Existence and geometry of Legendre moduli spaces
One of the most useful characteristics of an affine connection on a manifold \(M\) is its (restricted) holonomy group which is defined, up to conjugation, as a subgroup of \(GL (T_tM)\) consisting of all automorphisms of the tangent space \(T_tM\) at a point \(t\in M\) induced by parallel translations along the \(t\)-based contractible loops in \(M\). Which groups can occur as holonomies of affine connections? It is well-known that any closed subgroup of a general linear group can be realised as holonomy of some affine connection (which in general has a non-vanishing torsion tensor). The same question, when posed in the class of torsion-free affine connections only, is not yet answered. Long ago, Berger presented a very restricted list of possible irreducibly acting holonomies of torsion-free affine connections. His list was complete in the part of metric connections (and later much work has been done to refine this ``metric'' part of his list), while the situation with holonomies of non-metric torsion-free affine connections was and remains very unclear. As usual in representation theory, in order to get a deeper understanding of all irreducible real holonomies it is worthwhile trying to address a complex version of the problem. The main result of this paper says that any holomorphic irreducible 1-flat \(G\)-structure as well as any holomorphic torsion-free affine connection with irreducibly acting holonomy group can, in principle, be constructed by twistor methods. The first part of the paper solves a moduli problem which is of interest on its own, without references to differential geometry or twistor theory, and which is a generalization of the moduli problem solved by Kodaira in 1962. Kodaira's initial data is a pair, \(X\hookrightarrow Y\), consisting of a compact complex submanifold \(X\) of a complex manifold \(Y\). The object of study is the set \(M\) of complex submanifolds \(X_t\hookrightarrow Y\) obtained from \(X\) by all possible holomorphic deformations of \(X\) inside the fixed ambient manifold \(Y\). Kodaira showed that if the normal bundle \(N\) of the initial submanifold \(X\hookrightarrow Y\) is such that \(H^1(X,N) =0 \), then the moduli set \(M\) has very nice properties: first, it is a manifold with \(\dim M=h^0 (X,N) \); second, a tangent vector at any point \(t\in M\) can be realized canonically as a global section of the normal bundle \(N_t\) of the associated submanifold \(X_t \hookrightarrow Y\), i.e., there is a canonical isomorphism \(k_t:T_t M\to H^0 (X_t,N_t)\). The manifold \(M\) is called a Kodaira moduli space. This paper studies the following specialization (which turns out to be a generalization!) of the Kodaira relative deformation problem. The initial data is again a compact complex submanifold \(X\hookrightarrow Y\) of a complex manifold \(Y\), but now \(Y\) is assumed to have a complex contact structure and \(X\) is assumed to be a Legendre submanifold. It is shown that if \(H^1(X, L|_X)=0\), where \(L\) is the contact line bundle on \(Y\), then there exists a complete and maximal analytic family \(\{X_t \hookrightarrow Y| t\in M\}\) of compact Legendre submanifolds containing \(X\). The moduli space \(M\), called a Legendre moduli space, is an \(h^0 (X,L |_X)\)-dimensional complex manifold. The second part of the paper studies Legendre moduli spaces of compact complex homogeneous manifolds. It is shown that such moduli spaces always come equipped with an induced \(G\)-structure \({\mathcal G}_{\text{ind}}\) whose basic geometric invariants can be computed directly from the embedding ``initial data'' \(X \hookrightarrow Y\). This analysis is then used to show that if \(\nabla\) is a holomorphic torsion-free affine connection on a complex manifold \(M\) with irreducibly acting reductive holonomy group \(G\), then there exists a complex contact manifold \((Y,L)\) and a Legendre submanifold \(X\hookrightarrow Y\) with \(X=G_s/P\) for some parabolic subgroup \(P\) of the semisimple quotient \(G_s\) of \(G\) and with \(L|_X\) being ample, such that, at least locally, \(M\) is canonically isomorphic to the associated Legendre moduli space and \(\nabla\) is an induced torsion-free affine connection in \({\mathcal G}_{\text{ind}}\).
holonomy of torsion-free affine connections
scientific article; zbMATH DE number 1096018