Almost Kahler forms on rational 4-manifolds

Proceedings of the American Mathematical Society, 2014

T.-J. Li and W. Zhang defined an almost complex structure J on a manifold X to be C ∞ -pure-and-full, if the second de Rham cohomology group can be decomposed as a direct sum of the subgroups whose elements are cohomology classes admitting J-invariant and J-anti-invariant representatives. It turns out (see T. Draghici, T.-J. Li and W. Zhang ) that any almost complex structure on a 4-dimensional compact manifold is C ∞ -pure-and-full. We study the J-invariant and J-anti-invariant cohomology subgroups on almost complex manifolds, possibly non compact. In particular, we prove an analytic continuation result for anti-invariant forms on almost complex manifolds.

International Journal of Mathematics, 2012

Following T.-J. Li, W. Zhang , 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 and the results obtained by D. Sullivan in to study the cone of semi-Kähler structures on a compact semi-Kähler manifold.

arXiv (Cornell University), 2011

For a compact almost complex 4-manifold (M, J), we study the subgroups H ± J of H 2 (M, R) consisting of cohomology classes representable by J-invariant, respectively, J-anti-invariant 2-forms. If b + = 1, we show that for generic almost complex structures on M , the subgroup H - J is trivial. Computations of the subgroups and their dimensions h ± J are obtained for almost complex structures related to integrable ones. We also prove semi-continuity properties for h ± J .

Annals of Global Analysis and Geometry, 1998

Let (M4n,g,Q) be a quaternion Kähler manifold with reduced scalar curvature ? = K/4n(n + 2). Suppose J is an almost complex structure which is compatible with the quaternionic structure Q and let ? = - d F ° J be the Lee form of J. We prove the following local results: (1) if J is conformally symplectic, then it

arXiv (Cornell University), 2008

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.

2007

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 J-compatible symplectic cones over a large class of almost complex manifolds, including all Kähler manifolds, almost Kähler 4-manifolds and complex surfaces. Definition 1.1. The J-tamed symplectic cone is |ω is tamed by J}, and the J-compatible symplectic cone is Recall that J is an automorphism of the tangent bundle T M satisfying J 2 = -id, and J is said to tame ω if ω is positive on any J-line span(v, Jv) Kähler criterion of [6] 1 . We also point out the parallel to some classical results in algebraic geometry and Kähler geometry. Further, if we let HC(J) be the cone of homology complex cycles, in the sense of Sullivan [30], then we can describe it using the J-compatible symplectic cone and the analytic subsets of M . Moreover, for complex surfaces, we confirm Question 1.2. Theorem 1.5. Let J be a complex structure on a 4-manifold M . Then It is a direct consequence of several remarkable results in complex surface theory: the Kodaira classification [4], the Kähler criterion of b + being odd ([29], [31], [26], [3]), and the analysis of complex curves in non-Kähler elliptic surfaces ([17]). Finally we compare the union of tamed cones and the union of compatible cones over all complex structures in dimension 4.

Journal of Geometry and Physics, 2012

Let J be an almost complex structure on a 4-dimensional and unimodular Lie algebra g. We show that there exists a symplectic form taming J if and only if there is a symplectic form compatible with J. We also introduce groups H + J (g) and H − J (g) as the subgroups of the Chevalley-Eilenberg cohomology classes which can be represented by J-invariant, respectively J-anti-invariant, 2-forms on g. and we prove a cohomological J−decomposition theorem following [9]: H 2 (g) = H + J (g) ⊕ H − J (g). We discover that tameness of J can be characterized in terms of the dimension of H ± J (g), just as in the complex surface case. We also describe the tamed and compatible symplectic cones respectively. Finally, two applications to homogeneous J on 4−manifolds are obtained.

Annals of Global Analysis and Geometry - ANN GLOB ANAL GEOM, 1997

In this note we prove that half of all homotopy classes of almost complex structures on M is not compatible with any symplectic structure for a certain class of oriented compact 4-manifolds M. In particular, half of all homotopy classes of almost complex structures on an oriented 4-manifold is not compatible to any Kähler structure.

International Mathematics Research Notices, 2006