The Frölicher spectral sequence can be arbitrarily non-degenerate

Abstract

The Frölicher spectral sequence of a compact complex manifold X measures the difference between Dolbeault cohomology and de Rham cohomology. If X is Kähler then the spectral sequence collapses at the E 1 term and no example with d n = 0 for n > 3 has been described in the literature.We construct for n ≥ 2 nilmanifolds with left-invariant complex structure X n such that the n-th differential d n does not vanish. This answers a question mentioned in the book of Griffiths and Harris.

Invariant Complex Structures on 6-Nilmanifolds: Classification, Frölicher Spectral Sequence and Special Hermitian Metrics

“…Notice also that this basis defines implicitly an invariant complex structure J a on M just by declaring that the forms µ 1 a , µ 2 a , µ 3 a are of type (1,0) with respect to J a . Moreover, a direct calculation shows that the complex structure equations for J a , with respect to this basis, are (25) dµ Recall that by Corollary 5.2 and Table 3, if a = 0 then the complex nilmanifold (M, J 0 ) does not admit sG metrics because J 0 is abelian.…”

“…The examples in [9] are complex nilmanifolds of complex dimension 3, which is the lowest possible dimension for which the Frölicher sequence can be non-degenerate at E 2 . More recently, Rollenske has constructed in [25] complex nilmanifolds for which the sequence {E r } can be arbitrarily non-degenerate. The behaviour of the Frölicher sequence has been studied for some other complex manifolds [14,29], but as far as we know its general behaviour for complex nilmanifolds has not been studied, although some partial results can be found in [10,11,12].…”

Lie-algebra Dolbeault-cohomology and small deformations of nilmanifolds

“…Remark 2.2 -(i ) A nilmanifold M J of type (g, Γ) with left-invariant complex structure is Kählerian if and only if g is abelian and M is a complex torus [BG88,Has89]. It can be arbitrarily far from being Kähler in the sense that the Frölicher spectral sequence may be arbitrarily non-degenerate [Rol07b]. (ii ) If [g 1,0 , g 0,1 ] = 0 then J is called complex parallelisable (bi-invariant), (G, J) is a complex Lie group and M J is complex parallelisable.…”

Geometry of nilmanifolds with left-invariant complex structure and deformations in the large

“…Among these are the so-called Kodaira-Thurston manifolds, historically the first examples known to admit both a complex structure and a symplectic structure but no Kähler structure. In fact, a nilmanifold M admits a Kähler structure if and only if it is a complex torus [BG88,Has89] and the author showed [Rol07c] that nilmanifolds can be arbitrarily far from Kähler manifolds in the sense that the Frölicher spectral sequence may be arbitrarily non-degenerate.…”