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The Sheaf of Lie Algebras \(\mathcal{G}_{\Gamma }\)

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2072)

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

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

  1. 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. 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. 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. 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. 5.

    Since \(\mathcal{Z}_{c}^{{\prime}}\) is smooth, this is the same as the set of irreducible components of \(\mathcal{Z}_{c}^{{\prime}}\).

  6. 6.

    The degree assumption is needed for the same reason as in the footnote in Theorem 4.25, (2).

  7. 7.

    A study of the quasi-abelian configurations will appear elsewhere.

References

  1. N. Bourbaki, Lie Groups and Lie Algebras, Chaps. 1–3 (Springer, Berlin, 1989)

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  2. P. Griffiths, J. Harris, Principles of Algebraic Geometry (Wiley, New York, 1978)

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  3. C. Okonek, M. Schneider, H. Spindler, in Vector Bundles on Complex Projective Space. Progress in Mathematics, vol. 3 (Birkhäuser, Boston, 1980)

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  4. 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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