A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations
R. P. Thomas
TL;DR
This work develops a holomorphic analogue of the Casson invariant for Calabi–Yau 3‑folds by constructing a deformation theory that yields virtual moduli cycles for stable sheaves and defines the holomorphic Casson invariant as the length of a zero‑dimensional virtual cycle, with deformation invariance guaranteed by Li–Tian/Fulton–Behrend–Fulton theory. It advances a two‑step tangent–obstruction framework, uses higher Ext vanishing to obtain perfect obstruction theories, and derives two holomorphic Chern–Simons descriptions that motivate counting integrable holomorphic structures. The theory is then specialized to $K3$ fibrations, where explicit moduli computations give the invariant in several ranks/Chern classes and reveal connections to Gromov–Witten invariants of Mukai‑dual Calabi–Yau threefolds; in particular, the invariant can distinguish certain diffeomorphic Calabi–Yau 3‑folds and the Mukai dual is shown to be Calabi–Yau with cohomology constrained by that of the original space. The paper also discusses the potential appearance of modular forms via the generating functions of these invariants and provides detailed examples to illustrate the interplay between bundle moduli and curve counting in this holomorphic setting.
Abstract
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 \cite{LT}, \cite{BF} in moduli spaces of stable sheaves whose higher obstruction groups vanish. This gives, for instance, virtual moduli cycles in Hilbert schemes of curves in $\Pee^3$, 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 $X$.