Abstract
The preceding sections show that the Lie theoretic aspects of the Jacobian J(X; L, d) provide new methods and insights in the study of geometry of surfaces. Starting from this section we change the logic of our investigations—we make use of the sheaves of Lie algebras \(\boldsymbol{\mathcal{G}}_{\Gamma }\), for admissible components \(\Gamma \in {C}^{r}(L,d)\), to construct various objects (sheaves, complexes of sheaves, constructible functions), either on J(X; L, d) or on the Hilbert scheme X [d], which can serve as new invariants for vector bundles on X as well as for X itself. Our basic tool for this will be the morphisms d ± encountered in §5.1, (5.26), (5.35).
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Notes
- 1.
In this case it is the usual conjugation.
- 2.
For this and other basic facts of the geometric representation theory our reference is [C-Gi].
- 3.
Recall: \(b_{\mu } = dim_{\mathbf{C}}({\sigma }^{-1}(x))\).
- 4.
The set \(\mathbf{B}(\mathfrak{h}_{{Z}^{{\prime}}})\) can be identified with the set of orderings of eigen spaces of \(\mathfrak{h}_{{Z}^{{\prime}}}\)-action on \({H}^{0}(\mathcal{O}_{{Z}^{{\prime}}})\). Those eigen spaces are generated by the delta-functions \(\delta _{{z}^{{\prime}}}\), for \({z}^{{\prime}}\in {Z}^{{\prime}}\). Thus \(\mathbf{B}(\mathfrak{h}_{{Z}^{{\prime}}})\) has a natural identification with the set of orderings of Z ′.
- 5.
The self-duality of \(\mathcal{T}_{\pi ,\Gamma }\) comes from the isomorphism M in (5.23) and the identification of the quotient-sheaf \(\mathbf{\tilde{H}}/\mathcal{O}_{\mathbf{\breve{J}}_{\Gamma }}\) with the orthogonal complement \(\mathbf{H} = {(\mathcal{O}_{\mathbf{\breve{J}}_{\Gamma }})}^{\perp }\) of \(\mathcal{O}_{\mathbf{\breve{J}}_{\Gamma }}\) in \(\mathbf{\tilde{H}}\). Now the quadratic form \(\mathbf{\tilde{q}}\) in (2.42) restricts to a non-degenerate quadratic form on H, thus making it self-dual.
- 6.
For basic facts and terminology concerning symmetric functions our reference is [Mac].
References
N. Chriss, V. Ginzburg, Representation Theory and Complex Geometry (Birkhauser, Boston, 1997)
M. Haiman, Hilbert schemes, polygraphs and the Macdonald positivity conjecture. J. Am. Math. Soc. 14(4), 941–1006 (2001)
I. Macdonald, in Symmetric Functions and Hall Polynomials, 2nd edn. Oxford Mathematical Monographs (Oxford Science Publications/The Clarendon Press, Oxford University Press/New York, 1995)
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Reider, I. (2013). Representation Theoretic Constructions. In: Nonabelian Jacobian of Projective Surfaces. Lecture Notes in Mathematics, vol 2072. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35662-9_11
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DOI: https://doi.org/10.1007/978-3-642-35662-9_11
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