Hodge De Rham Spectral Sequence Research Articles

On the de Rham homology of affine varieties in characteristic 0

Let K be a field of characteristic 0, let S be a complete local ring with coefficient field K, let K[[x1,…,xn]] be the ring of formal power series in variables x1,…,xn with coefficients from K, let K[[x1,…,xn]]→S be a K-algebra surjection and let E••,• be the associated Hodge-de Rham spectral sequence for the computation of the de Rham homology of S. Nicholas Switala [12] proved that this spectral sequence is independent of the surjection beginning with the E2 page, and the groups E2p,q are all finite-dimensional over K.In this paper we extend this result to affine varieties. Namely, let Y be an affine variety over K, let X be a non-singular affine variety over K, let Y⊂X be an embedding over K and let E••,• be the associated Hodge-de Rham spectral sequence for the computation of the de Rham homology of Y. Then this spectral sequence is independent of the embedding beginning with the E2 page, and the groups E2p,q are all finite-dimensional over K.

Integral $p$-adic Hodge filtrations in low dimension and ramification

Given an integral $p$-adic variety, we observe that if the integral Hodge–de Rham spectral sequence behaves nicely, then the special fiber knows the Hodge numbers of the generic fiber. Applying recent advancements of integral $p$-adic Hodge theory, we show that such a nice behavior is guaranteed if the $p$-adic variety can be lifted to an analogue of second Witt vectors and satisfies some bound on dimension and ramification index. This is a (ramified) mixed characteristic analogue of results due to Deligne–Illusie, Fontaine–Messing, and Kato. Lastly, we discuss an example illustrating the necessity of the aforementioned lifting condition, which is of independent interest.

On the blow-up formula of twisted de Rham cohomology

We derive a blow-up formula for the de Rham cohomology of a local system of complex vector spaces on a compact complex manifold. As an application, we obtain the blow-up invariance of $$E_{1}$$ -degeneracy of the Hodge–de Rham spectral sequence associated with a local system of complex vector spaces.

$E_1$-degeneration and $d'd''$-lemma

For a double complex $(A, d', d'')$, we show that if it satisfies the $d'd''$-lemma and the spectral sequence $\{E^{p, q}_r\}$ induced by $A$ does not degenerate at $E_0$, then it degenerates at $E_1$. We apply this result to prove the degeneration at $E_1$ of a Hodge-de Rham spectral sequence on compact bi-generalized Hermitian manifolds that satisfy a version of $d'd''$-lemma.

De Rham Theory for Tame Stacks and Schemes with Linearly Reductive Singularities

We prove that the Hodge-de Rham spectral sequence for smooth proper tame Artin stacks in characteristic p (as defined by Abramovich, Olsson, and Vistoli) which lift mod p degenerates. We push the result to the coarse spaces of such stacks, thereby obtaining a degeneracy result for schemes which are etale locally the quotient of a smooth scheme by a finite linearly reductive group scheme. Given a scheme X smooth and proper over a field k, the cohomology of the algebraic de Rham complex ΩX/k is an important invariant of X, which, when k = C, recovers the singular cohomology of X(C). When the Hodge-de Rham spectral sequence E 1 = H (ΩX/k)⇒ H (ΩX/k) degenerates, the invariants dimkH (ΩX/k) break up into sums of the finer invariants dimkH (ΩX/k). The degeneracy of this spectral sequence for smooth proper schemes in characteristic 0 was first proved via analytic methods. It was not until much later that Faltings [Fa] gave a purely algebraic proof by means of p-adic Hodge Theory. Soon afterwards, Deligne and Illusie [DI] gave a substantially simpler algebraic proof by showing that the degeneracy of the Hodge-de Rham spectral sequence in characteristic 0 is implied by its degeneracy for smooth proper schemes in characteristic p that lift mod p2. Their method therefore extends de Rham Theory to the class of smooth proper schemes in positive characteristic which lift. A version of de Rham Theory also exists for certain singular schemes. Steenbrink showed [St, Thm 1.12] that if k is a field of characteristic 0, M a proper k-scheme with quotient singularities, and j : M0 ↪→ M its smooth locus, then the hypercohomology spectral sequence E 1 = H (j∗Ω s M0/k)⇒ H (j∗Ω • M0/k) degenerates and H(j∗Ω • M0/k) agrees with H n(M(C),C) when k = C. As we explain in this paper, a version of this theorem is true in positive characteristic as well: if k has characteristic p and M is proper with quotient singularities by groups whose orders are prime to p, then the above spectral sequence degenerates for s+ t < p provided a certain liftability criterion is satisfied (see Theorem 1.15 for precise hypotheses). As a warm-up for the rest of the paper, we begin by showing how Steenbrink’s result can be reproved using the theory of stacks. The idea is as follows. Every scheme M as above is the coarse space of a smooth Deligne-Mumford stack X whose stacky structure is supported at the singular locus of M . We show that the de Rham cohomology H(ΩX/k) of the stack agrees with H(j∗Ω • M0/k). After checking that the method of Deligne-Illusie extends to Deligne-Mumford stacks, we recover Steenbrink’s result as a consequence of the degeneracy of the Hodge-de Rham spectral sequence for X. The above extends de Rham Theory to the class of schemes with quotient singularities by groups whose orders are prime to the characteristic, but in positive characteristic this class of schemes contains certain “gaps” and it is natural to ask if de Rham Theory can be 2010 Mathematics Subject Classification. 14A20, 14F40.

Homological vanishing theorems for locally analytic representations

Let G be the group of rational points of a split connected reductive group over a p-adic local field, and let Γ be a discrete and cocompact subgroup of G. Motivated by questions on the cohomology of p-adic symmetric spaces, we investigate the homology of Γ with coefficients in locally analytic principal series and related representations of G. The vanishing and finiteness results that we find partially rely on the compactness of certain Banach–Hecke operators. We also give a new construction of P. Schneider’s reduced Hodge–de Rham spectral sequence and show that the induced filtration is the Hodge–de Rham filtration. In a previously unknown case, our vanishing theorems then also imply two other of P. Schneider’s conjectures.

Cohomology of bundles on homological Hopf manifolds

We discuss the properties of complex manifolds having rational homology of S 1 × S 2n−1 including those constructed by Hopf, Kodaira and Brieskorn-van de Ven. We extend certain previously known properties of cohomology of bundles on such manifolds. As an application we consider degeneration of Hodge-de Rham spectral sequence in this non Kahler setting.

Limits of Hodge Structure in Several Variables

We define the notion of a morphism of generalized semi-stable type, which is a generalization of the notion of a semistable degeneration over a curve. We partially generalize Steenbrink‘s results on the limit of Hodge structures to the case of such a morphism. As an application we prove the E1-degeneration of the relative Hodge–De Rham spectral sequence for this case.

On Infinitesimal Liftings and Degenerations of Hodge–de Rham Spectral Sequences

We prove the degeneration of the Hodge–de Rham spectral sequence at E 1of a log smooth proper scheme over a p-primary torsion log scheme under certain condition.

The nondegeneration of the Hodge-de Rham spectral sequence

Using an integrable, homogeneous complex structure on the compact group SO(9), we show that the Hodge-de Rham spectral sequence for this non-Kahler compact complex manifold does not degenerate at Ei, contrary to a well-known conjecture. On any (compact) complex manifold M, the algebra of global complexvalued C°°-differential forms a*(M) has a bigrading given by the Hodge type; and the corresponding decomposition of the de Rham differential d = d + d gives rise to a double complex (a*'*(M),d, d). The spectral sequence corresponding to the first (holomorphic) degree is E = H<*{M, QM) =» HPp(M) with dx = d : this is the Hodge-de Rham (HdR) spectral sequence. When M is compact and Kahler, E = Eoo by Hodge theory. A folklore conjecture of about thirty years' standing says that for any compact complex manifold one should have E2 = E^. We will give an example to show that this conjecture is false. 1. There is an old observation of H. Samelson (see Wang [4]) that every compact Lie group G of even dimension (equivalently even rank) can be made into a complex manifold in such a way that all left-translations by elements g e G are holomorphic maps: we call such a structure an LICS on G (= left invariant, integrable complex structure on G). If G is in addition semisimple, then no LICS can be Kahlerian because H(G : R) = 0. Our example is a particular LICS on SO(9) (equivalently Spin(9): see §3 below), for which E2 has complex dimension 26. Since one knows that E^ has complex dimension 16, we obtain the required nondegeneration of HdR. Received by the editors November 23, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 32C10, 58A14, 55J05, 32M10. 1 Samelson's paper appeared in a Portuguese journal, which is perhaps understandably not available in Indian libraries. ©1989 American Mathematical Society 0273-0979/89 $1.00 + $.25 per page 19