The nondegeneration of the Hodge-de Rham spectral sequence

Abstract

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