, arXiv:1510.01260.
  7. Boucksom S., Cônes positifs des variétés complexes compactes, Ph.D. Thesis, Université Joseph-Fourier-Grenoble I, 2002.
  8. Boucksom S., Divisorial Zariski decompositions on compact complex manifolds, Ann. Sci. École Norm. Sup. (4) 37 (2004), 45-76.
  9. Boucksom S., Eyssidieux P., Guedj V., Zeriahi A., Monge-Ampère equations in big cohomology classes, Acta Math. 205 (2010), 199-262, arXiv:0812.3674.
  10. Chen X., Cheng J., On the constant scalar curvature Kähler metrics (I)—A priori estimates, J. Amer. Math. Soc. 34 (2021), 909-936, arXiv:1712.06697.
  11. Chen X., Cheng J., On the constant scalar curvature Kähler metrics (II)—Existence results, J. Amer. Math. Soc. 34 (2021), 937-1009, arXiv:1801.00656.
  12. Darvas T., Di Nezza E., Lu C.H., $L^1$ metric geometry of big cohomology classes, Ann. Inst. Fourier (Grenoble) 68 (2018), 3053-3086, arXiv:1802.00087.
  13. Darvas T., Di Nezza E., Lu C.H., Monotonicity of nonpluripolar products and complex Monge-Ampère equations with prescribed singularity, Anal. PDE 11 (2018), 2049-2087, arXiv:1705.05796.
  14. Darvas T., Di Nezza E., Lu C.H., On the singularity type of full mass currents in big cohomology classes, Compos. Math. 154 (2018), 380-409, arXiv:1606.01527.
  15. Darvas T., Di Nezza E., Lu C.H., The metric geometry of singularity types, J. Reine Angew. Math. 771 (2021), 137-170, arXiv:1909.00839.
  16. Darvas T., Di Nezza E., Lu C.H., Relative pluripotential theory on compact Kähler manifolds, arXiv:2303.11584.
  17. Darvas T., Rubinstein Y.A., Kiselman's principle, the Dirichlet problem for the Monge-Ampère equation, and rooftop obstacle problems, J. Math. Soc. Japan 68 (2016), 773-796, arXiv:1405.6548.
  18. Di Nezza E., Stability of Monge-Ampère energy classes, J. Geom. Anal. 25 (2015), 2565-2589, arXiv:1311.7301.
  19. Di Nezza E., Guedj V., Lu C.H., Finite entropy vs finite energy, Comment. Math. Helv. 96 (2021), 389-419, arXiv:2006.07061.
  20. Di Nezza E., Lu C.H., Geodesic distance and Monge-Ampère measures on contact sets, Anal. Math. 48 (2022), 451-488, arXiv:2112.09627.
  21. Di Nezza E., Trapani S., The regularity of envelopes, Ann. Sci. Éc. Norm. Supér., to appear, arXiv:2110.14314.
  22. Di Nezza E., Trapani S., Monge-Ampère measures on contact sets, Math. Res. Lett. 28 (2021), 1337-1352, arXiv:1912.12720.
  23. Guedj V., Zeriahi A., Intrinsic capacities on compact Kähler manifolds, J. Geom. Anal. 15 (2005), 607-639, arXiv:math.CV/0401302.
  24. Guedj V., Zeriahi A., The weighted Monge-Ampère energy of quasiplurisubharmonic functions, J. Funct. Anal. 250 (2007), 442-482, arXiv:math.CV/0612630.
  25. Guedj V., Zeriahi A., Degenerate complex Monge-Ampère equations, EMS Tracts in Math., Vol. 26, European Mathematical Society (EMS), Zürich, 2017, arXiv:2305.17955.
  26. Gupta P., A complete metric topology on relative low energy spaces, Math. Z. 303 (2023), 56, 27 pages, arXiv:2206.03999.
  27. Trusiani A., Kähler-Einstein metrics with prescribed singularities on Fano manifolds, J. Reine Angew. Math. 793 (2022), 1-57, arXiv:2006.09130.
  28. Trusiani A., $L^1$ metric geometry of potentials with prescribed singularities on compact Kähler manifolds, J. Geom. Anal. 32 (2022), 37, 37 pages, arXiv:1909.03897.
  29. Trusiani A., Continuity method with movable singularities for classical complex Monge-Ampère equations, Indiana Univ. Math. J. 72 (2023), 1577-1625, arXiv:2006.09120.
  30. Trusiani A., The strong topology of $\omega$-plurisubharmonic functions, Anal. PDE 16 (2023), 367-405, arXiv:2002.00665.
  31. Zeriahi A., Volume and capacity of sublevel sets of a Lelong class of plurisubharmonic functions, Indiana Univ. Math. J. 50 (2001), 671-703.

Previous article  Next article  Contents of Volume 20 (2024)