Nauk 33 1978 91105 mr511883 80j58008 7 v arnold s

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Unformatted text preview: H(χ) over U whose fibres are the spaces H 1 (X (z ), C)χ and the monodromy map π1 (U ; z (0 ) → U(H 1 (X (z (0) ), C)χ ). Denote the monodromy group by Γ(µ). The important case for us is when |µ| = 2. In this case the signature is (1, n), and we can consider the image of the monodromy group Γ(µ) in PU (1, n)) which n acts in the complex hyperbolic space HC . Here is the main theorem from [28], [80]. Theorem 10.2. The image of each generator si of π1 (U ; z (0) ) in Γ(µ) acts as a n complex reflection in the hyperbolic space HC . The group Γ(µ) is a crystallographic n reflection group in HC if and only if one of the following conditions is satisfied: m • (1 − mi − dj )−1 ∈ Z, i = j, mi + mj < 1; d m mi • 2(1 − d − dj )−1 ∈ Z, if mi = mj , i = j. All possible µ satisfying the conditions from the theorem can be enumerated. We have 59 cases if n = 2, 20 cases if n = 3, 10 cases if n = 4, 6 cases when n = 5, 3 cases if n = 6, 2 cases when n = 7, and 1 case if n = 8 or n = 9. There are several cases when the monodromy group is cocompact. It does not happen in dimension n > 7. n The orbit spaces HC /Γ(µ) of finite volume have a moduli theoretical interpretation. It is isomorphic to the geometric invariant theory quotient (P1 )n+3 //SL(2) with respect to an appropriate choice of linearization of the action. We refer to Mostow’s survey paper [81], where he explains a relation between the monodromy groups Γ(µ) and the monodromy groups of hypergeometric integrals. 10.2. Moduli space of Del Pezzo surfaces as complex ball quotients. In the last section we will discuss some recent work on complex ball uniformization of some moduli spaces in algebraic geometry. It is well-known that a nonsingular cubic curve in the projective plane is isomorphic as a complex manifold to a complex torus C/Z + Zτ , where τ belongs to the REFLECTION GROUPS IN ALGEBRAIC GEOMETRY 53 upper-half plane H = {a + bi ∈ C : b > 0}. Two such tori are isomorphic if and only if the corresponding τ ’s belong to the same orbit of the group Γ = SL(2, Z) which acts on the H by M¨bius transformations z → (az + b)/(cz + d). This result o implies that the moduli space of plane cubic curves is isomorphic to the orbit space H/Γ. Of course, the upper-half plane is a model of the one-dimensional complex 1 hyperbolic space HC and the group Γ acts as a crystallographic reflection group. In a beautiful paper of D. Allcock, J. Carlson and D. Toledo [2], the complex ball uniformization of the moduli space of plane cubics is generalized to the case of the moduli space of cubic surfaces in P3 (C). It has been known since the last century that the linear space V of homogeneous forms of degree 3 in 4 variables admits a natural action of the group SL(4) such that the algebra of invariant polynomial functions on V of degree divisible by 8 is freely generated by invariants I8 , I16 , I24 , I32 , I40 of degrees indicated by the subscript. This can be...
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This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of Michigan-Dearborn.

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