Remarks on Donaldson's symplectic submanifolds (Q1623082)

If \(V\) is a compact projective manifold with a Kähler form \(\omega\) with integral periods, the integral cohomology class \(k[\omega]\) is realized as the dual class of a transversal hyperplane section \(W=V\cap H\). \(V\setminus W\) is a Stein manifold of finite type and \(\omega|_{V\setminus W}=dd^c\phi\), where \(\phi=\frac{1}{2k\pi}\log|s|\), \(s\) is the restriction to \(V\subset \mathbb{CP}^m\) of complex linear function. It is an exhausting function having no critical points near \(W\). When \(V\) is a closed manifold with a symplectic form \(\omega\) having integral periods, the following analogy of the above properties of algebraic manifolds was proved in [\textit{S. K. Donaldson}, J. Differ. Geom. 44, No. 4, 666--705 (1996; Zbl 0883.53032)]. Theorem 0. Let \(V\) be a closed manifold with a symplectic form \(\omega\) having integral periods, then if \(k\) is a sufficiently large integer, there exists a symplectic submanifold \(W\) of codimension 2 in \((V,\omega)\) whose homology class is the Poincaré dual to \(k[\omega]\) and whose inclusion into \(V\) is an \((n-1)\)-connected map where \(n=\frac{1}{2}\dim_\mathbb{R}V\). Here, a map \(X\to Y\) is said to be \(m\)-connected, if it induces a bijection \(\pi_j(X)\to \pi_j(Y)\) and a surjection \(\pi_m(X)\to \pi_m(Y)\). As variants of Theorem 0, for the same class of symplectic manifolds, the following results are proved. Theorem 1. Let \(V\), \(\omega\) and \(W\) be the same as in Theorem 0. Then we can take \(W\) so that \(V\setminus W\) has a complex structure \(J\) such that \(\omega_{V\setminus W}=dd'\phi\) for some exhausting function \(\phi:V\setminus W\to\mathbb{R}\) having no critical points near \(W\). In particular, \((V\setminus W,J)\) is a Stein manifold of finite type. Theorem 2. Let \((V,\omega)\) be as in Theorem 0. Then for sufficiently large \(k\), there exist a Weinstein domain \((F,\lambda)\) and a map \(q:F\to V\) such that {\parindent=0.7cm\begin{itemize}\item[1.] \(q(\partial F)\) is a symplectic hyperplane section \(W\) of degree \(k\) in \((V,\omega)\) and \(\partial F\) is the normal circle bundle of \(W\) projecting to \(W\) by \(q\).\item[2.] \(q|_{F\setminus\partial F}:F\setminus\partial F\to V\setminus W\) is a diffeomorphism with \(q^\ast\omega=d\lambda\). \end{itemize}} Here, a Weinstein domain is a Liouville domain whose Liouville field \(\underrightarrow{\lambda}\) is gradient-like for some Morse function \(\phi:F\to\mathbb{R}\); \(\underrightarrow{\lambda}\cdot\phi\geq c|\underrightarrow{\lambda}|^2\). A Liouville domain is a domain \(F\) endowed with a Liouville form \(\lambda\), that means \(d\lambda\) is a symplectic form on \(F\) and induces a contact form on \(K=\partial W\) orienting \(K\) as the boundary of \((F,d\lambda)\). Since Weinstein and Stein domains are essentially the same objects [\textit{K. Cieliebak} and \textit{Y. Eliashberg}, From Stein to Weinstein and back. Symplectic geometry of affine complex manifolds. Providence, RI: American Mathematical Society (AMS) (2012; Zbl 1262.32026)], Theorem 1 follows from Theorem 2. A statement similar to Theorem 2, but the condition \((F, \lambda)\) to be a Weinstein domain being relaxed to be a Liouville domain, is proved in \S A (Proposition 5). Then, following Donaldson's method ([loc. cit.], see also, [\textit{S. K. Donaldson}, J. Differ. Geom. 53, No. 2, 205--236 (1999; Zbl 1040.53094)]), Theorem 2 is proved from Proposition 5 and a generalization of Biran's decomposition; namely if \((V,\omega)\) and \(k\) are the same as above, then there exists an isotropic skeleton \(\bigtriangleup\subset V\) whose complement \(V\setminus \bigtriangleup\) has the structure of a standard symplectic disk bundle of area \(1/k\) over a symplectic manifold \(W\) (Proposition 4. Proved in \S A. [\textit{P. Biran}, Geom. Funct. Anal. 11, No. 3, 407--464 (2001; Zbl 1025.57032)], in \S B. In \S C, the last section, a variant of Cielieback-Eliashberg theorem (reviewed as Theorem 17) is proved, that asserts for a Weinstein domain \((F,\lambda)\) with a complex structure \(J\) and a \(J\)-convex Morse function \(\phi\) with regular level set \(\partial F=\{\phi=0\}\) with \(d\lambda=dd'\phi\) (Corollary 18). By this corollary, the proof of Theorem 1 is completed.