A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on $K3$ fibrations

We briefly review the formal picture in which a Calabi-Yau n-fold is the complex analogue of an oriented real n-manifold, and a Fano with a fixed smooth anticanonical divisor is the analogue of a manifold with boundary, motivating a holomorphic Casson invariant counting bundles on a Calabi-Yau 3-fold. We develop the deformation theory necessary to obtain the virtual moduli cycles of [31], This gives, for instance, virtual moduli cycles in Hilbert schemes of curves in P 3 , and Donaldson-and Gromov-Witten-like invariants of Fano 3-folds. It also allows us to define the holomorphic Casson invariant of a Calabi-Yau 3-fold X, prove it is deformation invariant, and compute it explicitly in some examples. Then we calculate moduli spaces of sheaves on a general K3 fibration X, enabling us to compute the invariant for some ranks and Chern classes, and equate it to Gromov-Witten invariants of the "Mukai-dual" 3-fold for others. As an example the invariant is shown to distinguish Gross' diffeomorphic 3-folds. Finally the Mukai-dual 3-fold is shown to be Calabi-Yau and its cohomology is related to that of X.