Abstract
Based on the results of Berndtsson on the positivity of holomorphic vector bundles associated to holomorphic fibration, we prove that, under the setting of the compact Lie group actions, the character subbundles of such bundles are holomorphic vector bundles and positive in the sense of Nakano. The second main purpose of the present paper is to establish Berndtsson’s integral form of Kiselman’s minimum principle in the setting of the noncompact Lie group actions.
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References
Berman, R.J., Berndtsson, B.: The volume of Kähler-Einstein Fano varieties and convex bodies. Preprint arXiv:1204.1308
Berndtsson, B.: Prekopa’s theorem and Kiselman’s minimum principle for plurisubharmonic functions. Math. Ann. 312, 785–792 (1998)
Berndtsson, B.: Curvature of vector bundles associated to holomorphic fibrations. Ann. Math. (2) 169(2), 531–560 (2009)
Berndtsson, B.: The openness conjecture and complex Brunn–Minkowski inequalities. arXiv:1405.0989
Bröcker, T., Dieck, T.T.: Representations of Compact Lie Groups, GTM98. Springer, Berlin (1985)
Chafi, B.: Principe du Minimum pour les fonctions plurisousharmoniques. Thèse de \(3^e\) cycle, Université de Lille 1 (1983)
Deng, F., Zhang, H., Zhou, X.: Positivity of direct images of positively curved volume forms. Math. Z. 278, 347–362 (2014)
Heinzner, P.: Geometric invariant theory on Stein spaces. Math. Ann. 281, 631–662 (1991)
Kiselman, C.: The partial Legendre transformation for plurisubharmonic functions. Invent. Math. 49(2), 137–148 (1978)
Knapp, W.: Lie Groups Beyond an Introduction. In: Progress in Math. 140. Birkhäuser, Boston (1996)
Loeb, J.J.: Action d’une forme réelle d’un groupe de Lie complexe sur les fonctions plurisousharmoniques. Ann. Inst. Fourier 35, 59–97 (1994)
Onishchik, A.L., Vinberg, E.B. (eds.): Lie Groups and Lie Algebras II: Discrete Subgroups of Lie Groups and Cohomologies of Lie Groups and Lie Algebras. Springer, Berlin (2000)
Simmons, D.: Conditional measures and conditional expectation; Rohlin’s disintegration theorem. Discrete Contin. Dyn. Syst. 32(7), 2565–2582 (2012)
Snow, D.M.: Reductive group action on Stein spaces. Math. Ann. 259, 79–97 (1982)
Acknowledgments
The first author would like to thank Professor Bo Berndtsson for helpful discussions. The authors were partially supported by NSFC.
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Deng, F., Zhang, H. & Zhou, X. Positivity of character subbundles and minimum principle for noncompact group actions. Math. Z. 286, 431–442 (2017). https://doi.org/10.1007/s00209-016-1767-x
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DOI: https://doi.org/10.1007/s00209-016-1767-x