A singular Demailly-Păun theorem. (Un théorème de Demailly-Păun singulier.)(English. French summary)Zbl 1344.32005
The Demailly-Păun theorem [J.-P. Demailly andM. Paun, Ann. Math. (2) 159, No. 3, 1247–1274 (2004;Zbl 1064.32019)] gives a precise numerical description of the Kähler cone \(\mathcal{K}(X)\) of a compact Kähler manifold \(X\). The cone \(\mathcal{K}(X)\) is a connected component of the set of all real \((1,1)\)-cohomology classes \(\{\alpha\}\) on \(X\) with the property \(\int_Y\alpha^{\dim Y}>0\) for every positive-dimensional irreducible analytic subvariety \(Y\) in \(X\). Moreover \(\{\alpha\}\) lies in the closure of \(\mathcal{K}(X)\) if and only if there exists a Kähler metric \(\omega\) on \(X\) such that \(\int_Y\alpha^k\wedge\omega^{\dim Y-k}\geq 0\) for all those \(Y\) and \(1\leq k\leq \dim Y\).
The authors of the article under review present an extension of this theorem to compact analytic subvarieties \(E\) of (not necessarily compact) Kähler manifolds \((M,\omega)\). Let \(\alpha\) be a closed smooth real \((1,1)\)-form on \(M\) with the property \(\int_V\alpha^k\wedge\omega^{\dim V-k}>0\) for all positive-dimensional irreducible analytic subvarieties \(V\subset E\) and for all \(1\leq k\leq\dim V\). Then there exists an open neighborhood \(U\) of \(E\) in \(M\) and a smooth function \(\varphi:U\rightarrow\mathbb R\) such that \(\alpha + i\partial\overline{\partial}\varphi\) is a Kähler metric on \(U\). If, in addition, \(M\) is an open subset of some projective variety, then the condition \(\int_V\alpha^{\dim V}>0\) is sufficient for the conclusion.
The proof uses induction on \(\dim E\) and the resolution of the singularities of \(E\) by a suitable modification \(\hat{M}\rightarrow M\). Main tools are the Demailly-Păun theorem and results and techniques from the authors’ paper [Invent. Math. 202, No. 3, 1167–1198 (2015;Zbl 1341.32016)]. The final construction of \(\varphi\) uses a refinement of a gluing procedure which goes back toR. Richberg [Math. Ann. 175, 257–286 (1968;Zbl 0153.15401)].
The authors of the article under review present an extension of this theorem to compact analytic subvarieties \(E\) of (not necessarily compact) Kähler manifolds \((M,\omega)\). Let \(\alpha\) be a closed smooth real \((1,1)\)-form on \(M\) with the property \(\int_V\alpha^k\wedge\omega^{\dim V-k}>0\) for all positive-dimensional irreducible analytic subvarieties \(V\subset E\) and for all \(1\leq k\leq\dim V\). Then there exists an open neighborhood \(U\) of \(E\) in \(M\) and a smooth function \(\varphi:U\rightarrow\mathbb R\) such that \(\alpha + i\partial\overline{\partial}\varphi\) is a Kähler metric on \(U\). If, in addition, \(M\) is an open subset of some projective variety, then the condition \(\int_V\alpha^{\dim V}>0\) is sufficient for the conclusion.
The proof uses induction on \(\dim E\) and the resolution of the singularities of \(E\) by a suitable modification \(\hat{M}\rightarrow M\). Main tools are the Demailly-Păun theorem and results and techniques from the authors’ paper [Invent. Math. 202, No. 3, 1167–1198 (2015;Zbl 1341.32016)]. The final construction of \(\varphi\) uses a refinement of a gluing procedure which goes back toR. Richberg [Math. Ann. 175, 257–286 (1968;Zbl 0153.15401)].
Reviewer: Eberhard Oeljeklaus (Bremen)
MSC:
| 32J25 | Transcendental methods of algebraic geometry (complex-analytic aspects) |
Cite
References:
| [1] | Campana, F.; Peternell, T., Algebraicity of the ample cone of projective varieties, J. Reine Angew. Math., 407, 160-166 (1990) ·Zbl 0728.14004 |
| [2] | Chiose, I., The Kähler rank of compact complex manifolds, J. Geom. Anal. (2015), in press |
| [3] | Collins, T.; Tosatti, V., Kähler currents and null loci, Invent. Math., 202, 3, 1167-1198 (2015) ·Zbl 1341.32016 |
| [4] | Conlon, R. J.; Hein, H.-J., Asymptotically conical Calabi-Yau manifolds, I, Duke Math. J., 162, 15, 2855-2902 (2013) ·Zbl 1283.53045 |
| [5] | Conlon, R. J.; Hein, H.-J., Asymptotically conical Calabi-Yau manifolds, III, preprint ·Zbl 1283.53045 |
| [7] | Demailly, J.-P.; Păun, M., Numerical characterization of the Kähler cone of a compact Kähler manifold, Ann. of Math. (2), 159, 3, 1247-1274 (2004) ·Zbl 1064.32019 |
| [8] | Kleiman, S. L., Toward a numerical theory of ampleness, Ann. of Math. (2), 84, 293-344 (1966) ·Zbl 0146.17001 |
| [9] | Păun, M., Sur l’effectivité numérique des images inverses de fibrés en droites, Math. Ann., 310, 3, 411-421 (1998) ·Zbl 1023.32014 |
| [10] | Richberg, R., Stetige streng pseudokonvexe Funktionen, Math. Ann., 175, 257-286 (1968) ·Zbl 0153.15401 |
| [11] | Smith, P. A.N., Smoothing plurisubharmonic functions on complex spaces, Math. Ann., 273, 3, 397-413 (1986) ·Zbl 0562.31007 |
| [12] | Varouchas, J., Kähler spaces and proper open morphisms, Math. Ann., 283, 1, 13-52 (1989) ·Zbl 0632.53059 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
