Roles of Plurisubharmonic Functions

Xiangyu Zhou

Informatics and Automation, Mathematical physics and applications, Tome 306 (2019), pp. 304-312.

Résumé

We recall the basics of multiplier ideal sheaves and formulate our recent solution of Demailly's strong openness conjecture on multiplier ideal sheaves and related results. Then we present some applications in complex geometry, including some new results related to the vanishing and finiteness theorems for analytic cohomology groups with multiplier ideal sheaves in the case of pseudo-effective line bundles over holomorphically convex manifolds, and to the generalized Siu lemma and pseudo-effectiveness of the twisted relative pluricanonical bundles and their direct images.

Bibliographie

[1] Berndtsson B., “The openness conjecture and complex Brunn–Minkowski inequalities”, Complex geometry and dynamics: The Abel Symposium 2013, Abel Symp., 10, Springer, Cham, 2015, 29–44 | DOI | MR | Zbl

[2] Berndtsson B., Păun M., “Bergman kernels and the pseudoeffectivity of relative canonical bundles”, Duke Math. J., 145:2 (2008), 341–378 | DOI | MR | Zbl

[3] Blel M., Mimouni S.K., “Singularité et intégrabilité des fonctions plurisousharmoniques”, Ann. Inst. Fourier, 55:2 (2005), 319–351 | DOI | MR | Zbl

[4] Boucksom S., Favre C., Jonsson M., “Valuations and plurisubharmonic singularities”, Publ. Res. Inst. Math. Sci., 44:2 (2008), 449–494 | DOI | MR | Zbl

[5] Demailly J.-P., “Kähler manifolds and transcendental techniques in algebraic geometry”, Proc. Int. Congr. Math. 2006, v. I, Plenary lectures and ceremonies, Eur. Math. Soc., Zürich, 2007, 153–186 | MR | Zbl

[6] Demailly J.-P., Analytic methods in algebraic geometry, Surv. Mod. Math., 1, Int. Press, Somerville, MA, 2012 | MR | Zbl

[7] Demailly J.-P., Complex analytic and differential geometry, Univ. Grenoble I, Inst. Fourier, Grenoble, 2012 https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf

[8] Demailly J-P., Kollár J., “Semi-continuity of complex singularity exponents and Kähler–Einstein metrics on Fano orbifolds”, Ann. sci. Éc. norm. supér. Sér. 4, 34:4 (2001), 525–556 | MR | Zbl

[9] Favre C., Jonsson M., “Valuative analysis of planar plurisubharmonic functions”, Invent. math., 162:2 (2005), 271–311 | DOI | MR | Zbl

[10] Favre C., Jonsson M., “Valuations and multiplier ideals”, J. Amer. Math. Soc., 18:3 (2005), 655–684 | DOI | MR | Zbl

[11] Grauert H., “On Levi's problem and the imbedding of real-analytic manifolds”, Ann. Math. Ser. 2, 68 (1958), 460–472 | DOI | MR | Zbl

[12] Grauert H., Remmert R., Theory of Stein spaces, Grundl. Math. Wiss., 236, Springer, Berlin, 1979 | MR | Zbl

[13] Grauert H., Remmert R., Coherent analytic sheaves, Grundl. Math. Wiss., 265, Springer, Berlin, 1984 | MR | Zbl

[14] Guan Q., Zhou X., “Optimal constant problem in the $L^2$ extension theorem”, C. r. Math. Acad. sci. Paris, 350:15–16 (2012), 753–756 | DOI | MR | Zbl

[15] Guan Q., Zhou X., “Optimal constant in an $L^2$ extension problem and a proof of a conjecture of Ohsawa”, Sci. China. Math., 58:1 (2015), 35–59 | DOI | MR | Zbl

[16] Guan Q., Zhou X., “A solution of an $L^2$ extension problem with an optimal estimate and applications”, Ann. Math. Ser. 2, 181:3 (2015), 1139–1208 | DOI | MR | Zbl

[17] Guan Q., Zhou X., “A proof of Demailly's strong openness conjecture”, Ann. Math. Ser. 2, 182:2 (2015), 605–616 | DOI | MR | Zbl

[18] Guan Q., Zhou X., “Effectiveness of Demailly's strong openness conjecture and related problems”, Invent. math., 202:2 (2015), 635–676 | DOI | MR | Zbl

[19] Guan Q., Zhou X., “Characterization of multiplier ideal sheaves with weights of Lelong number one”, Adv. Math., 285 (2015), 1688–1705 | DOI | MR | Zbl

[20] Guan Q.A., Zhou X.Y., “Strong openness of multiplier ideal sheaves and optimal $L^2$ extension”, Sci. China. Math., 60:6 (2017), 967–976 | DOI | MR | Zbl

[21] Hacon C., Popa M., Schnell C., “Algebraic fiber spaces over abelian varieties: around a recent theorem by Cao and Păun”, Local and global methods in algebraic geometry, Contemp. Math., 712, Amer. Math. Soc., Providence, RI, 2018, 143–195 | DOI | MR | Zbl

[22] Hörmander L., An introduction to complex analysis in several variables, North-Holland Math. Libr., 7, 3rd ed., North-Holland, Amsterdam, 1990 | MR | Zbl

[23] Jonsson M., Mustaţă M., “An algebraic approach to the openness conjecture of Demailly and Kollár”, J. Inst. Math. Jussieu, 13:1 (2014), 119–144 | DOI | MR | Zbl

[24] Jonsson M., Mustaţă M., “Valuations and asymptotic invariants for sequences of ideals”, Ann. Inst. Fourier, 62:6 (2012), 2145–2209 | DOI | MR | Zbl

[25] Kiselman C.O., “Plurisubharmonic functions and potential theory in several complex variables”, Development of mathematics 1950–2000, Birkhäuser, Basel, 2000, 655–714 | DOI | MR | Zbl

[26] Meng X., Zhou X., “Pseudo-effective line bundles over holomorphically convex manifolds”, J. Algebr. Geom., 28:1 (2019), 169–200 | DOI | MR | Zbl

[27] Ohsawa T., “Finiteness theorems on weakly 1-complete manifolds”, Publ. Res. Inst. Math. Sci., 15 (1979), 853–870 | DOI | MR | Zbl

[28] Ohsawa T., $L^2$ approaches in several complex variables: Development of Oka–Cartan theory by $L^2$ estimates for the $\bar \partial $ operator, Springer Monogr. Math., Springer, Tokyo, 2015 | DOI | MR | Zbl

[29] Păun M., Takayama S., “Positivity of twisted relative pluricanonical bundles and their direct images”, J. Algebr. Geom., 27:2 (2018), 211–272 | MR

[30] Peternell T., “On strongly pseudo-convex Kähler manifolds”, Invent. math., 70 (1982), 157–168 | DOI | MR | Zbl

[31] Peternell T., “Der Kodairasche Verschwindungssatz auf streng pseudokonvexen Räumen. I”, Math. Ann., 270 (1985), 87–96 | DOI | MR | Zbl

[32] Phong D.H., Sturm J., “Algebraic estimates, stability of local zeta functions, and uniform estimates for distribution functions”, Ann. Math. Ser. 2, 152:1 (2000), 277–329 | DOI | MR | Zbl

[33] Prill D., “The divisor class groups of some rings of holomorphic functions”, Math. Z., 121 (1971), 58–80 | DOI | MR | Zbl

[34] A. G. Sergeev and X. Zhou, “Invariant domains of holomorphy: Twenty years later”, Proc. Steklov Inst. Math., 285 (2014), 241–250 | DOI | DOI | MR | Zbl

[35] Siu Y.-T., “The Fujita conjecture and the extension theorem of Ohsawa–Takegoshi”, Geometric complex analysis, Proc. 3rd Int. Res. Inst. (Math. Soc. Japan, Hayama, 1995), World Scientific, Singapore, 1996, 577–592 | MR | Zbl

[36] Siu Y.-T., “Invariance of plurigenera”, Invent. math., 134:3 (1998), 661–673 | DOI | MR | Zbl

[37] Siu Y.-T., “Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type”, Complex geometry: Collection of papers dedicated to Hans Grauert, Springer, Berlin, 2002, 223–277 | MR | Zbl

[38] Siu Y.-T., “Some recent transcendental techniques in algebraic and complex geometry”, Proc. Int. Congr. Math. 2002, v. I, Plenary lectures and ceremonies, High. Educ. Press, Beijing, 2002, 439–448 | MR | Zbl

[39] Siu Y.-T., “Invariance of plurigenera and torsion-freeness of direct image sheaves of pluricanonical bundles”, Finite or infinite dimensional complex analysis and applications, Adv. Complex Anal. Appl., 2, Kluwer, Boston, MA, 2004, 45–83 | MR | Zbl

[40] Siu Y.-T., “Multiplier ideal sheaves in complex and algebraic geometry”, Sci. China. Ser. A, 48 (Suppl.) (2005), 1–31 | DOI | MR | Zbl

[41] V. S. Vladimirov, Methods of the Theory of Functions of Many Complex Variables, Nauka, Moscow, 1964 (in Russian) | MR

[42] X.-Y. Zhou, “A proof of the extended future tube conjecture”, Izv. Math., 62:1 (1998), 201–213 | DOI | DOI | MR | MR | Zbl

[43] Zhou X., “Some results related to group actions in several complex variables”, Proc. Int. Congr. Math. 2002, v. II, Invited lectures, High. Educ. Press, Beijing, 2002, 743–753 | MR | Zbl

[44] Zhou X., “A survey on $L^2$ extension problem”, Complex geometry and dynamics: The Abel Symposium 2013, Abel Symp., 10, Springer, Cham, 2015, 291–309 | DOI | MR | Zbl

[45] Zhou X., Zhu L., “A generalized Siu's lemma”, Math. Res. Lett., 24:6 (2017), 1897–1913 | DOI | MR | Zbl

[46] Zhou X., Zhu L., “An optimal $L^2$ extension theorem on weakly pseudoconvex Kähler manifolds”, J. Diff. Geom., 110:1 (2018), 135–186 | DOI | MR | Zbl

[47] Zhou X., Zhu L., “Siu's lemma, optimal $L^2$ extension and applications to twisted pluricanonical sheaves”, Math. Ann., 2018 | DOI