## 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 $$\mathrm{SL}(2,\mathbb R)$$.

### 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

### Citations:

Zbl 0125.10702; Zbl 0446.14002