[Paper Review] Submanifolds of symplectic manifolds with contact border

Francisco Presas|ArXiv.org|Jul 6, 2000

Geometry and complex manifolds6 references3 citations

TL;DR

This paper constructs symplectic submanifolds in symplectic manifolds with contact boundary, showing their boundaries are contact submanifolds. Using approximately holomorphic techniques and a refined Jackson-type approximation theorem, it proves a relative Lefschetz hyperplane theorem: inclusion induces isomorphisms in relative homology up to dimension $n - r$ and an epimorphism at $n - r$, with submanifolds Poincaré dual to $c_r(L^{igotimes k} \otimes E)$ for large $k$. The results provide a topological characterization and extend symplectic invariants to the sym-con category.

ABSTRACT

We construct symplectic submanifolds of symplectic manifolds with contact border. The boundary of such submanifolds is shown to be a contact submanifold of the contact border. We also give a topological characterization of the constructed submanifolds by means of a ``relative Lefschetz hyperplane Theorem''. We sketch some of the applications of the results.

Motivation & Objective

• To construct symplectic submanifolds in symplectic manifolds with contact boundary, termed the sym-con category, where the submanifold's boundary is a contact submanifold of the ambient contact boundary.
• To provide a topological characterization of such submanifolds via a relative Lefschetz hyperplane theorem in the sym-con setting.
• To extend techniques from approximately holomorphic symplectic topology to the contact-boundary case, overcoming difficulties in extending contact structures near the boundary.
• To establish a Chern class formula for the constructed submanifolds, linking their geometry to characteristic classes of line and vector bundles.

Proposed method

• Uses approximately holomorphic sections of the prequantizable line bundle $L^{igotimes k} \otimes E$ to construct symplectic submanifolds $W_k \subset M$.
• Applies a refined version of Jackson's theorem to control approximation errors in the derivatives of sections, ensuring transversality near the contact boundary.
• Constructs a double of the symplectic manifold $M^d = M^1 \cup M^2$ to analyze relative homology, using Morse theory on a perturbed function $\hat{f}_k$ with critical points of index $\geq n - r + 1$.
• Employs excision and chain decomposition techniques to show that relative homology classes in $H_j(\bar{W}_k, \bar{W}_k \cap C)$ map isomorphically to $H_j(\bar{M}, C)$ for $j \leq n - r$, using symmetry under the involution $e$.
• Derives the Chern class formula for the submanifold $W_k$ via the normal bundle relation $i^*c(TX) = i^*c(E \otimes L^{\bigotimes k}) \cdot c(TW_k)$.
• Reduces the proof to a transversality result near the boundary by constructing global sections that solve the problem in a neighborhood of $C$.

Experimental results

Research questions

• RQ1 Can symplectic submanifolds with contact boundary be constructed in symplectic manifolds with contact boundary, such that their boundary is a contact submanifold of the ambient boundary?
• RQ2 Does a relative Lefschetz hyperplane theorem hold in the sym-con category, relating the relative homology of the submanifold to that of the ambient manifold?
• RQ3 Can the topological invariants of such submanifolds be characterized via characteristic classes, particularly Chern classes?
• RQ4 How can the approximately holomorphic method be adapted to handle the boundary conditions in the sym-con setting, where canonical constructions are absent?

Key findings

• For sufficiently large $k$, there exists a symplectic submanifold $W \subset M$ Poincaré dual to $c_r(L^{\bigotimes k} \otimes E)$, with $\bar{W} \cap C$ a contact submanifold of $C$.
• The inclusion $i: \bar{W} \to \bar{M}$ induces an isomorphism in relative homology $H_p(\bar{W}, \bar{W} \cap C) \to H_p(\bar{M}, C)$ for $p < n - r$, and an epimorphism for $p = n - r$, establishing a relative Lefschetz hyperplane theorem.
• The Chern classes of the constructed submanifold $W_k$ satisfy $c_l(TW_k) = (-1)^l \binom{r + l - 1}{l} (k[\omega/2\pi])^l + O(k^{l-1})$, linking geometry to characteristic classes.
• The method avoids direct extension of contact structures near the boundary by constructing global sections that are transverse and controlled via approximation theory.
• The topological results are new even in the integrable complex case, revealing deeper structure in sym-con manifolds.
• The proof relies on a double construction $M^d = M^1 \cup M^2$ and symmetry under the involution $e$, enabling homological comparison via chain decomposition and excision.