Journal of the Mathematical Society of Japan

Vanishing theorems of $L^2$-cohomology groups on Hessian manifolds


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We show vanishing theorems of $L^2$-cohomology groups of Kodaira–Nakano type on complete Hessian manifolds by introducing a new operator $\partial'_F$. We obtain further vanishing theorems of $L^2$-cohomology groups $L^2H^{p,q}_{\bar{\partial}}(\Omega)$ on a regular convex cone $\Omega$ with the Cheng–Yau metric for $p>q$.

Article information

J. Math. Soc. Japan, Volume 71, Number 1 (2019), 65-89.

Received: 22 February 2017
Revised: 4 June 2017
First available in Project Euclid: 26 October 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C55: Hermitian and Kählerian manifolds [See also 32Cxx]

Hessian manifolds Hesse–Einstein Monge–Ampère equation Laplacians $L^2$-cohomology groups regular convex cones


AKAGAWA, Shinya. Vanishing theorems of $L^2$-cohomology groups on Hessian manifolds. J. Math. Soc. Japan 71 (2019), no. 1, 65--89. doi:10.2969/jmsj/77397739.

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  • [1] H. Shima, The geometry of Hessian structures, World Scientific, Singapore, 2007.
  • [2] H. Shima, Vanishing theorems for compact Hessian manifolds, Ann. Inst. Fourier, 36 (1986), 183–205.
  • [3] S. Y. Cheng and S. T. Yau, The real Monge–Ampére equation and affine flat structures, Proc. the 1980 Beijing symposium of differential geometry and differential equations, Science Press, Beijing, China, Gordon and Breach, Science Publishers Inc., New York, 1982, 339–370.
  • [4] J.-P. Demailly, 𝐿2 estimates for the $\bar{∂}$ operator on complex manifolds, Notes de cours, Ecole d'été de Mathématiques (Analyse Complexe), Institut Fourier, Grenoble, Juin 1996.
  • [5] L. Hörmander, An introduction to complex analysis in several variables, Third edition, North-Holland Mathematical Library, 7, North-Holland Publishing Co., Amsterdam, 1990. xii+254 pp.