Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-22T21:27:13.720Z Has data issue: false hasContentIssue false
  • English
  • Français

Positivity properties of the bundle of logarithmic tensors on compact Kähler manifolds

Part of: Complex manifolds Pluripotential theory

Published online by Cambridge University Press:  21 September 2016

Frédéric Campana  and
Mihai Păun
Frédéric Campana
Affiliation:
Institut Elie Cartan, Nancy, Université de Lorraine, France email [email protected] KIAS scholar, KIAS, 85 Hoegiro, Dongdaemun-gu, Seoul 130-722, South Korea
Mihai Păun
Affiliation:
Korea Institute for Advanced Study, School of Mathematics, 85 Hoegiro, Dongdaemun-gu, Seoul 130-722, Korea email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let $X$ be a compact Kähler manifold, endowed with an effective reduced divisor $B=\sum Y_{k}$ having simple normal crossing support. We consider a closed form of $(1,1)$ -type $\unicode[STIX]{x1D6FC}$ on $X$ whose corresponding class $\{\unicode[STIX]{x1D6FC}\}$ is nef, such that the class $c_{1}(K_{X}+B)+\{\unicode[STIX]{x1D6FC}\}\in H^{1,1}(X,\mathbb{R})$ is pseudo-effective. A particular case of the first result we establish in this short note states the following. Let $m$ be a positive integer, and let $L$ be a line bundle on $X$ , such that there exists a generically injective morphism $L\rightarrow \bigotimes ^{m}T_{X}^{\star }\langle B\rangle$ , where we denote by $T_{X}^{\star }\langle B\rangle$ the logarithmic cotangent bundle associated to the pair $(X,B)$ . Then for any Kähler class $\{\unicode[STIX]{x1D714}\}$ on $X$ , we have the inequality

$$\begin{eqnarray}\displaystyle \int _{X}c_{1}(L)\wedge \{\unicode[STIX]{x1D714}\}^{n-1}\leqslant m\int _{X}(c_{1}(K_{X}+B)+\{\unicode[STIX]{x1D6FC}\})\wedge \{\unicode[STIX]{x1D714}\}^{n-1}.\end{eqnarray}$$
If $X$ is projective, then this result gives a generalization of a criterion due to Y. Miyaoka, concerning the generic semi-positivity: under the hypothesis above, let $Q$ be the quotient of $\bigotimes ^{m}T_{X}^{\star }\langle B\rangle$ by $L$ . Then its degree on a generic complete intersection curve $C\subset X$ is bounded from below by
$$\begin{eqnarray}\displaystyle \biggl(\frac{n^{m}-1}{n-1}-m\biggr)\int _{C}(c_{1}(K_{X}+B)+\{\unicode[STIX]{x1D6FC}\})-\frac{n^{m}-1}{n-1}\int _{C}\unicode[STIX]{x1D6FC}.\end{eqnarray}$$
 As a consequence, we obtain a new proof of one of the main results of our previous work [F. Campana and M. Păun, Orbifold generic semi-positivity: an application to families of canonically polarized manifolds, Ann. Inst. Fourier (Grenoble) 65 (2015), 835–861].

Keywords

tensorscurvatureMonge–Ampère equations

MSC classification

Primary: 32Q05: Negative curvature manifolds 32Q10: Positive curvature manifolds 32Q15: Kähler manifolds 32Q20: Kähler-Einstein manifolds 32U05: Plurisubharmonic functions and generalizations
Secondary: 32U15: General pluripotential theory
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 11 , November 2016 , pp. 2350 - 2370
Copyright
© The Authors 2016 

References

Birkar, C., Cascini, P., Hacon, C. and Mckernan, J., Existence of minimal models for varieties of log general type , J. Amer. Math. Soc. 23 (2010), 405–468.CrossRefGoogle Scholar
Boucksom, S., Demailly, J.-P., Păun, M. and Peternell, T., The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension , J. Algebraic Geom. 22 (2013), 201–248.CrossRefGoogle Scholar
Campana, F., Guenancia, H. and Păun, M., Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields , Ann. Sci. Éc. Norm. Supér (4) 46 (2013), 879–916.CrossRefGoogle Scholar
Campana, F. and Păun, M., Orbifold generic semi-positivity: an application to families of canonically polarized manifolds , Ann. Inst. Fourier (Grenoble) 65 (2015), 835–861.CrossRefGoogle Scholar
Demailly, J.-P., Estimations L 2 pour l’opérateur ∂̄ d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète , Ann. Sci. Éc. Norm. Supér (4) 15 (1982), 457–511.CrossRefGoogle Scholar
Demailly, J.-P., Regularization of closed positive currents and intersection theory , J. Algebraic Geom. 1 (1992), 361–409.Google Scholar
Donaldson, S., Anti self-dual Yang Mills connections over complex algebraic surfaces and stable vector bundles , Proc. Lond. Math. Soc. (3) 50 (1985), 1–26.CrossRefGoogle Scholar
Enoki, I., Stability and negativity for tangent sheaves of minimal Kähler spaces , in Geometry and analysis on manifolds (Katata/Kyoto, 1987), Lecture Notes in Mathematics, vol. 1339 (Springer, Berlin, 1988), 118–126.CrossRefGoogle Scholar
Guenancia, H., Kähler–Einstein metrics with cone singularities on klt pairs , Int. J. Math. 24 (2013), 1350035.CrossRefGoogle Scholar
Guenancia, H. and Păun, M., Conic singularities metrics with prescribed Ricci curvature: the case of general cone angles along normal crossing divisors , J. Differential Geom. 103 (2016), 15–57.CrossRefGoogle Scholar
Kobayashi, S., Differential geometry of complex vector bundles (Princeton University Press, Princeton, NJ).CrossRefGoogle Scholar
Kollár, J., Lectures on resolution of singularities , Ann. of Math. Stud. (2007).Google Scholar
Miyaoka, Y., The Chern classes and Kodaira dimension of a minimal variety , in Algebraic geometry, Sendai, 1985, ed. Oda, T. (North-Holland, Amsterdam, 1987).Google Scholar
Siu, Y.-T., Lectures on Hermitian–Einstein metrics for stable bundles and Kähler–Einstein metrics (Birkhäuser, Basel, 1987).CrossRefGoogle Scholar
Yau, S.-T., On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation. I. , Comm. Pure Appl. Math. 31 (1978), 339–411.CrossRefGoogle Scholar