Vanishing theorems for compact hessian manifolds
Shima, Hirohiko
Annales de l'Institut Fourier, Tome 36 (1986), p. 183-205 / Harvested from Numdam
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Une variété est dite hessienne si elle admet une connexion plate D et une métrique riemannienne g telle que g=D 2 u où u est une fonction locale. On étudie la cohomologie des variétés hessiennes et on montre un théorème de dualité et des “vanishing theorems”.

A manifold is said to be Hessian if it admits a flat affine connection D and a Riemannian metric g such that g=D 2 u where u is a local function. We study cohomology for Hessian manifolds, and prove a duality theorem and vanishing theorems.

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     author = {Shima, Hirohiko},
     title = {Vanishing theorems for compact hessian manifolds},
     journal = {Annales de l'Institut Fourier},
     volume = {36},
     year = {1986},
     pages = {183-205},
     doi = {10.5802/aif.1065},
     mrnumber = {88f:53059},
     zbl = {0586.57013},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1986__36_3_183_0}
}

Shima, Hirohiko. Vanishing theorems for compact hessian manifolds. Annales de l'Institut Fourier, Tome 36 (1986) pp. 183-205. doi : 10.5802/aif.1065. http://gdmltest.u-ga.fr/item/AIF_1986__36_3_183_0/

[1] Y. Akizuki and S. Nakano, Note on Kodaira-Spencer's proof of Lefschetz theorems, Proc. Japan Acad., 30 (1954), 266-272. | MR 16,619a | Zbl 0059.14701

[2] S.Y. Cheng and S.T. Yau, The real Monge-Ampère equation and affine flat structures, Proceedings of the 1980 Beijing symposium of differential geometry and differential equations, Science Press, Beijing, China, 1982, Gordon and Breach, Science Publishers, Inc., New York, 339-370. | MR 85c:53103 | Zbl 0517.35020

[3] K. Kodaira, On cohomology groups of compact analytic varieties with coefficients in some analytic faisceaux, Proc. Nat. Acad. Sci., U.S.A., 39 (1953), 865-868. | MR 16,74b | Zbl 0051.14502

[4] K. Kodaira, On a differential-geometric method in the theory of analytic stacks, Proc. Nat. Acad. Sci., U.S.A., 39 (1953), 1268-1273. | MR 16,618b | Zbl 0053.11701

[5] J.L. Koszul, Domaines bornés homogènes et orbites de groupes de transformations affines, Bull. Soc. Math. France, 89 (1961), 515-533. | Numdam | MR 26 #3090 | Zbl 0144.34002

[6] J.L. Koszul, Variétés localement plates et convexité, Osaka J. Math., 2 (1965), 285-290. | MR 33 #4849 | Zbl 0173.50001

[7] J.L. Koszul, Déformations de connexions localement plates, Ann. Inst. Fourier, Grenoble, 18-1 (1968), 103-114. | Numdam | MR 39 #886 | Zbl 0167.50103

[8] J. Morrow and K. Kodaira, Complex manifolds, Holt, Rinehart and Winston, Inc., 1971. | MR 46 #2080 | Zbl 0325.32001

[9] J.P. Serre, Une théorème de dualité, Comm. Math. Helv., 29 (1955), 9-26. | MR 16,736d | Zbl 0067.16101

[10] H. Shima, On certain locally flat homogeneous manifolds of solvable Lie groups, Osaka J. Math., 13 (1976), 213-229. | MR 54 #1131 | Zbl 0332.53032

[11] H. Shima, Symmetric spaces with invariant locally Hessian structures, J. Math. Soc. Japan, 29 (1977), 581-589. | MR 56 #9462 | Zbl 0349.53036

[12] H. Shima, Compact locally Hessian manifolds, Osaka J. Math., 15 (1978), 509-513. | MR 80e:53054 | Zbl 0415.53032

[13] H. Shima, Homogeneous Hessian manifolds, Ann. Inst. Fourier, Grenoble, 30-3 (1980), 91-128. | Numdam | MR 82a:53054 | Zbl 0424.53023

[14] H. Shima, Hessian manifolds and convexity, in Manifolds, and Lie groups, Papers in honor of Y. Matsushima, Progress in Mathematics, vol. 14, Birkhäuser, Boston, Basel, Stuttgart, 1981, 385-392. | MR 83h:53066 | Zbl 0481.53038

[15] K. Yagi, On Hessian structures on an affine manifold, in Manifolds and Lie groups. Papers in honor of Y. Matsushima, Progress in Mathematics, vol. 14, Birkhäuser, Boston, Basel, Stuttgart, 1981, 449-459. | MR 83h:53067 | Zbl 0495.53011