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

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dc.creatorThomas, R. P.
dc.date1998-06-19
dc.date2001-06-11
dc.date.accessioned2026-07-07T05:25:08Z
dc.date.available2026-07-07T05:25:08Z
dc.descriptionWe 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 $X$.
dc.description65 pages
dc.identifierhttps://arxiv.org/abs/math/9806111
dc.identifierhttp://arxiv.org/abs/math/9806111
dc.identifierJour. Diff. Geom. 54, no. 2, 367-438, 2000
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/77068
dc.subjectAlgebraic Geometry
dc.subjectHigh Energy Physics - Theory
dc.subject14D20
dc.titleA holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations
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