sl2-Structures on \({\mathcal{F}}^{{\prime}}\)
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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sl2 -Structures on FQ 0
The morphism d C (resp. d ) considered in 5, (5.26) [resp. (5.35)], attaches intrinsically the nilpotent endomorphism d C .v/ (resp. d .v/) to every tangent vector v in T . We have seen their importance with respect to the period maps r defined for JM , for every component in Cadm .L; d /. In this section we explore more subtle representation theoretic aspects of this assignment by completing d C .v/ to an sl2 -triple. This is made possible by the well-known Jacobson-Morozov theorem.1 Once such a triple is chosen, we look at its representation on the fibre of the sheaf FQ 0 at the point of JM underlying the tangent vector v. This yields further, finer, decomposition of the orthogonal decomposition of FQ 0 in (5.4). The resulting structure is somewhat reminiscent of the linear algebra data arising in the theory of Mixed Hodge structure. r We fix a component in Cadm .L; d / and assume it to be simple.2 As before the morphism W JM ! M stands for the natural projection and T denotes its relative tangent bundle (recall our convention in 2.7 of distinguishing locally free sheaves and the corresponding vector bundle). We begin by considering the situation at a closed point of T and then give a sheaf version of our construction.
6.1 Constructions on a Fibre of FQ 0 Fix a point .ŒZ; Œ˛/ 2 JM and let v be a tangent vector in T lying over .ŒZ; Œ˛/. Evaluating the morphism d C (resp. d ) in (5.26) [resp. (5.35)] at the point .ŒZ; Œ˛; v/ 2 T , we obtain the endomorphism d C .v/ (resp. d .v/) of
1
For this and other standard facts about such triples we refer to [Kos]. See Definition 4.22; from the results in 4.2, Theorem 4.26, it follows that this assumption is inessential.
2
I. Reider, Nonabelian Jacobian of Projective Surfaces, Lecture Notes in Mathematics 2072, DOI 10.1007/978-3-642-35662-9 6, © Springer-Verlag Berlin Heidelberg 2013
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6 sl2 -Structures on FQ 0
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FQ 0 .ŒZ/, the fibre3 of FQ 0 at .ŒZ; Œ˛/. By definition of G the endomorphisms d ˙ .v/ are nilpotent elements of G .ŒZ; Œ˛/, the fibre of G at .ŒZ; Œ˛/. By Jacobson-Morozov theorem, d C .v/ (resp. d .v/) can be completed to an sl2 -triple fd C .v/; h; yg (resp. fy 0 ; h0 ; d .v/g), where h (resp. h0 ) is a semisimple element of G .ŒZ; Œ˛/ subject to the standard relations Œh; d C .v/ D 2d C .v/ Œh; y D 2y Œd C .v/; y D h ; (resp. Œh0 ; d .v/ D 2d .v/ Œh0 ; y 0 D 2y 0 Œy 0 ; d .v/ D h0 / : It is well-known that semisimple elements coming along with d C .v/ (resp. d .v/) in an sl2 -triple form a homogeneous space modeled on the nilpotent Lie algebra
gC .v/ D ker.ad.d C .v/// \ im.ad.d C .v/// .resp. g .v/ D ker.ad.d .v/// \ im.ad.d .v/// / ;
(6.1)
i.e. two choices for a semisimple element in an sl2 -triple for d ˙ .v/ differ by an element in g˙ .v/. In fact it is known that if h (resp. h0 ) is a semisimple element which goes along with d C .v/ (resp. d .v/) in an sl2 -triple, then any other semisimple element hQ (resp. hQ0 ) for d C .v/ (resp. d .v
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