Twistor transform of vector bundles

Abstract

Let {Xt ↪→ Y | t ∈ M} be a family of compact complex submanifolds Xt of a complex manifold Y parametrised by a complex manifold M . In this paper we study vector bundles and linear connections on the moduli space M induced from holomorphic vector bundles on the space Y . The origin of this approach lies in the Ward transform [W] of holomorphic vector bundles E on Y which are trivial when restricted to each submanifoldXt of the family {Xt ↪→ Y | t ∈M}. We consider a more general class of holomorphic vector bundles E on twistor spaces Y which, when restricted to a submanifold Xt of the family, have the same integer h(Xt, E|Xt) for all t ∈M . With such a vector bundle E on Y there is an associated vector bundle E on M whose fibre at t ∈ M is isomorphic to H(Xt, E|Xt). It is shown that provided the cohomology groups H ( Xt, E|Xt ⊗N ∗ t ) and H ( Xt, E|Xt ⊗N ∗ t ) vanish (where Nt is the normal bundle of Xt ↪→ Y ), the vector bundle E induced on the parameter space M comes equipped with a distinguished linear connection ∇ satisfying some natural integrability conditions (cf. [L, E, B-E]). The curvature tensor of ∇ is represented, at each t ∈ M , by a cohomology class in H ( Xt, E|Xt ⊗ 2N∗ ) ⊗ ( H(Xt, E|Xt) )∗ . This theorem-construction gives a very simple way to estimate the curvature tensor of an induced connection directly from the original twistor data. As an application, we consider a pair X ↪→ Y consisting of a complex contact 3-fold (Y, L) and a Legendre submanifold X = CP such that the contact line bundle L restricts to X as O(3). Then the moduli space M of all holomorphic Legendre deformations of X inside Y is a 4-dimensional manifold which comes equipped with an induced 1-flat G3structure [Br2], and any such structure arises locally in this way. Suppose that there exists a line bundle E → Y such that L = E⊗3 (such an E always exists on a sufficiently small tubular neighbourhood of X in Y ). Since the normal bundle of X is isomorphic to JO(3) = C ⊗ O(2), the above cohomology conditions on E|Xt ⊗ N ∗ t are satisfied. Then the induced rank 2 vector bundle E comes equipped with a distinguished linear connection which, as easily follows from its integrability properties, is a “spinor” version of the torsion-free connection on M with holonomy in G3. In fact, any torsion-free connection with holonomy in G3 can be constructed, at least locally, in this way (cf. [Me1]). The point is that the way we prove the above mentioned theorem-construction gives a rather