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**Locally homogeneous variations of Hodge structure.**
*(English)*
Zbl 0584.14003

It was my goal to show the link between the Hodge theory of vector-valued forms on (compact) quotients of Hermitian symmetric spaces to the Hodge theory for local systems that underlie polarized variations of Hodge structure.

Let \({\mathcal V}\) be a locally constant sheaf on the compact Kähler manifold \(S\), with \({\mathcal V}\) underlying a polarized variation of Hodge structure. One can decompose the \({\mathcal V}\)-valued forms on \(S\) into components of type \((p,q);(r,s)\): \((p,q)\)-forms with values in the \((r,s)\) Hodge decomposition bundle. [Note that \(r+s\) will be equal to the weight of the variation of Hodge structure, and \(p+q\) will usually be held fixed (though arbitrary), so there are really only two independent parameters.] Then, according to Deligne, the harmonic forms, and therefore the cohomology \(H^{\bullet}(S,V)\), decompose according to “total” bidegree \((P,Q)\), where \(P=p+r\) and \(Q=q+s\).

Now let \(G\) be a real semisimple Lie group with finite center, \(K\) a maximal compact subgroup, \(\Gamma\) a cocompact discrete subgroup of \(G\), \(S=\Gamma \setminus G/K\), and \((\rho,V)\) a finite-dimensional representation of \(G\). There is a simple construction of a locally constant sheaf \({\mathcal V}\) on \(S\), such that there is a natural isomorphism \(H^{\bullet}(\Gamma,V)\simeq H^{\bullet}(S,{\mathcal V})\). If \(G/K\) is Hermitian symmetric, then according to Y. Matsushima and S. Murakami [Ann. Math. (2) 78, 364–416 (1963; Zbl 0125.10702)] the harmonic forms decompose according to \((p,q)\) type.

There is a natural “locally homogeneous” (complex) variation of Hodge structure on \(S\), determined by the decomposition of \(V\) into character spaces under the center of \(K\). The variation is real if \((\rho,V)\) is. By combining Deligne’s theory and that of the cited paper by Y. Matsushima and S. Murakami, we see that there is a complete decomposition of harmonic forms into \((p,q);(r,s)\) components in this case.

Ultimately, one would like to understand the cohomology in the case of noncompact locally symmetric varieties \(S\) (of finite volume). The Hodge-theoretic techniques yield a decomposition theorem for the intrinsic \(L_ 2\) cohomology \(H^{\bullet}_{(2)}(S,V).\)

The use of Deligne’s Hodge decomposition in the locally homogeneous case permitted us in Ann. Math. (2) 109, 415–476 (1979; Zbl 0446.14002), § 12 to arrive at an explanation of the “mysterious” isomorphism of (Eichler-) Shimura. In the paper under review, we prove for arbitrary \(G\) a cohomological result which underlies the Shimura isomorphism in the case of \(\mathrm{SL}(2,\mathbb R)\).

Let \({\mathcal V}\) be a locally constant sheaf on the compact Kähler manifold \(S\), with \({\mathcal V}\) underlying a polarized variation of Hodge structure. One can decompose the \({\mathcal V}\)-valued forms on \(S\) into components of type \((p,q);(r,s)\): \((p,q)\)-forms with values in the \((r,s)\) Hodge decomposition bundle. [Note that \(r+s\) will be equal to the weight of the variation of Hodge structure, and \(p+q\) will usually be held fixed (though arbitrary), so there are really only two independent parameters.] Then, according to Deligne, the harmonic forms, and therefore the cohomology \(H^{\bullet}(S,V)\), decompose according to “total” bidegree \((P,Q)\), where \(P=p+r\) and \(Q=q+s\).

Now let \(G\) be a real semisimple Lie group with finite center, \(K\) a maximal compact subgroup, \(\Gamma\) a cocompact discrete subgroup of \(G\), \(S=\Gamma \setminus G/K\), and \((\rho,V)\) a finite-dimensional representation of \(G\). There is a simple construction of a locally constant sheaf \({\mathcal V}\) on \(S\), such that there is a natural isomorphism \(H^{\bullet}(\Gamma,V)\simeq H^{\bullet}(S,{\mathcal V})\). If \(G/K\) is Hermitian symmetric, then according to Y. Matsushima and S. Murakami [Ann. Math. (2) 78, 364–416 (1963; Zbl 0125.10702)] the harmonic forms decompose according to \((p,q)\) type.

There is a natural “locally homogeneous” (complex) variation of Hodge structure on \(S\), determined by the decomposition of \(V\) into character spaces under the center of \(K\). The variation is real if \((\rho,V)\) is. By combining Deligne’s theory and that of the cited paper by Y. Matsushima and S. Murakami, we see that there is a complete decomposition of harmonic forms into \((p,q);(r,s)\) components in this case.

Ultimately, one would like to understand the cohomology in the case of noncompact locally symmetric varieties \(S\) (of finite volume). The Hodge-theoretic techniques yield a decomposition theorem for the intrinsic \(L_ 2\) cohomology \(H^{\bullet}_{(2)}(S,V).\)

The use of Deligne’s Hodge decomposition in the locally homogeneous case permitted us in Ann. Math. (2) 109, 415–476 (1979; Zbl 0446.14002), § 12 to arrive at an explanation of the “mysterious” isomorphism of (Eichler-) Shimura. In the paper under review, we prove for arbitrary \(G\) a cohomological result which underlies the Shimura isomorphism in the case of \(\mathrm{SL}(2,\mathbb R)\).

### MSC:

14C30 | Transcendental methods, Hodge theory (algebro-geometric aspects) |

57T15 | Homology and cohomology of homogeneous spaces of Lie groups |

32J25 | Transcendental methods of algebraic geometry (complex-analytic aspects) |

32M10 | Homogeneous complex manifolds |

14M17 | Homogeneous spaces and generalizations |