Flat Hermitian forms for reflection arrangements

It may very well hold in a much greater generality.

Theorem 3.7. — Suppose that H is the reflection arrangement of a finite complex reflection group G. Then there exists a map from (R^H)^G to the space of nonzero Hermitian forms on the tangent bundle of V^◦ (denoted κ→h^κ) with the following properties: for every κ∈(R^H)^G,

(i)h^κ is flat for ∇^κ and invariant under G.

(ii)t ∈ R→ h^tκ is smooth (notice that h⁰ was already defined) and the associated curve of projectivized forms, t → [h^tκ] is real-analytic.

Moreover, h^κ is unique up to scalar multiple (relative to (i) and (ii)).

Likewise there is a map from (R^H)^G to the space of nonzero Hermitian forms on the cotan-gent bundle of V^◦ with analogous properties.

Example 3.8. — For V =C and Ω = κz⁻¹dz, we can take h^κ(z):= |z|^−2κ|dz|². Notice that we can expand this in powers of κ as

h^κ(z)= ∞

k=0

κ^k(−log|z|²)^k k! |dz|².

We shall first prove that in the situation of Theorem 3.7 we can find such an h^κ formally at κ = 0. For this we need the following notion, suggested by Example 3.8.

Let be given a complex manifold M and a smooth hypersurface D⊂M. We say that a function f on a neighborhood of p in M−D is logsingular of order ≤ k along D if there is a defining equation φ ofD at p such that f can be written as a polynomial of degree ≤k in log|φ| with coefficients that are real-analytic on Mp. Since φ is unique up to a unit as a factor, log|φ| is unique up to the addition of a real-analytic function on Mp and so any generator of OM,p(−D) will do for this purpose.

We say that a differential η on a neighborhood of p in M−D is logsingular of order ≤ k along D at p if it is a linear combination of real-analytic forms on Mp and

|φ|⁻¹d|φ| =Re(dφ/φ) and with logsingular functions of order ≤k as coefficients, but with the coefficient of |φ|⁻¹d|φ| being of order ≤ k −1 (read equal to zero when k=0).

Lemma 3.9. — In this situation we have:

(i)log|w| is algebraically independent over the ring of real-analytic functions on Mp. (ii) Any logsingular differential of order ≤k atp that is closed is the differential of a

logsin-gular function of order ≤k at p.

Proof. — The proof of (i) is left to the reader. For the proof of (ii) we use a re-traction Mp →Dp in order to identify Mp withDp×C0 and employ polar coordinates (r, θ) on the second coordinate. Ifη is a logsingular differential of order≤k at p, then we can write

η= k

i=0

(logr)ⁱ(αi+girdθ)+

k−1

i=0

(logr)ⁱfir⁻¹dr,

withαi a differential on Dp whose coefficients are real-analytic functions on Mp and fi

andgi real-analytic on Dp×R0×S¹. Since limr↓0r(logr)ⁱ =0 fori ≥0, the integral of such a form over the circle{(p,r)}×S¹ tends to zero as r tends to zero. Hence, ifη is closed, then the integral over any such circle is closed and the form can be integrated to a function on (M−D)p. A straightforward verification shows that this function is

logsingular of order ≤k.

Lemma 3.10. — In the situation of Theorem 3.7, let κ∈(R^H)^G. Then there exists a for-mal expansion h^sκ = _∞

k=0s^khk in G-invariant Hermitian forms with initial coefficient h0=h⁰

and such thathk is logsingular of order ≤k along the smooth part of the arrangement and with the property that h^sκ is flat for ∇^sκ.

Proof. — The flatness of h^sκ means that for every pair v,v ∈V (when regarded as translation invariant vector fields onV) we have

d(h^sκ(v,v))= −sh^sκ(Ω^κ(v),v)−sh^sκ(v,Ω^κ(v)), whereΩ^κ(v)=

HκHπH(v)⊗ωH, which boils down to

d(hk+1(v,v))= −hk(Ω^κ(v),v)−hk(v,Ω^κ(v)), k=0,1,2, ... . (∗)

In other words, we must show that we can solve (∗)inductively by G-invariant forms.

In case we can solve (∗), then it is clear that a solution will be unique up to a con-stant.

The first step is easy: if we choose our defining equation φH ∈ V^∗ for H to be such that φH, φH =1, then

h1(v,v):= −κH

H

πH(v), πH(v)log|φH|²

will do. Suppose that for some k ≥ 1 the forms h0, ...,hk have been constructed. In order that (∗) has a solution for hk+1 we want the right-hand side (which we shall denote by ηk(v,v)) to be exact. It is certainly closed: if we agree that h(ω⊗v, ω⊗v) stands for h(v,v)ω∧ω, then

dηk(v,v)=hk−1(Ω^κ∧Ω^κ(v),v)−hk−1(Ω^κ(v),Ω^κ(v)) +hk−1(Ω^κ(v),Ω^κ(v))+hk−1(v,Ω^κ∧Ω^κ(v))

=hk−1(Ω^κ∧Ω^κ(v),v)+hk−1(v,Ω^κ∧Ω^κ(v))=0

(since Ω^κ∧Ω^κ = 0). So in order to complete the induction step, it suffices by Lem-ma 3.9 that to prove that ηk is logsingular of order ≤ k+1 along the arrangement:

since the complement in V of the singular part of the arrangement is simply con-nected, we then write ηk as the differential of a Hermitian form hk+1 on V that is logsingular of order ≤ k+1 along the arrangement and averaging such hk+1 over its G-transforms makes it G-invariant as well.

Our induction assumption says that near H^◦ we can expand hk in log|φH| uniquely as:

hk = k

i=0

(log|φ^H|)ⁱhk,i

wherehk,i is a real-analytic Hermitian form on TVH^◦. Now ηk = −

H∈H

k i=0

κH(log|φH|)ⁱ hk,i

πH(v),v

ωH+hk,i

v, πH(v) ωH

.

The terms indexed byH =Hare linear combinations of differentials that are regular atH^◦ with logsingular coefficients of order ≤k and hence are logsingular of order≤k alongH^◦. So it remains to show thathk,i(π^H(v),v)ω^H+hk,i(v, π^H(v))ω^H is logsingular of order≤k+1alongH^◦. To see this, let GH be the group of complex reflections in G with mirrorH. Each termhk,i is invariant underGH (acting on the germVH^◦), because hk and |φH| are (and the uniqueness of the expansion in powers of log|φH|). Since πH

is the projection onto an eigenspace ofGH, we find that if a⊥H andb∈H, then the real-analytic function hk,i(a,b) on Mp transforms under GH with the same character as φH. Hence this function is divisible by φH with real-analytic quotient. So

α:=h_k,i(πH(v),v)ωH−h_k,i(πH(v), πH(v))ωH

=hk,i(πH(v),v−πH(v))dφH

φ^H is real-analytic. Hence

hk,i(π^H(v),v)ω^H+hk,i(v, π^H(v)ω^H

=hk,i(πH(v), πH(v))(ωH+ωH)+α+α

=2hk,i(πH(v), πH(v))d|φH|

|φ^H| +α+α.

This shows that ηk is logsingular of order ≤k+1.

In order to prove Theorem 3.7, we begin with a few generalities regarding con-jugate complex structures. Denote by V^† the complex vector space V with its conju-gate complex structure: scalar multiplication by λ∈C acts on V^† as scalar multiplica-tion byλ∈C in V. ThenV⊕V^† has a natural real structure for which complex con-jugation is simply interchanging arguments. The ensuing concon-jugation onGL(V⊕V^†) is, when restricted toGL(V)×GL(V^†), also interchanging arguments, whereas on the space of bilinear forms onV×V^†, it is given by h^†(v,v):=h(v,v). So a real point of (V⊗V^†)^∗ is just a Hermitian form on V.

Fix a base point ∗ ∈ V^◦ and identify T_∗V^◦ with V. For κ∈ (C^H)^G, we denote the monodromy representation of∇^κ by ρ^κ∈Hom(π¹(V^◦,∗),GL(V)). Notice thatρ^κ depends holomorphically on κ. Then the same property must hold for

κ∈(C^H)^G→(ρ^κ)^†∈Hom

π1(V^◦,∗),GL(V^†) .

Recall from 2.17 that (C^H)^G is invariant under complex conjugation.

Lemma 3.11. — Let H be the set of pairs (κ,[h])∈ (C^H)^G×P((V⊗V^†)^∗), where h ∈V×V^† → C is invariant under ρ^κ⊗(ρ^κ)^† and let p1 : H→ (C^H)^G be the projection.

Then H resp. p1(H) is a complex-analytic set defined over R (in (C^H)^G×P((V⊗V^†)^∗) resp.

(C^H)^G) and we have p1(H(R))=(R^H)^G.

Proof. — That H is complex-analytic and defined over R is clear. Since p1 is proper and defined over R, p1(H) is also complex-analytic and defined over R.

If κ ∈ (R^H)^G is in the image of H, then there exists a nonzero bilinear map h : V×V^† → C invariant under ρ^κ⊗(ρ^κ)^†. But then both the ‘real part’ ¹₂(h+h^†) and the ‘imaginary part’ ₂^√1₋₁(h −h^†) of h are Hermitian forms invariant under ρ^κ and clearly one of them will be nonzero. The lemma follows.

Proof of Theorem 3.7. — Now let L ⊂ (C^H)^G be a line defined over R. By the preceding discussion, there is a unique irreducible component L˜ of the preimage of L in H which contains (0,[h⁰]). The map L˜ → L is proper and the preimage of 0 is a singleton. Hence L˜ →L is an complex-analytic isomorphism. Since L is defined over R, so are L˜ and the isomorphismL˜ →L. The forms parametrized by L˜(R) de-fine a real line bundle overL(R). Such a line bundle is trivial in the smooth category and hence admits a smooth generating section with prescribed value in 0. We thus find a map κ →h^κ with the stated properties. The proof for the map the Hermitian

form on the cotangent bundle is similar.

If h is a nondegenerate Hermitian form on the tangent bundle of V^◦ which is flat for the Dunkl connection, then ∇ must be its Levi-Civita connection of h (for ∇ is torsion free); in particular, h determines ∇. Notice that to give a flat Hermitian form h amounts to giving a monodromy invariant Hermitian form on the translation space ofA. So h will be homogeneous in the sense that the pull-back ofh under scalar multiplication onV^◦ by λ∈C^× is |λ|^2−2Re(κ0)h.

3.4. The hyperbolic exponent of a complex reflection group. — In case H is a