Symplectic submanifolds in special position (Q5939284)

A compact \(2n\)-dimensional symplectic manifold \((M, \omega)\) is said to be almost Hodge if the cohomology class \([\omega]/2\pi\) lies in the image of the integral lattice in \(H^2 (M;\mathbb{R})\). \textit{S. K. Donaldson} [J. Differ. Geom. 44, No. 4, 666-705 (1996; Zbl 0883.53032)] showed that for sufficiently large integers \(k\) there exists a connected symplectic submanifold \(V\) of \(M\) such that \(k[\omega]/2\pi\) is Poincaré dual to \(V\). In this paper, the author provides a refinement of Donaldson's original theorem, enabling one to construct symplectic hypersurfaces which lie in special position with respect to a given symplectic submanifold. For a compact \(2d\)-dimensional symplectic submanifold \(V\) of a closed \(2n\)-dimensional almost Hodge manifold \((M, \omega)\) let \(q: \widetilde M\to M\) be the blow-up of \(V\) in \(M\), \(E=q^{-1}(V)\), and \(\varepsilon\in H^2(\widetilde M;\mathbb{R})\) the Poincaré dual class of \(E\). The author shows that, for \(k\) sufficiently large, there are symplectic submanifolds \(\widetilde Z_k\) dual to the symplectic form \(\widetilde \omega = k[q^*\omega]-\varepsilon\in H^2 (\widetilde M;\mathbb{R})\) meeting every fiber \(E_x=q^{-1}(x)\), \(x\in V\), in the zero locus of some holomorphic section of the hyperplane bundle. If \(2d<n\), then there exists a symplectic submanifold \(Z_k\) of \(M\) Poincaré dual to \([k\omega]\) with \(V\subset Z_k\). In a complementary direction, the author, following Donaldson's methods, shows that for \(k\gg 0\) the de Rham cohomology class \([k\omega]\) is Poincaré dual to a submanifold \(Z\subset M\) transversal to \(V\) and such that \(V\cap Z\subset M\) is symplectic.