Entropy for Monge–Ampère Measures in the Prescribed Singularities Setting

Eleonora Di Nezza ; Stefano Trapani ; Antonio Trusiani

Symmetry, integrability and geometry: methods and applications, Tome 20 (2024)

Résumé

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 $n=1$.

Détail

Keywords: Kähler manifolds, Monge–Ampère energy, entropy, big classes.

Bibliographie

[1] Bedford E., Taylor B.A., “Fine topology, Šilov boundary, and $(dd^c)^n$”, J. Funct. Anal., 72 (1987), 225–251 | DOI | MR | Zbl

[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 | DOI | MR

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[16] Darvas T., Di Nezza E., Lu C.H., Relative pluripotential theory on compact Kähler manifolds, arXiv: 2303.11584 | MR

[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 | DOI | MR | Zbl

[18] Di Nezza E., “Stability of Monge–Ampère energy classes”, J. Geom. Anal., 25 (2015), 2565–2589, arXiv: 1311.7301 | DOI | MR | Zbl

[19] Di Nezza E., Guedj V., Lu C.H., “Finite entropy vs finite energy”, Comment. Math. Helv., 96 (2021), 389–419, arXiv: 2006.07061 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[23] Guedj V., Zeriahi A., “Intrinsic capacities on compact Kähler manifolds”, J. Geom. Anal., 15 (2005), 607–639, arXiv: math.CV/0401302 | DOI | MR

[24] Guedj V., Zeriahi A., “The weighted Monge–Ampère energy of quasiplurisubharmonic functions”, J. Funct. Anal., 250 (2007), 442–482, arXiv: math.CV/0612630 | DOI | MR | Zbl

[25] Guedj V., Zeriahi A., Degenerate complex Monge–Ampère equations, EMS Tracts in Math., 26, European Mathematical Society (EMS), Zürich, 2017, arXiv: 2305.17955 | DOI | MR | Zbl

[26] Gupta P., “A complete metric topology on relative low energy spaces”, Math. Z., 303 (2023), 56, 27 pp., arXiv: 2206.03999 | DOI | MR | Zbl

[27] Trusiani A., “Kähler–Einstein metrics with prescribed singularities on Fano manifolds”, J. Reine Angew. Math., 793 (2022), 1–57, arXiv: 2006.09130 | DOI | MR | Zbl

[28] Trusiani A., “$L^1$ metric geometry of potentials with prescribed singularities on compact Kähler manifolds”, J. Geom. Anal., 32 (2022), 37, 37 pp., arXiv: 1909.03897 | DOI | MR | Zbl

[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 | DOI | MR

[30] Trusiani A., “The strong topology of $\omega$-plurisubharmonic functions”, Anal. PDE, 16 (2023), 367–405, arXiv: 2002.00665 | DOI | MR | Zbl

[31] Zeriahi A., “Volume and capacity of sublevel sets of a Lelong class of plurisubharmonic functions”, Indiana Univ. Math. J., 50 (2001), 671–703 | DOI | MR | Zbl