Symplectic maps to projective spaces and symplectic invariants (Q2719816)

By \textit{S. K. Donaldson}'s paper [J. Differ. Geom. 44, 666-705 (1996; Zbl 0883.53032)], every symplectic closed 4-manifold admits a Lefschetz pencil. Auroux, and Katzarkov have shown that every symplectic 4-dimensional manifold is a quasiholomorphic branched covering of \(CP^2\) and braid monodromy invariants arising from construction (see the review of Zbl 0961.57019 for the background). The paper under review mainly extends the previous case in 4-dimension to the case of higher-dimensional symplectic manifolds. The key relies on \textit{S. K. Donaldson}'s extension to higher dimensional symplectic compact manifold and transversality results in [J. Differ. Geom. 53, 205-236 (1999; Zbl 1040.53094)]. NEWLINENEWLINENEWLINEThe general principle follows from (1) local construction of almost holomorphic sections and effective transversality for those sections, (2) passage from local to global with transversality everywhere, (3) successive perturbations to control transversality on various strata and (4) control the anti-holomorphic parts of the derivatives near a singularity. Section 2 of the paper under review describes symplectic Lefschetz pencils and their monodromy, Section 3 recalls the results in \textit{D. Auroux} and \textit{L. Katzarkov} [Invent. Math. 142, 631-673 (2000; Zbl 0961.57019)] and connections with Lefschetz pencils. Section 4 contains the author's higher dimensional extension and Section 5 shows that any dimensional compact symplectic manifold can be described by a series of braid words and symmetric group words from dimension induction. NEWLINENEWLINENEWLINEFor a compact symplectic manifold \((X^{2n}, \omega)\) and a complex line bundle \(L\) with \(c_1(L) = \frac{1}{2 \pi} [\omega]\), Theorem 4.1 of the paper states that for sufficiently large \(k\), there exist asymptotic holomorphic sections of \(C^3 \otimes L^{\otimes k}\) such that \(f_k : X^{2n} \to \mathbb{C}\mathbb{P}^2\)'s are quasiholomorphic and these \(f_k\)'s are unique up to isotopy and cancellations of pairs of nodes in the critical curves. Subsection 4.1 occupies the proof of Theorem 4.1 which uses essentially the same arguments as in the 4-dimensional case. Subsection 4.2 describes the local model which characterizes quasiholomorphic maps near critical points. Then the monodromy invariants from quasihololomorphic maps \(f_k : X^{2n} \to \mathbb{C}\mathbb{P}^2\) are naturally extended from the 4-dimensional case as well in Subsection 4.3. The relation with higher dimensional version of Lefschetz pencils is given in Subsection 4.4. The paper is a parallel version of the author's previous work in 4-dimensional symplectic compact manifolds.