Hyperbolic structures on a toric arrangement complement

115 Pages·2015·0.71 MB·English

Hyperbolic structures
on a toric arrangement complement
Dali Shen
Thesis committee:
Prof. dr. F. Beukers, Universiteit Utrecht
Prof. dr. G.J. Heckman, Radboud Universiteit Nijmegen
Prof. dr. P. Py, Universit´e de Strasbourg & CNRS
Prof. dr. J.V. Stokman, Universiteit van Amsterdam
Prof. dr. I. Strachan, University of Glasgow
ISBN: 978-90-393-6296-9
Cover designed by Qinqin Xie
Printed by Smart Printing Solutions
Copyright (cid:13)c 2015 by Dali Shen
Hyperbolic structures
on a toric arrangement complement
Hyperbolische structuren
op het complement van een torische schikking
(met een samenvatting in het Nederlands)
Proefschrift
ter verkrijging van de graad van doctor aan de Universiteit Utrecht
op gezag van de rector magnificus, prof. dr. G.J. van der Zwaan,
ingevolge het besluit van het college voor promoties in het openbaar
te verdedigen op woensdag 4 februari 2015 des ochtends te 10.30 uur
door
Dali Shen
geboren op 15 mei 1981 te Zhejiang, China
Promotor: Prof. dr. E.J.N. Looijenga
This thesis was accomplished with financial support from China Scholarship
Council and Mathematical Sciences Center of Tsinghua University.
dedicated to the memory of my father

Contents
Introduction............................................................ ix
Chapter 1. Preliminary theories........................................ 1
1.1. The Deligne-Mostow theory of Lauricella functions.............. 1
1.2. Geometric structures on projective arrangement complements... 8
1.3. Torus embeddings.............................................. 14
Chapter 2. Affine structures........................................... 21
2.1. Projective structures............................................ 21
2.2. Differential operators and connections on tori................... 26
2.3. Flatness of ∇˜κ.................................................. 32
Chapter 3. Hyperbolic structures...................................... 45
3.1. Geometric structures with logarithmic singularities.............. 45
3.2. Eigenvalues of the residue endomorphisms ...................... 48
3.3. Reflection representation........................................ 56
3.4. The evaluation map ............................................ 62
3.5. Hyperbolic complex ball........................................ 67
3.6. An example of ball quotients.................................... 69
Chapter 4. Frobenius structures ....................................... 73
4.1. Frobenius manifolds............................................ 73
4.2. Frobenius algebras on H◦×C×................................. 78
Appendix A. Classification of (extended) Dynkin diagrams............. 83
A.1. Root systems .................................................. 83
A.2. Proof of Theorem 3.18......................................... 87
Bibliography............................................................ 91
Index................................................................... 95
Samenvatting........................................................... 97
Acknowledgments.......................................................101
Curriculum Vitae....................................................... 103
vii

Introduction
This thesis could be viewed as a mergence of two lines of research. One
line is the study on the hypergeometric functions associated with a root sys-
tem. This is actually a multivariable analogue of the classical Euler-Gauss
hypergeometric functions. These functions were introduced and studied by
Heckman and Opdam in a series of papers [16, 17, 29, 30] in the eighties
and nineties of the last century. The other one is the work by Couwenberg,
Heckman and Looijenga (2005) in [6], which studies the geometric structures
on the projective arrangement complements. This work actually includes the
theory of Deligne-Mostow (1986) [11] on the Lauricella functions as a special
case, which providesball quotientstructuresonPn minus ahyperplaneconfig-
uration of type A . This thesis adopts Couwenberg-Heckman-Looijenga’s
n+1
point of view to investigate the root system hypergeometric functions and
hence studies the hyperbolic structure on the corresponding toric arrange-
ment complements. This is why we think of this thesis as a mergence of the
aforementioned two lines of research.
We first construct a projective structure on a toric arrangement comple-
ment. The basic idea is that we can write a projective structure on a complex
manifold M in terms of an affine structure on M × C×. It is well-known
that an affine structure on a complex manifold is given by a torsion free and
flat connection on its (co)tangent bundle, and vice versa. So constructing a
projective structure on M is equivalent to producing a torsion free and flat
connection on M × C×. We start with an adjoint torus H := Hom(Q,C×)
given by a root lattice Q := ZR where R is a reduced irreducible root system.
Denote the Lie algebra of H by h and the Weyl group of R by W. We are also
given a toric arrangement associated with a root system R, that is, a finite
collection of hypertori each of which is defined by H := {h ∈ H | eα(h) = 1}
α
where eα is a character of H. We write H◦ for the complement of the union
of these hypertori. Let κ be a W-invariant multiplicity parameter for R de-
fined by κ := (k ) ∈ CR. Inspired by the special hypergeometric system
α α∈R
constructed by Heckman and Opdam, we consider for u,v ∈ h, such a second
ix
x Introduction
order differential operator on O :
H◦
1 (cid:88) eα+1
Dκ := ∂ ∂ + k α(u)α(v) ∂ +∂ +aκ(u,v)
u,v u v 2 α eα−1 α∨ bκ(u,v)
α>0
where∂ denotestheassociatedtranslationinvariantvectorfieldonH forany
u
u ∈ h and
aκ : h×h → C, bκ : h×h → h
areaW-invariantbilinearformandaW-equivariantbilinearmaprespectively.
We want this system to define a projective structure on H◦. That means for
each multiplicity parameter κ and each W-equivariant bilinear map bκ, there
existsaW-invariantbilinearformaκ suchthatthesystemofdifferentialequa-
tions Dκ f = 0 for all u,v ∈ h is integrable. In order to see the integrability
u,v
of the system, we treat it from a different point of view, i.e., the one from
the work of Couwenberg-Heckman-Looijenga. Now we associate to these data
connections ∇κ = ∇0 +Ωκ and ∇˜κ = ∇˜0 +Ω˜κ on the cotangent bundles of
H◦ and H◦×C× with Ωκ ∈ Hom(Ω ,Ω ⊗Ω ) given by
H◦ H◦ H◦
1 (cid:88) eα+1
Ωκ : ζ ∈ Ω (cid:55)→ k ζ(∂ )dα⊗dα+(Bκ)∗(ζ)
H◦ 2 αeα−1 α∨
α>0
and Ω˜κ ∈ Hom(Ω ,Ω ⊗Ω ) given by
H◦×C× H◦×C× H◦×C×
 1 (cid:88) eα+1
 ζ ∈ ΩH◦ (cid:55)→2 kαeα−1ζ(∂α∨)dα⊗dα+(Bκ)∗(ζ)

 α>0

Ω˜κ : dt dt
−ζ ⊗ − ⊗ζ,
 t t

 dt dt dt
 ∈ Ω (cid:55)→Aκ− ⊗ .
C×
t t t
Here ∇0 and ∇˜0 denote the (flat) translation invariant connections on H
and H ×C× respectively, t is the coordinate for C×, and Aκ and Bκ denote
the translation invariant tensor fields on H or H ×C× defined by aκ and bκ
respectively. WecanshowthatthesystemdefinedbyDκ f = 0forallu,v ∈ h
u,v
is integrable if and only if the connection ∇˜κ given above defines an affine
structure, i.e., the connection ∇˜κ is torsion free and flat. The torsion freeness
of ∇˜κ comes directly from the torsion freeness of ∇κ while the flatness of ∇˜κ
needs more effort. In order to check the flatness of ∇˜κ, we need to invoke a
flatnesscriterionsetupbyLooijenga[21], orbyKohno[19]atanearliertime.
This criterion requires us to compactify H◦ ×C× and compute the residues
of Ω˜κ along those added mirrors and boundary divisors. Then by applying
the criterion to our situation, we can obtain the conditions for ∇˜κ being flat.
According to these conditions, we can find an appropriate bilinear form aκ
so that the connection ∇˜κ is indeed flat and hence a W-invariant projective