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

Part of: Pluripotential theory Complex manifolds

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 Frederic.Campana@univ-lorraine.fr 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 paun@kias.re.kr
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Abstract

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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 

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