Positivity and vanishing theorems for ample vector bundles

Liu, Kefeng; Sun, Xiaofeng; Yang, Xiaokui

doi:10.1090/S1056-3911-2012-00588-8

Authors: Kefeng Liu, Xiaofeng Sun and Xiaokui Yang
Journal: J. Algebraic Geom. 22 (2013), 303-331
DOI: https://doi.org/10.1090/S1056-3911-2012-00588-8
Published electronically: August 9, 2012
MathSciNet review: 3019451
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Abstract | References | Additional Information

Abstract: In this paper, we study the Nakano-positivity and dual-Nakano- positivity of certain adjoint vector bundles associated to ample vector bundles. As applications, we get new vanishing theorems about ample vector bundles. For example, we prove that if $E$ is an ample vector bundle over a compact Kähler manifold $X$, $S^kE\otimes \det E$ is both Nakano-positive and dual-Nakano-positive for any $k\geq 0$. Moreover, $H^{n,q}(X,S^kE\otimes \det E)=H^{q,n}(X,S^kE\otimes \det E)=0$ for any $q\geq 1$. In particular, if $(E,h)$ is a Griffiths-positive vector bundle, the naturally induced Hermitian vector bundle $(S^kE\otimes \det E, S^kh\otimes \det h)$ is both Nakano-positive and dual-Nakano-positive for any $k\geq 0$.

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Additional Information

Kefeng Liu
Affiliation: Center of Mathematical Science, Zhejiang University, Hangzhou 310027, P. R. China
Address at time of publication: Department of Mathematics, University of California Los Angeles, Los Angeles, California 90095
Email: liu@math.ucla.edu

Xiaofeng Sun
Affiliation: Department of Mathematics, Lehigh University, Bethlehem, Pennsylvania 18015
Email: xis205@lehigh.edu

Xiaokui Yang
Affiliation: Department of Mathematics, University of California Los Angeles, Los Angeles, California 90095
Email: xkyang@math.ucla.edu

Received by editor(s): July 2, 2010
Received by editor(s) in revised form: January 5, 2011, January 25, 2011, and February 15, 2011
Published electronically: August 9, 2012
Additional Notes: The first author was supported in part by NSF Grant #1007053.