Nonabelian Jacobian of Projective Surfaces: Geometry and Representation Theory (Lecture Notes in Mathematics)

By Igor Reider

The Jacobian of a gentle projective curve is surely essentially the most amazing and gorgeous items in algebraic geometry. This paintings is an try and strengthen an identical concept for gentle projective surfaces - a idea of the nonabelian Jacobian of gentle 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 built with the adaptation of Hodge-like buildings, which produces a sheaf of reductive Lie algebras clearly hooked up to our Jacobian. This constitutes a nonabelian analogue of the (abelian) Lie algebra constitution of the classical Jacobian. this option obviously relates geometry of surfaces with the illustration conception of reductive Lie algebras/groups. This work's major concentration is on delivering an in-depth research of varied features of this relation. It provides a considerable physique of proof that the sheaf of Lie algebras at the nonabelian Jacobian is an effective software for utilizing the illustration thought to systematically handle numerous algebro-geometric difficulties. It additionally indicates easy methods to build new invariants of illustration theoretic foundation on gentle projective surfaces.

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F zero /: (4. seventy eight) Our first step in figuring out the grading (4. seventy two) when it comes to € being easy is to calculate G zero (resp. s G zero ). € € r Proposition four. 27. enable € be an easy part in Cad m . L; d /. Then s G zero€ D lM € 1 sl. Hp / pD0 and the guts C zero is the subsheaf of G zero , whose neighborhood sections € € shape lX € 1 D cp idHp ; have the subsequent pD0 the place cp ’s are neighborhood sections of OJM € such that lX € 1 cp D zero pD0 evidence. the end result follows instantly from (4. seventy seven) and the orthogonal decomposit u tion of FQ zero in (4. 70). to explain different graded items G i of G€ within the decomposition (4. seventy two) it truly is necessary € to make a normal commentary: each one G€i i s a G€0 -module and; particularly; it's C€0 -module: We go back now to the case of € being easy and describe G i , for i ¤ zero, jointly € with its weight decomposition below the motion of C zero . € Set ep D idHp to be the identification endomorphism of Hp and allow C zero D Cfe0 ; : : : ; el€ 1g (4. seventy nine) be the complicated vector area spanned via e0 ; : : : ; el€ 1 . Denote via . C zero / the twin of C zero , built with the foundation zero ; : : : ; l€ 1 twin to e0 ; : : : ; el€ 1 and set ij D i j : (4. eighty) 4 The Sheaf of Lie Algebras GQ€ seventy two zero Then ij , for i ¤ j , is definitely obvious to be the burden of C€ -action on Hom. Hj ; hello /. this offers the next. r Proposition four. 28. allow € be an easy part in Cad m . L; d /. Then for each i ¤ zero, one has lM € 1 Gi D Hom. Hp ; HpCi / ; € pD0 the place the direct sum at the correct hand part is the load decomposition of G i below € the motion of C zero with the summand Hom. Hp ; HpCi / being the weight-subsheaf € resembling the load pCi;p in (4. 80), for p D zero; : : : ; l€ 1. additionally, the summands Hom. Hp ; HpCi / are irreducible s G zero -modules. € facts. all of the assertions are fast from (4. 77), the orthogonal decomposition of FQ zero in (4. 70) and Proposition four. 27. t u Substituting the decompositions of Proposition four. 28 into (4. seventy two) yields zero G€ D G zero ˚ @ € M 1 Hom. Hj ; hello /A ; (4. eighty one) i ¤j the decomposition of G€ into the weight-sheaves of C zero -action. € comment four. 29. become aware of that the set Rl€ D ˚ ij ˇ ˇ i ¤ j 2 f0; : : : ; l€ 1g « (4. eighty two) can be pointed out with the set of roots of sll€ . C/. environment W ij D Hom. Hj ; hello / (4. eighty three) the decomposition in (4. eighty one) might be rewritten as follows zero G€ D G zero ˚ @ € M 1 W A: (4. eighty four) 2Rl€ allow us to additionally become aware of that the looks of sll€ . C/ with a distinct set of roots Rl€ (and its polarization) depends upon the orthogonal decomposition (4. 70) and the triangular decomposition (2. 63). certainly, the decomposition (4. 70) and the operators D ˙ in (2. sixty three) may be considered because the following quiver zero 1 2 l€ 1 (4. eighty five) 4. three A traditional Grading of G€ seventy three The vertices are classified by means of integers f0; 1; : : : ; l€ 1g from left to correct and symbolize the summands Hp . p D zero; : : : ; l€ 1/ of the decomposition in (4. 70) and the arrows among the neighboring vertices p and . p C 1/ symbolize the motion of operators DpC (the higher arrow) and DpC1 (lower arrow), the place Dp˙ is the limit to Hp of the operators D ˙ in (2.

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