Constructs nested non-commutative Hilbert scheme on C^n, equips nested Hilbert scheme on C^2 with equivalent obstruction theory, and derives closed formula for equivariant virtual Euler characteristic generating series via torus localization and bundle maps.
A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations
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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 $X$.
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UNVERDICTED 3representative citing papers
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Virtual K-theoretic invariants of the nested Hilbert scheme on $\mathbb{C}^2$
Constructs nested non-commutative Hilbert scheme on C^n, equips nested Hilbert scheme on C^2 with equivalent obstruction theory, and derives closed formula for equivariant virtual Euler characteristic generating series via torus localization and bundle maps.
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Shell formulas for instantons and gauge origami
A new shell formula unifies and delivers explicit closed-form expressions plus recursions for instanton partition functions in 5d SYM and multiple gauge origami configurations using arbitrary-dimensional Young diagrams.
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