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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 wellknown 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 upperhalf 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 upperhalf plane is a model of the onedimensional 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 MichiganDearborn.
 Fall '04
 IgorDolgachev
 Algebra, Geometry, The Land

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