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In this article we prove that locally Griffiths' horizontal distribution on the period domain is given by a generalized version of the familiar contact differential system. As a consequence of this description we obtain strong local rigidity properties of maximal dimensional variations of Hodge structure. For example, we prove that if the weight is odd then there is a unique germ of maximal dimensional variation of Hodge stucture though every point of the period domain. Similar results hold if the weight is even with the exception of one case., Comment: 24 pages, latex2e uses amsart.cls

Give the smooth subset of a normal singular complex projective surface the metric induced from the ambient projective space. The L 2 {L^2} cohomology of this incomplete manifold is isomorphic to the surface’s intersection cohomology, which has a natural Hodge decomposition. This paper identifies Dolbeault complexes whose ∂ ¯ \bar \partial -closed and ∂ ¯ \bar \partial -coclosed forms represent the classes of pure type in the corresponding Hodge decomposition of L 2 {L^2} cohomology.

We study the Hodge spectral sequence of a local system on a compact, complex torus by means of the theory of harmonic integrals. It is shown that, in some cases, Baker's theorems concerning linear forms in the logarithms of algebraic numbers may be applied to obtain vanishing theorems in cohomology. This is applied to the study of Betti and Hodge numbers of compact analytic threefolds which are analogues of hyperelliptic surfaces. Among other things, it is shown that, in contrast to the two-dimensional case, some of these varieties are nonalgebraic. 0. Introduction. (0.0) The purpose of this paper is to determine the Hodge spectral sequence associated with a vector bundle with integrable connection on a complex torus. Such vector bundles typically arise as the hypercohomology sheaves attached to a proper and smooth morphism f: Y -* X where X is a complex torus. Consequently our results will have application to the study of those manifolds which admit such fiberings over tori. (0.1) In outline, the main result is as follows: Let (l, V) be a holomorphic vector bundle with integrable connection on a complex torus X = V/L, where L C V is a lattice in a complex vector space. Let W = QlV be the corresponding C-local system. We compute Hq(X, S2P(qll)) and Hk(X, W) in terms of harmonic differential forms via Hodge's theorem. First we show that qtS admits a global real analytic frame. Relative to this frame, C? differential forms in qIS may be represented by C?r m-vectors of differential forms p on V which are L-periodic (m = rank qLS'). Using the frame, a Hermitian metric may be introduced into the fibers of qLS in such a way that the harmonic theory takes on a simple form. Specifically, the Laplace equations lA 0 become, upon Fourier transform, AX(m).j4m) 0, m E Z2n (n = dimV). Each of these is a finite-dimensional matrix equation. We prove (for the choices made): (a) For Hk(X W), LA(m) j(m) 0 O has only the zero solution if m #7 0. Hence Hk(X, W) ker A(0). (b) For Hq(X, iP(qLS')), only a finite number of m can occur for which det i\(m) = 0 and these are identified by a diophantine condition involving the logarithms of the periods of X. These m are "singular". Therefore, modulo determining the singular m, all computations are reduced to a finite amount of linear algebra. Received by the editors October 28, 1980. 1980 Mathematics Subject Classification. Primary 14C30; Secondary 32J25, 14K20. 'Partially supported by NSF Grant MCS-800223 1. ?D 982 American Mathematical Society 0002-9947/82/0000-1051 /$04.75

The paper deals mainly with combinatorial structures; in some cases we need refinements of combinatorial structures. Riemannian metrics are defined on any combinatorial manifold M. The existence of distance functions and of Riemannian metrics with “constant volume density” implies smoothing. A geometric realization of PL ( m ) /O ( m ) {\text {PL}}\left ( m \right ){\text {/O}}\left ( m \right ) is given in terms of Riemannian metrics. A graded differential complex Ω ∗ ( M ) {\Omega ^ {\ast } }( M ) is constructed: it appears as a subcomplex of Sullivan’s complex of piecewise differentiable forms. In the complex Ω ∗ ( M ) {\Omega ^{\ast }}( M ) the operators d d , ∗ \ast , δ \delta , Δ \Delta are defined. A Rellich chain of Sobolev spaces is presented. We obtain a Hodge-type decomposition theorem, and the Hodge homomorphism is defined and studied. We study also the combinatorial analogue of the signature operator.