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Kähler currents and null loci. (English) Zbl 1341.32016

The article under review investigates the singular loci of cohomology classes sitting at the boundary of the Kähler cone \(\mathcal K_X\) of a compact Kähler manifold \(X\), i.e., the nef cone. There are at least two natural notions of singular locus associated with a nef class \(\alpha\in H^{1,1}(X,\mathbb R)\) such that \(\alpha^{\dim(X)}>0\) (a big class). First, according to J.-P. Demailly [J. Algebr. Geom. 1, No. 3, 361–409 (1992; Zbl 0777.32016)], such a class can be represented by a Kähler current with analytic singularities and it is thus Kähler away from a proper subvariety of \(X\). The smallest subvariety as before is called the non-Kähler locus and is denoted by \(E_{nK}(\alpha)\). Alternatively, we can look at the null locus \(\mathrm{Null}(\alpha)\) of \(\alpha\) which is defined as the union of subvarieties \(V\subset X\) such that \(\alpha^{\dim(V)}\cdot [V]=0\). The main theorem of this article states that \(E_{nK}(\alpha)=\mathrm{Null}(\alpha)\) for a nef and big class \(\alpha\).
Let us note here that the algebraic case (when \(X\) is smooth projective and the class \(\alpha\) given by \(\alpha=[D]\) where \(D\) is a nef \(\mathbb R\)-divisor) has been considered in [M. Nakamaye, Math. Ann. 318, No. 4, 837–847 (2000; Zbl 1063.14008)] and [L. Ein et al., Am. J. Math. 131, No. 3, 607–651 (2009; Zbl 1179.14006)]. In this case, the null locus (and the non Kähler locus) equals to the augmented base locus \(\mathbb B_+(D)\).
The authors are then able to derive the celebrated result of [J.-P. Demailly and M. Paun, Ann. Math. (2) 159, No. 3, 1247–1274 (2004; Zbl 1064.32019)]: the Kähler cone is a connected component of the positive cone \[ \mathcal P_X:=\left\{\alpha\in H^{1,1}(X,\mathbb R)\mid \alpha^{\dim(V)}\cdot [V]>0,\,\forall\,V\subset X\right\}. \]
They also establish the following striking result concerning singularities of the Ricci flow \[ \frac{\partial}{\partial t}\omega=-\mathrm{Ric}(\omega)\quad\mathrm{and}\quad \omega(0)=\omega_0. \] It is known [G. Tian and Z. Zhang, Chin. Ann. Math., Ser. B 27, No. 2, 179–192 (2006; Zbl 1102.53047)] that it has a solution for \(t<T:=\sup\{s>0\mid[\omega_0]-sc_1(X)\in \mathcal K_X\}\). If \(T<\infty\), the limiting class \(\alpha:=[\omega_0]-Tc_1(X)\) is nef but not Kähler. Let us consider now the points where the flow develops singularities: they are the points \(x\in X\) where the curvature of \(\omega_t\) is not bounded near \(x\) when \(t\to T\). It is proven there that the Ricci flow develops singularities exactly along the null locus of \(\alpha\). The end of the article also contains results about degenerations of Ricci-flat metrics on Calabi-Yau manifolds.
The techniques used in this paper belong to pluripotential theory: the main technical tool is an extension result for quasi-plurisubharmonic functions defined on subvarieties of compact Kähler manifolds (see also [T. C. Collins and V. Tosatti, Ann. Fac. Sci. Toulouse, Math. (6) 23, No. 4, 893–905 (2014; Zbl 1333.32023)]). Its proof relies on resolution of singularities and gluing techniques à la Richberg. In addition to its very interesting content, it is worth noting that this article is very clear and well written.

MSC:

32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
32J27 Compact Kähler manifolds: generalizations, classification
32U05 Plurisubharmonic functions and generalizations
32Q15 Kähler manifolds
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[1] Barth, W.P., Hulek, K., Peters, C.A.M., Van de Ven, A.: Compact Complex Surfaces, 2nd edn. Springer, Berlin (2004) · Zbl 1036.14016 · doi:10.1007/978-3-642-57739-0
[2] Berman, R.J., Demailly, J.-P.: Regularity of plurisubharmonic upper envelopes in big cohomology classes. In: Proceedings of Perspectives in Analysis, Geometry, and Topology, Progress in Mathematics, vol. 296, pp. 39-66. Birkhäuser, New York (2012) · Zbl 1258.32010
[3] Boucksom, S.: Cônes positifs des variétés complexes compactes, Ph.D. Thesis, Institut Fourier Grenoble (2002) · Zbl 0574.53042
[4] Boucksom, S.: On the volume of a line bundle. Internat. J. Math. 13(10), 1043-1063 (2002) · Zbl 1101.14008 · doi:10.1142/S0129167X02001575
[5] Boucksom, S.: Divisorial Zariski decompositions on compact complex manifolds. Ann. Sci. École Norm. Sup. (4) 37(1), 45-76 (2004) · Zbl 1054.32010
[6] Boucksom, S., Demailly, J.-P., Păun, M., Peternell, T.: The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension. J. Algebr. Geom. 22(2), 201-248 (2013) · Zbl 1267.32017 · doi:10.1090/S1056-3911-2012-00574-8
[7] Boucksom, S., Eyssidieux, P., Guedj, V., Zeriahi, A.: Monge-Ampère equations in big cohomology classes. Acta Math. 205(2), 199-262 (2010) · Zbl 1213.32025 · doi:10.1007/s11511-010-0054-7
[8] Boucksom, S., Guedj, V.: Regularizing properties of the Kähler-Ricci flow. In: An Introduction to the Kähler-Ricci Flow, Lecture Notes in Mathematcs, vol. 2086, pp. 189-237. Springer, Switzerland (2013) · Zbl 1283.53061
[9] Cacciola, S., Lopez, A.F.: Nakamaye’s theorem on log canonical pairs. Ann. Inst. Fourier (Grenoble) 64(6), 2283-2298 (2014) · Zbl 1325.14017 · doi:10.5802/aif.2913
[10] Cantat, S., Zeghib, A.: Holomorphic actions, Kummer examples, and Zimmer program, Ann. Sci. École Norm. Sup. (4) 45(3), 447-489 (2012) · Zbl 1280.22015
[11] Cao, H.-D.: Deformation of Kähler metrics to Kähler-Einstein metrics on compact Kähler manifolds. Invent. Math. 81(2), 359-372 (1985) · Zbl 0574.53042 · doi:10.1007/BF01389058
[12] P. Cascini, J. McKernan, M. Mustaţă The augmented base locus in positive characteristic. Proc. Edinb. Math. Soc. (2) 57 1, 79-87 · Zbl 1290.14006
[13] Collins, T.C., Greenleaf, A., Pramanik, M.: A multi-dimensional resolution of singularities with applications to analysis. Am. J. Math. 135(5), 1179-1252 (2013) · Zbl 1281.32026 · doi:10.1353/ajm.2013.0042
[14] Collins, T.C., Tosatti, V.: An extension theorem for Kähler currents with analytic singularities. Ann. Fac. Sci. Toulouse Math. 23(4), 893-905 (2014) · Zbl 1333.32023 · doi:10.5802/afst.1429
[15] Coltoiu, M.: Traces of Runge domains on analytic subsets. Math. Ann. 290, 545-548 (1991) · Zbl 0747.32005 · doi:10.1007/BF01459259
[16] Coman, D., Guedj, V., Zeriahi, A.: Extension of plurisubharmonic functions with growth control. J. Reine Angew. Math. 676, 33-49 (2013) · Zbl 1269.32018
[17] Demailly, J.-P.: Singular Hermitian metrics on positive line bundles. In: Proceedings of Complex Algebraic Varieties (Bayreuth, 1990), Lecture Notes in Mathematics, vol. 1507, pp. 87-104. Springer, Berlin (1992) · Zbl 0784.32024
[18] Demailly, J.-P.: Regularization of closed positive currents and intersection theory. J. Algebr. Geom. 1(3), 361-409 (1992) · Zbl 0777.32016
[19] Demailly, J.-P.: Complex Analytic and Differential Geometry. (available on the author’s webpage) · Zbl 0289.32003
[20] Demailly, J.-P., Dinew, S., Guedj, V., Hiep, P.H., Kołodziej, S. Zeriahi, A.: Hölder continuous solutions to Monge-Ampère equations, J. Eur. Math. Soc. (JEMS) 16 4, 619-647 (2014) · Zbl 1296.32012
[21] Demailly, J.-P., Păun, M.: Numerical characterization of the Kähler cone of a compact Kähler manifold. Ann. Math. 159 3, 1247-1274 (2004) · Zbl 1064.32019
[22] Ein, L., Lazarsfeld, R., Mustaţă, M., Nakamaye, M., Popa, M.: Asymptotic invariants of base loci. Ann. Inst. Fourier (Grenoble) 56 6, 1701-1734 · Zbl 1127.14010
[23] Ein, L., Lazarsfeld, R., Mustaţă, M., Nakamaye, M., Popa, M.: Restricted volumes and base loci of linear series. Am. J. Math. 131(3), 607-651 (2009) · Zbl 1179.14006 · doi:10.1353/ajm.0.0054
[24] Enders, J., Müller, R., Topping, P.M.: On type-I singularities in Ricci flow. Comm. Anal. Geom. 19(5), 905-922 (2011) · Zbl 1244.53074 · doi:10.4310/CAG.2011.v19.n5.a4
[25] Eyssidieux, P., Guedj, V., Zeriahi, A.: Singular Kähler-Einstein metrics. J. Am. Math. Soc. 22, 607-639 (2009) · Zbl 1215.32017 · doi:10.1090/S0894-0347-09-00629-8
[26] Feldman, M., Ilmanen, T., Knopf, D.: Rotationally symmetric shrinking and expanding gradient Kähler-Ricci solitons. J. Differ. Geom. 65(2), 169-209 (2003) · Zbl 1069.53036
[27] Fujiki, A.: Closedness of the Douady spaces of compact Kähler spaces, Publ. Res. Inst. Math. Sci. 14 (1978/79) 1, 1-52 · Zbl 0409.32016
[28] Hironaka, H.: Bimeromorphic smoothing of a complex-analytic space. Acta Math. Vietnam 2(2), 103-168 (1977) · Zbl 0407.32006
[29] Hisamoto, T.: Remarks on \[L^2\] L2-jet extension and extension of singular Hermitian metric with semi positive curvature. arXiv:1205.1953 · Zbl 1262.53056
[30] Kołodziej, S.: The complex Monge-Ampère equation. Acta Math. 180, 69-117 (1998) · Zbl 0913.35043 · doi:10.1007/BF02392879
[31] La Nave, G., Tian, G.: Soliton-type metrics and Kähler-Ricci flow on symplectic quotients. J. Reine Angew. Math. (to appear) · Zbl 1339.53069
[32] Lazarsfeld, R.: Positivity in Algebraic Geometry I and II. Springer, Berlin (2004) · Zbl 1066.14021
[33] Nakamaye, M.: Stable base loci of linear series. Math. Ann. 318(4), 837-847 (2000) · Zbl 1063.14008 · doi:10.1007/s002080000149
[34] Păun, M.: Sur l’effectivité numérique des images inverses de fibrés en droites. Math. Ann. 310(3), 411-421 (1998) · Zbl 1023.32014 · doi:10.1007/s002080050154
[35] Phong, D.H., Stein, E.M.: J. Sturm On the growth and stability of real-analytic functions. Am. J. Math. 121(3), 519-554 (1999) · Zbl 1015.26031 · doi:10.1353/ajm.1999.0023
[36] Phong, D.H., Sturm, J.: Algebraic estimates, stability of local zeta functions, and uniform estimates for distribution functions. Ann. Math. (2) 152 1, 277-329 (2000) · Zbl 0995.11065
[37] Phong, D.H., Sturm, J.: On the algebraic constructibility of varieties of integrable rational functions on \[\mathbb{C}^nCn\]. Math. Ann. 323(3), 453-484 (2002) · Zbl 1010.32017 · doi:10.1007/s002080100310
[38] Phong, D.H., Sturm, J.: On the singularities of the pluricomplex Green’s function. In: Proceedings of Advances in Analysis. The Legacy of Elias M. Stein, pp. 419-435. Princeton University Press, Princeton (2014) · Zbl 1329.32020
[39] Richberg, R.: Stetige streng pseudokonvexe Funktionen. Math. Ann. 175, 257-286 (1968) · Zbl 0153.15401 · doi:10.1007/BF02063212
[40] Rong, X., Zhang, Y.: Continuity of extremal transitions and flops for Calabi-Yau manifolds. J. Differ. Geom. 89(2), 233-269 (2011) · Zbl 1264.32021
[41] Rong, X., Zhang, Y.: Degenerations of Ricci-flat Calabi-Yau manifolds. Commun. Contemp. Math. 15(4), 8 (2013) · Zbl 1275.32021 · doi:10.1142/S0219199712500575
[42] Schumacher, G.: Asymptotics of Kähler-Einstein metrics on quasi-projective manifolds and an extension theorem on holomorphic maps. Math. Ann. 311(4), 631-645 (1998) · Zbl 0915.32002 · doi:10.1007/s002080050203
[43] Sherman, M., Weinkove, B.: Interior derivative estimates for the Kähler-Ricci flow. Pac. J. Math. 257(2), 491-501 (2012) · Zbl 1262.53056 · doi:10.2140/pjm.2012.257.491
[44] Siu, Y.-T.: Analyticity of sets associated to Lelong numbers and the extension of closed positive currents. Invent. Math. 27, 53-156 (1974) · Zbl 0289.32003 · doi:10.1007/BF01389965
[45] Siu, Y.-T.: Lectures on Hermitian-Einstein metrics for stable bundles and Kähler-Einstein metrics. In: DMV Seminar, 8. Birkhäuser, Basel, Boston (1987) · Zbl 0631.53004
[46] Song, J.: Finite time extinction of the Kähler-Ricci flow. arXiv:0905.0939 · Zbl 1319.53076
[47] Song, J.: Ricci flow and birational surgery. arXiv:1304.2607 · Zbl 1267.32017
[48] Song, J., Székelyhidi, G., Weinkove, B.: The Kähler-Ricci flow on projective bundles. Int. Math. Res. Not. 2013, no. 2, 243-257 · Zbl 1315.53077
[49] Song, J., Tian, G.: The Kähler-Ricci flow on surfaces of positive Kodaira dimension. Invent. Math. 170(3), 609-653 (2007) · Zbl 1134.53040 · doi:10.1007/s00222-007-0076-8
[50] Song, J., Weinkove, B.: An introduction to the Kähler-Ricci flow. In: An Introduction to the Kähler-Ricci Flow, Lecture Notes in Mathematics, vol. 2086, pp. 89-188. Springer, Switzerland (2013) · Zbl 1288.53065
[51] Song, J., Weinkove, B.: Contracting exceptional divisors by the Kähler-Ricci flow. Duke Math. J. 162(2), 367-415 (2013) · Zbl 1266.53063 · doi:10.1215/00127094-1962881
[52] Tian, G.: New results and problems on Kähler-Ricci flow. In: Proceedings of Géométrie différentielle, physique mathématique, mathématiques et société. II, Astérisque No. 322 pp. 71-92 (2008) · Zbl 1180.53067
[53] Tian, G.: Finite-time singularity of Kähler-Ricci flow. Discret. Contin. Dyn. Syst. 28(3), 1137-1150 (2010) · Zbl 1193.53144 · doi:10.3934/dcds.2010.28.1137
[54] Tian, G., Zhang, Z.: On the Kähler-Ricci flow on projective manifolds of general type. Chin. Ann. Math. Ser. B 27(2), 179-192 (2006) · Zbl 1102.53047 · doi:10.1007/s11401-005-0533-x
[55] Tosatti, V.: Limits of Calabi-Yau metrics when the Kähler class degenerates. J. Eur. Math. Soc. (JEMS) 11(4), 755-776 (2009) · Zbl 1177.32015 · doi:10.4171/JEMS/165
[56] Tosatti, V.: Degenerations of Calabi-Yau metrics, in geometry and physics in Cracow. Acta Phys. Polon. B Proc. Suppl. 4(3), 495-505 (2011) · doi:10.5506/APhysPolBSupp.4.495
[57] Tosatti, V.: Calabi-Yau manifolds and their degenerations. Ann. N. Y. Acad. Sci. 1260, 8-13 (2012) · doi:10.1111/j.1749-6632.2011.06259.x
[58] Varouchas, J.: Kähler spaces and proper open morphisms. Math. Ann. 283(1), 13-52 (1989) · Zbl 0632.53059 · doi:10.1007/BF01457500
[59] Voisin, C.: Hodge Theory and Complex Algebraic Geometry I. Cambridge University Press, Cambridge (2007) · Zbl 1129.14019
[60] Włodarczyk, J.: Resolution of singularities of analytic spaces. In: Proceedings of Gökova Geometry-Topology Conference 2008, pp. 31-63. International Press, Somerville, MA (2009) · Zbl 1187.32023
[61] Wu, D.: Higher canonical asymptotics of Kähler-Einstein metrics on quasi-projective manifolds. Comm. Anal. Geom. 14(4), 795-845 (2006) · Zbl 1116.32019 · doi:10.4310/CAG.2006.v14.n4.a8
[62] Yau, S.-T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation I. Comm. Pure Appl. Math. 31(3), 339-411 (1978) · Zbl 0369.53059 · doi:10.1002/cpa.3160310304
[63] Zhang, Y.: Convergence of Kähler manifolds and calibrated fibrations, Ph.D. thesis, Nankai Institute of Mathematics (2006)
[64] Zhang, Z.: Scalar curvature behavior for finite-time singularity of Kähler-Ricci flow. Mich. Math. J. 59(2), 419-433 (2010) · Zbl 1198.53079 · doi:10.1307/mmj/1281531465
[65] Zhang, Z.: General weak limit for Kähler-Ricci flow. arXiv:1104.2961 · Zbl 1345.53071
[66] Zhang, Z.: Ricci lower bound for Kähler-Ricci flow. Commun. Contemp. Math. 16(2), 11 (2014) · Zbl 1288.53067 · doi:10.1142/S0219199713500533
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