On the construction of contact submanifolds with prescribed topology. (Q1609802)

Let \(C\) be a compact contact manifold of dimension \(2n+1\) with a completely nonintegrable hyperplane distribution \(D\) on \(C\). Locally, \(D\) is defined by the kernel of a 1-form \(\theta\), that is \(D_x= \text{Ker}(\theta(x))\), and the locally defined 2-form \(d\theta\) is nondegenerate when restricted to \(D\). A contact submanifold of the contact manifold \((C,D_C)\) is a triple \((N,D_N, i)\), where \((N,D_N)\) is a contact manifold and \(i: N\to C\) is an embedding such that \(D_N= i^{-1}_*(D_C)\). The authors prove the following theorem: Let \(E\to C\) be a rank \(r\) complex vector bundle over \(C\) \((r\leq n)\) with top Chern class \(c_r(E)\). Then, there exists a contact submanifold \(W\) of \(C\) realizing the Poincaré dual of \(C_r(E)\) on \(H_{2n+1-2r}(C,\mathbb{Z})\). This result is analogue in the contact category to \textit{S. K. Donaldson's} construction of symplectic submanifolds [J. Differ. Geom. 44, 666--705 (1996; Zbl 0883.53032)]. The main tool in the proof is to show the existence of sequences of sections which are asymptotically holomorphic in an appropriate sense, and follows in some part the proof of \textit{D. Auroux} [Geom. Funct. Anal. 7, 971--995 (1997; Zbl 0912.53020)].