Geometry of tamed almost complex structures on 4-dimensional manifolds (Q2900287)

In [Nankai Tracts Math. 11, 153--172 (2006; Zbl 1140.58018)], \textit{S. K. Donaldson} asked the following ``tamed vs. compatible'' question: Suppose \(J\) is an almost complex structure on a compact 4-manifold \(M\), tamed by a symplectic form \(\omega\). Is there a symplectic form \(\tilde{\omega}\) compatible with \(J\)? This paper describes recent upshots concerning this question and provides a few new results. If \((M,J,g)\) is an almost Hermitian manifold of dimension \(4\), the space \(H^2(M,\mathbb{R})\) can be decomposed in \(J\)-invariant and \(J\)-anti-invariant forms, giving two subspaces of dimension \(h^+_J\) and \(h^-_J\) respectively. Such a decomposition, which is related to the standard decomposition of \(H^2(M,\mathbb{R})\) given by the Hodge-star operator \(\ast_g\), allows one to state Donaldson's question in cohomological terms. There exist basic estimates relating the numbers \(h^{\pm}_J\) with \(b^{\pm}\), which can be refined in some special cases (\(J\) integrable, well balanced almost Hermitian \(4\)-manifolds, \(J\) tamed). The authors study how these numbers change under deformation of \(J\). They also give possible extensions of this approach to higher dimensions. They show how to give a positive answer to Donaldson's question in some particular cases, for instance, rational surfaces. Using ideas of \textit{C. H. Taubes} [J. Symplectic Geom. 9, No. 2, 161--250 (2011; Zbl 1232.57021)], they construct almost Kähler forms as regularizations of currents given by integration over \(J\)-holomorphic submanifolds. Finally, they construct \(J\)-holomorphic submanifolds on almost complex \(4\)-manifolds \((M,J)\) as divisors of a closed, \(J\)-anti-invariant \(2\)-form.NEWLINENEWLINEFor the entire collection see [Zbl 1235.00045].