Asymptotically holomorphic embeddings of presymplectic manifolds
A closed 2-form \(\omega\) on a \(C^\infty\) manifold \(M\) of dimension \(2n+1\), \(n>0\), \(l\geq 0\), is called a presymplectic form of rank \(2n\) if \(\omega_x^n\neq 0\) and \(\omega_x^{n+1}=0\) for all \(x\in M\). Then \((M,\omega)\) is called a presymplectic manifold of rank \(2n\). Let \(L\) be a complex line bundle with \(c_1(L)\) an integral lift of \([\omega/2\pi]\) and \(U\to \text{Gr}(r,N)\) the universal bundle of rank \(r\) over the Grassmannian. Then asymptotically holomorphic sequences of maps to Grassmannians \(\text{Gr}(r,N)\) are constructed. Theorem 3. Let \((M,\omega)\) be a \((2n+l)\)-dimensional closed presymplectic manifold of rank \(2n\) with a pre-compatible pair \((J,g)\) and an integral form \(\omega\). Suppose that \(E\to M\) is a Hermitian vector bundle of rank \(r\) and \(N\) is a positive integer satisfying \(N\geq n+l+r\) and \(r(N-r)\geq 2n+1\). Then there exist asymptotically holomorphic sequences of immersions \(\varphi_k:M\to \text{Gr}(r,N)\) along \(W\) with \(\varphi_k^*U= E\oplus L^{\oplus k}\) for large integers \(k\). In particular, asymptotically holomorphic embeddings into the projective spaces such that the pullback of the Fubini-study form is cohomologous to \([k\omega/2\pi]\) for large integers \(k\) are constructed. The generalization of the Kodaira embedding theorem to presymplectic manifolds is obtained. Theorem 2. Let \((M,\omega)\) be a \((2n+l)\)-dimensional closed presymplectic manifold of rank \(2n\) with a pre-compatible pair \((J,g)\) and an integral form \(\omega\). Then there exist asymptotically holomorphic sequences of embeddings \(\varphi_k:M\to\mathbb C\mathbb P^n\) with \([\varphi_k^*\omega_{FS}]= [k\omega/2\pi]\) for large integers \(k\) where \(m=2n+ \max(2l,1)\).
scientific article; zbMATH DE number 5345243