sl2-Structures on \({\mathcal{G}}_{\Gamma }\)
The construction of sl 2-structure in the previous section could have been done for any locally free sheaf of graded modules equipped with a compatible graded action of the sheaf of graded Lie algebras \(\boldsymbol{\mathcal{G}}_{\Gamma }\) , where the gr
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sl2-Structures on G
The construction of sl2 -structure in the previous section could have been done for any locally free sheaf of graded modules equipped with a compatible graded action of the sheaf of graded Lie algebras G , where the grading on G is as described in 4.3, (4.72). We have chosen to do it for the sheaf FQ 0 , rather than to give a functorial treatment, since this sheaf arises naturally, it comes with the natural graded action of G and this action is closely tied with the geometry of X . However, there is another natural choice—the sheaf of graded Lie algebras G itself, equipped with the adjoint action. This object is “principal” in the category of graded sheaves of modules with graded G -action in a sense that all other objects in this category as well as their sl2 -structures are representations of this one. Thus it is important to have a good description of sl2 -structures on G . This is the subject of the present section. We begin by recall the grading G D
lM 1 i D.l 1/
Gi
(7.1)
introduced in 4.3, (4.72). Remark 7.1. This grading can be viewed as twisted orthogonal decomposition of G in the following sense. The sheaf G can be equipped with the quadratic form determined by the fibrewise Killing form. We will call it the Killing form of G . With respect to this j form the summands G i and G are orthogonal, unless j D i . In this, latter case, the Killing form induces a non-degenerate pairing G i G i ! OJM :
I. Reider, Nonabelian Jacobian of Projective Surfaces, Lecture Notes in Mathematics 2072, DOI 10.1007/978-3-642-35662-9 7, © Springer-Verlag Berlin Heidelberg 2013
(7.2)
113
7 sl2 -Structures on G
114
Next recall that G carries an involution ./ W G ! G
(7.3)
defined by taking the adjoint x of a local section x of G . This involution interchanges the summands in (7.1) of opposite sign, i.e. one has .G i / D G i ;
(7.4)
for every i 2 f.l 1/; : : : ; .l 1/g. Thus the decomposition (7.1) is orthogonal, up to the twist by the involution ./ . Our construction of a natural (positive) sl2 -structure on G follows the same steps as the considerations in 6.2. We begin by taking the pullback G of the sheaf G via the natural projection in (6.16). As it was pointed out in 6.2, the sheaf G comes together with the tautological section d˘C . Then we go further by lifting G from JM to h0 .d C /, the scheme defined in (6.22). The sheaf . C / G [see (6.24), for the definition of C ] continues to have the twisted orthogonal decomposition . C / G D
lM 1 i D.l 1/
. C / G i :
(7.5)
But in addition, it is equipped with the “universal” sl2 -triple fs C ; s 0 ; s g, where the sections s ˙ ; s 0 are defined in (6.25). Observe that these sections are the sections of the summands . C / G ˙1 and . C / G 0 respectively. We consider the decomposition of . C / G under the adjoint action of this sl2 -triple. The operator ad.s 0 / acts semisimply and gives the decomposition C
. / G D
2 .l 1/
M
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