Vanishing theorems for compact hessian manifolds

Shima, Hirohiko

Annales de l'Institut Fourier, Tome 36 (1986) no. 3, pp. 183-205.

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.

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

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

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

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

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

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

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

[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 | Zbl

[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 | Zbl