Nonabelian Jacobian of Projective Surfaces: Geometry and by Igor Reider

By Igor Reider

The Jacobian of a soft projective curve is absolutely the most notable and lovely items in algebraic geometry. This paintings is an try and increase the same idea for gentle projective surfaces - a concept of the nonabelian Jacobian of delicate projective surfaces. similar to its classical counterpart, our nonabelian Jacobian pertains to vector bundles (of rank 2) on a floor in addition to its Hilbert scheme of issues. however it additionally comes outfitted with the adaptation of Hodge-like buildings, which produces a sheaf of reductive Lie algebras obviously connected to our Jacobian. This constitutes a nonabelian analogue of the (abelian) Lie algebra constitution of the classical Jacobian. this option evidently relates geometry of surfaces with the illustration thought of reductive Lie algebras/groups. This work’s major concentration is on supplying an in-depth research of varied facets of this relation. It offers a considerable physique of facts that the sheaf of Lie algebras at the nonabelian Jacobian is a good device for utilizing the illustration conception to systematically handle quite a few algebro-geometric difficulties. It additionally exhibits how you can build new invariants of illustration theoretic beginning on gentle projective surfaces.

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32): Q Z ! ŒZ; Œ˛/ for which we use the same notation. 3. 7) In the sequel we use freely this identification, by switching frequently from one space to another, without explicitly invoking this isomorphism. OZ 0 / will be taken for granted. OZ 0 /. It is known that this action is semisimple. ŒZ; Œ˛/ is the weight space corresponding to a weight sum is taken over the weights of this action. 3. 8). 1 The Center C€ of GQ€ 41 Proof. OZ 0 /. OZ 0 /. f v/; for all h 2 c. OZ 0 /. OZ 0 /. OZ 0 /. h/vv0 : These two equalities yield .

So, to simplify the writing, we often omit tensoring with OJM € in the notation of restriction to JM € of a sheaf defined on a larger space. 23) will be considered over JM € , unless said for example, the filtration H Q otherwise, and we will continue to write H instead of the more cumbersome Q ˝ OM . 6. At certain parts of the paper we will need to distinguished between locally free sheaf on a variety and the corresponding vector bundle. To do this we use capital calligraphic letter for the former and the same capital letter, but in Roman type, for the latter.

0/ / ! 0/ . 5). 7. 49) holds. 3 it follows easily that this decomposition is the pullback by the orthogonal decomposition of F ˝ O€M D F 0 ˝ O€M ˚ F 0 ˝ O€M ? 12) ? e. the following holds F 0 ˝ O€M H l€ D FQ ˝ OJM € D ? 13) F 0 ˝ O€M ? 4. 5) the above decomposition and its constituents can be described as follows. 5) over it. ŒZ/ ! OZ 0 / ! 16) given by the pullback f of functions. 16), the summand H for every Œ˛ in JM € lying over ŒZ. ŒZ; Œ˛/ (resp. ŒZ/ (resp. 1, of Œ˛ and we denote it by H (2)).

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