Moduli of compact Legendre submanifolds of complex contact manifolds (Q1897742)

The author studies complete analytic families of compact complex Legendre submanifolds of complex contact manifolds. To fix notation, let \(Y\) be a complex \((2n + 1)\)-dimensional manifold. A complex contact structure on \(Y\) is a rank \(2n\) holomorphic subbundle \(D \subset TY\) such that \(\Phi : D \times D \to TY/D\) defined by \((v,w) \to [v,w]\mod D\) is non- degenerate. A complex \(n\)-dimensional submanifold \(X\) of \(Y\) is called a Legendre submanifold if \(TX \subset D\). Let the line bundle \(L_X\) be the restriction of \(TY/D\) to \(X\). The author shows an analogue of Kodaira's theorem, that if \(H^1(X,L_X) = 0\), then there exists a complete analytic family \(\{X_t \subset Y \mid t \in M\}\) of compact Legendre submanifolds containing \(X\), such that \(\dim M = \dim H^0 (X, LX)\). Let \(\{X_t \subseteq Y \mid t \in M\}\) be a complete analytic family of compact Legendre submanifolds. In the spirit of Penrose's twistor theory, the author constructs on \(M\) affine torsion free connections, called \(\Lambda\)-connections, and studies the holonomy of these induced connections. He indicates that almost all known torsion- free affine connections with irreducibly acting holonomy groups can be interpreted, at least locally, as induced connections on appropriate Legendre moduli spaces. In particular, he has an explicit example of the above whose holonomy group is Bryant's exotic group \(G_3 \subset\text{GL}(4, \mathbb{C})\) [\textit{R. L. Bryant}, Proc. Symp. Pure Math. 53, 33-88 (1991; Zbl 0758.53017)]. This leads him to conjecture that any torsion-free connection with irreducibly acting holonomy group can be constructed by this method.