Symplectic Forms and Cohomology Decomposition of almost Complex Four-Manifolds

Abstract

For any compact almost complex manifold (M, J), the last two authors [8] defined two subgroups H + J (M ), H − J (M ) of the degree 2 real de Rham cohomology group H 2 (M, R). These are the sets of cohomology classes which can be represented by J-invariant, respectively, Janti-invariant real 2−forms. In this note, it is shown that in dimension 4 these subgroups induce a cohomology decomposition of H 2 (M, R). This is a specifically 4-dimensional result, as it follows from a recent work of Fino and Tomassini [6]. Some estimates for the dimensions of these groups are also established when the almost complex structure is tamed by a symplectic form and an equivalent formulation for a question of Donaldson is given. 1 TEDI DRAGHICI, TIAN-JUN LI, AND WEIYI ZHANG is not C ∞ -pure (the intersection of H + J (M ) and H − J (M ) is non-empty). 1 Taking products of this example with arbitrary almost complex manifolds, one obtains examples in all dimensions ≥ 6 of almost complex structures which are not C ∞ -pure. Also in section 2, for a compact 4-manifold with an integrable J, we show that subgroups H + J (M ) and H − J (M ) relate naturally with the (complex) Dolbeault cohomology groups. We also show that a complex type decomposition for cohomology does not hold for non-integrable almost complex structures (see Lemma 2.12 and Corollary 2.14).In section 3 we focus on almost complex structures J which admit compatible or tame symplectic forms and we give estimates for the dimensions h ± J in this case. If there are J-compatible symplectic forms, then the collection of cohomology classes of all such forms, the so-called J−compatible cone, K c J (M ), is a subcone of H 2 (M ; R). In fact,is the collection of cohomology classes of J−tamed symplectic forms. Thus it is also important to understand the group H − J (M ). Our investigation of almost complex structures which are tamed by symplectic forms is also motivated by the following question of Donaldson ([4]).Question 1.1. If J is an almost complex structure on a compact 4-manifold M which is tamed by a symplectic form ω, is there a symplectic form compatible with J?In [8] it was shown that the question has an affirmative answer when J is integrable. For progress on a related problem proposed by Donaldson, the symplectic Calabi-Yau equation, and its relation to Question 1.1, the reader is referred to [4], [13], [12], [11].We observe in Theorem 3.3 that an estimate on h + J which is immediate for compatible J's can be carried over to the case of tamed J's as well. Section 3 ends with an equivalent formulation of Donaldson's Question 1.1.In a later paper [5] we will further study the group H − J . We appreciate V. Apostolov for his very useful comments, R. Hind, T. Perutz for their interest, A. Fino and A. Tomassini for sending us their paper [6], and NSF for the partial support. We also thank the referees for their careful reading of the manuscript and useful remarks. 1 We learned of the preprint [6] while putting together the final form of our paper. There are further interes...

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

“…We believe that they are important invariants of almost complex structures and deserve further study. We would like to mention that there are two recent papers [11,12] that are closely related to this work (see Remark 2.3). In particular, it is shown in [11] that any 4-dimensional almost complex structure is C ∞ pure and full.…”

“…That is to (M ) R is isomorphic to (H (M )) R [11]. For general almost complex structures the cohomology subgroups in (1.1) and their homology analogues in (1.2) seem to have not been systematically 1 The use of C ∞ here (as well as in Definitions 2.2, 2.3 and 2.6) indicates that we are dealing with forms rather than currents.…”

Cohomology of Almost-Complex Manifolds

“…In other words, we ask for a direction L along which we do not have the strong semi-continuity in the sense described above. Consider the almost-complex structure We need now to find a 2-form β such that (8) d β = b it is straightforward to check that no invariant β satisfying (8) could exist; therefore, by Lemma 4.7, also no non-invariant such β could exist.…”

“…Obviously, compact Kähler manifolds and compact complex surfaces are C ∞ -pure-and-full; moreover, T. Drǎghici, T.-J. 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 refer to [16], [8], [11] and [2] for a more detailed study of these and other properties. We recall that every compact complex surface and every compact Kähler manifold is C ∞ -pure-and-full (see [8]), since, in these cases, the Hodge-Frölicher spectral sequence degenerates at the first level and the trivial filtration on the space of differential forms induces a Hodge structure of weight 2 on H 2 dR , see, e.g., [4], [7] (in fact, a compact Kähler manifold is complex-C ∞ -pure-and-full at every stage); more in general, T. Drǎghici, T.-J.…”