Abstract
The morphism d + (resp. d −) considered in §5, (5.26) [resp. (5.35)], attaches intrinsically the nilpotent endomorphism d +(v) (resp. d −(v)) to every tangent vector v in T π.
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Notes
- 1.
For this and other standard facts about such triples we refer to [Kos].
- 2.
See Definition 4.22; from the results in §4.2, Theorem 4.26, it follows that this assumption is inessential.
- 3.
Recall, by (5.2), the fibre \(\tilde{{\mathcal{F}}}^{{\prime}}([Z])\) does not depend on [α].
- 4.
This terminology is explained in the footnote on page 7.
- 5.
Since the component \(\Gamma \) is assumed to be simple, the sheaf \(\boldsymbol{\mathcal{G}}_{\Gamma }\), by Corollary 4.23, is actually a sheaf of simple Lie algebras. But this will not matter in the constructions below.
- 6.
From Corollary 4.23 we know that \(\boldsymbol{\mathcal{G}}_{\Gamma } = \mathbf{sl}(\tilde{{\mathcal{F}}}^{{\prime}})\).
References
B. Kostant, The principal three-dimensional subgroups and the Betti numbers of a complex simple Lie group. Am. J. Math. 81, 973–1032 (1959)
I. Reider, Nonabelian Jacobian of smooth projective surfaces. J. Differ. Geom. 74, 425–505 (2006)
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Reider, I. (2013). sl2-Structures on \({\mathcal{F}}^{{\prime}}\) . 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_6
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