Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds

Abstract

We introduce certain homology and cohomology subgroups for any almost complex structure and study their pureness, fullness and duality properties. Motivated by a question of Donaldson, we use these groups to relate J-tamed symplectic cones and Jcompatible symplectic cones over a large class of almost complex manifolds, including all Kähler manifolds, almost Kähler 4-manifolds and complex surfaces.

On deformations of compact balanced manifolds

“…3.3], it follows that the linear map L given by (2.5) satisfies the necessary condition [14] and (2.6)), γ being the harmonic representative, and let V be as in (2.1). Then, defininĝ…”

Cohomological decomposition of complex nilmanifolds

“…Also similar subgroups H J (p,q) (M ) and definitions of pure or full almost complex structures can be given by using the space of currents instead of the space of differential forms, and the de Rham homology instead of the de Rham cohomology (for more details and related results see [2], [3], [4], [11], [12], [13], [14], [18]). …”

“…Let M be a 2n-dimensional manifold and J an almost complex structure on M . Following [2], [18], let H of bidegrees (p, q) and (q, p) with respect to J, one can consider the subgroups…”

Cohomology of Almost-Complex Manifolds

“…Li and W. Zhang proved in [8] that every 4-dimensional compact almost-complex manifold is C ∞ -pure-and-full. In [16], the notion of C ∞ -pure-and-full almost-complex structures arises in the study of the symplectic cones of an almost-complex manifold: more precisely, [16,Proposition 3.1] (which we quote in Theorem 2.1) proves that, if J is an almost-complex structure on a compact almost-Kähler manifold such that In §4, we consider the problem of the semi-continuity of h ) is upper-semicontinuous. We give some examples showing that the situation is more complicated in dimension greater than four: in Example 4.2 we present a curve of almost-complex structures on the manifold ηβ 5 as a counterexample to the upper-semi-continuity of h − Jt (ηβ 5 ) and in Example 4.4 we consider S 3 × T 3 as a counterexample to the lower-semi-continuity of h + Jt (S 3 × T 3 ); however, note that the C ∞ -pureness does not hold for all the structures of the curves in these examples, therefore one could ask for more fulfilling counterexamples.…”

“…Following T.-J. Li, W. Zhang [16], we continue to study the link between the cohomology of an almost-complex manifold and its almost-complex structure. In particular, we apply the same argument in [16] and the results obtained by D. Sullivan in [22] to study the cone of semi-Kähler structures on a compact semi-Kähler manifold.…”