A Holomorphic Casson invariant for Calabi-Yau three folds, and bundles on K3 fibrations
Jun, 199872 pages
Published in:
- J.Diff.Geom. 54 (2000) 2, 367-438
- Published: 2000
e-Print:
- math/9806111 [math.AG]
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Abstract:
We briefly review the formal picture in which a Calabi-Yau -fold is the complex analogue of an oriented real -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 , prove it is deformation invariant, and compute it explicitly in some examples. Then we calculate moduli spaces of sheaves on a general fibration , 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 .Note:
- 65 pages Subj-class: Algebraic Geometry MSC-class: 14D20 Journal-ref: Jour. Diff. Geom. 54, no. 2, 367-438, 2000
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