Vanishing theorems on cohomology associated to hermitian symmetric spaces
Annales de l'Institut Fourier, Volume 37 (1987) no. 4, p. 225-233

We consider the cohomoly groups of compact locally Hermitian symmetric spaces with coefficients in the sheaf of germs of holomorphic sections of those vector bundles over the spaces which are defined by canonical automorphic factors. We give a quick survey of the research on these cohomology groups, and then discuss vanishing theorems of the cohomology groups.

On considère le groupe de cohomologie d’un espace compact localement hermitien symétrique à coefficients dans le faisceau des germes de sections holomorphes d’un tel espace fibré holomorphe sur l’espace qui est défini par un facteur d’automorphie. On rappelle d’abord brièvement des résultats classiques sur ce groupe de cohomologie, et on discute des théorèmes d’annulation sur le groupe de cohomologie.

@article{AIF_1987__37_4_225_0,
     author = {Murakami, Shingo},
     title = {Vanishing theorems on cohomology associated to hermitian symmetric spaces},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {37},
     number = {4},
     year = {1987},
     pages = {225-233},
     doi = {10.5802/aif.1119},
     zbl = {0625.57023},
     mrnumber = {89j:32043},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1987__37_4_225_0}
}
Murakami, Shingo. Vanishing theorems on cohomology associated to hermitian symmetric spaces. Annales de l'Institut Fourier, Volume 37 (1987) no. 4, pp. 225-233. doi : 10.5802/aif.1119. http://www.numdam.org/item/AIF_1987__37_4_225_0/

[1] G. W. Anderson, Theta functions and holomorphic differential forms on compact quotients of bounded symmetric domains, Duke Math. J., 50 (1983), 1137-1170. | MR 85i:32047 | Zbl 0557.32006

[2] A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Princeton, Princeton University Press, 1980. (Annals of mathematical studies, 94.) | Zbl 0443.22010

[3] G. Faltings, On the cohomology of locally symmetric Hermitian symmetric spaces, Paul Dubriel and Marie-Paul Malliavin algebra seminar, 25th year (Paris, 1982), 55-98, Lecture Notes in Math., 1029 (1983). | Zbl 0539.22008

[4] R. Hotta and S. Murakami, On a vanishing theorem for certain cohomology groups, Osaka J. Math., 12 (1975), 555-564. | MR 57 #6291 | Zbl 0328.57020

[5] R. Hotta and N. Wallach, On Matsushima's formula for the Betti numbers of a locally symmetric space, Osaka J. Math., 12 (1975), 419-431. | MR 52 #661 | Zbl 0327.53032

[6] J. L. Koszul, Formes harmoniques vectorielles sur espaces localement symétriques, Geometry of Homogeneous Bounded Domains, C.I.M.E. 3 Ciclo, 1967, 197-260. | Zbl 0183.50903

[7] S. Kumaresan, On the canonical k-types in the irreducible unitary g-modules with non-zero relative cohomology, Inventiones Math., 59 (1980), 1-11. | MR 83c:17011 | Zbl 0442.22010

[8] Y. Matsushima, On the first Betti number of compact quotient spaces of higher dimensional symmetric spaces, Ann. of Math., 75 (1962), 312-330. | MR 28 #1629 | Zbl 0118.38303

[9] Y. Matsushima, On Betti numbers of compact, locally symmetric Riemannian manifolds, Osaka J. Math., 14 (1982), 312-330. | Zbl 0118.38401

[10] Y. Matsushima, A formula for the Betti numbers of locally symmetric Riemannian manifolds, J. Differential Geometry, 1 (1987), 99-109. | MR 222908 | MR 36 #5958 | Zbl 0164.22101

[11] Y. Matsushima and S. Murakami, On vector bundle valued harmonic forms and automorphic forms on symmetric Riemannian manifolds, Ann. of Math., 78 (1963), 365-416. | MR 153028 | MR 27 #2997 | Zbl 0125.10702

[12] Y. Matsushima and S. Murakami, On certain cohomology groups attached to hermitian symmetric spaces, Osaka J. Math., 2 (1965), 1-35. | MR 32 #1728 | Zbl 0142.19503

[13] Y. Matsushima and S. Murakami, On certain cohomology groups attached to Hermitian symmetric spaces (II), Osaka J. Math., 5 (1968), 223-241. | MR 42 #1145 | Zbl 0183.26103

[14] S. Murakami, Cohomology of vector-valued forms on compact, locally symmetric Riemannian manifolds, Proceeding Symposia in Pure Mathematics, vol. 9, Algebraic groups and discontinuous subgroups, 1966, 387-399. | MR 34 #6803 | Zbl 0208.24402

[15] S. Murakami, Cohomology groups of vector-valued forms on symmetric spaces, Lecture Notes, University of Chicago, 1966.

[16] S. Murakami, Facteur d'automorphie associé à un espace hermitien symétrique, Geometry of Homogeneous Bounded Domains, C.I.M.E. 3 Ciclo, (1967), 281-287. | Zbl 0187.02902

[17] S. Murakami, Certain cohomology groups attached to Hermitian symmetric spaces and unitary representations, Southeast Asian Bull. Math., 5 (1981), 39-44. | MR 83e:32034 | Zbl 0496.22020

[18] S. Murakami, Laplacians and cohomologies associated to locally symmetric Hermitian manifolds, Spectra of Riemannian Manifolds (Proceedings of the France-Japan Seminar, Kyoto, 1981), Kaigai Publ. Tokyo, (1983), 73-78.

[19] R. Parthasarathy, A note of the vanishing of L²-cohomologies, J. Math. Soc. Japan, 22 (1971), 1-30. | Zbl 0215.40703

[20] R. Parthasarathy, Criteria for the unitarizability of some highest weight modules, Proc. Indian Acad. Sci., 89 (1980), 1-24. | MR 82c:22020 | Zbl 0434.22011

[21] D. A. Vogan and G. J. Zuckerman, Unitary representations with non-zero cohomology, Compositio Mathematica, 53 (1984), 51-90. | Numdam | MR 86k:22040 | Zbl 0692.22008

[22] F. L. Williams, Vanishing theorems for type (0,q)-cohomology of locally symmetric spaces, Osaka J. Math., 18 (1981), 147-160. | MR 82k:22013 | Zbl 0469.32011

[23] F. L. Williams, Remarks on the unitary representations appearing in the Matsushima-Murakami formula, Proceeding of the Conference on Non-commutative Harmonic Analysis, Marseille-Luminy, France, Lecture Notes in Math., 880 (1981), 536-553. | MR 83e:22017 | Zbl 0508.22015

[24] F. L. Williams, Vanishing theorems for type (0,q)-cohomology of locally symmetric spaces II, Osaka J. Math., 20 (1983), 95-108. | MR 84h:22031 | Zbl 0523.22016

[25] S. Zucker, Locally homogeneous variations of Hodge structure, L'Enseignement Mathématiques, IIe série, 27 (1981), 243-276. | MR 83m:32034 | Zbl 0584.14003