Abstract
In this section we establish the basic properties of the sheaf \(\boldsymbol{\tilde{\mathcal{G}}}_{\Gamma }\) (see §2.6 for its definition). From [R1], Proposition 7.2, we know that it is a sheaf of reductive Lie algebras.
Keywords
- Root Vectors
- Subconfiguration
- Cartan Subalgebras
- Weight Decomposition
- Subsheaf
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- 1.
The rank of \(\boldsymbol{\mathcal{C}}_{\Gamma }\) is a priori constant on some non-empty Zariski open subset of \(\mathbf{\breve{J}}_{\Gamma }\).
- 2.
We use the fact, proved in Lemma 3.1, (2), that \(\mathbf{\tilde{H}}_{-l_{\Gamma }}\) is constant along \(\mathbf{\breve{J}}_{Z}\).
- 3.
The assumption \(Card(\Lambda _{[Z]}^{1}) \geq 2\) is needed to insure that the index of L-speciality δ(L,Z 1 ) ≥ 2 which is our convention in §2.7.
- 4.
The equality \(dim(V _{\lambda }([Z])) = degZ_{\lambda }^{{\prime}}\) follows from the fact that V λ([Z]) can be canonically identified with \({H}^{0}(\mathcal{O}_{Z_{\lambda }^{{\prime}}})\) as it was done in the proof of Corollary 4.13.
- 5.
Since \(\mathcal{Z}_{c}^{{\prime}}\) is smooth, this is the same as the set of irreducible components of \(\mathcal{Z}_{c}^{{\prime}}\).
- 6.
The degree assumption is needed for the same reason as in the footnote in Theorem 4.25, (2).
- 7.
A study of the quasi-abelian configurations will appear elsewhere.
References
N. Bourbaki, Lie Groups and Lie Algebras, Chaps. 1–3 (Springer, Berlin, 1989)
P. Griffiths, J. Harris, Principles of Algebraic Geometry (Wiley, New York, 1978)
C. Okonek, M. Schneider, H. Spindler, in Vector Bundles on Complex Projective Space. Progress in Mathematics, vol. 3 (Birkhäuser, Boston, 1980)
I. Reider, Nonabelian Jacobian of smooth projective surfaces. J. Differ. Geom. 74, 425–505 (2006)
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Reider, I. (2013). The Sheaf of Lie Algebras \(\mathcal{G}_{\Gamma }\) . 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_4
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DOI: https://doi.org/10.1007/978-3-642-35662-9_4
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