Aproximaciones a las implicaciones de los imaginarios en el sentido de participación

Pick an elementη∈t∗transverse to WI and define

τt := τ0+tη.

Fix a small enoughε >0 such thatτtis regular for allt∈[−ε, ε]\ {0}.

Now the set{(v, t)∈V ×[−ε, ε] |µ(v) =τt}provides a smooth compact cobor-

dism between the manifoldsµ−1(τ−ε) andµ−1(τε). If theT action on (the V

factor of) this cobordism was regular for all timest this would even establish a cobordism of the T-moduli problems associated to τ−ε and τε and the Euler

classesχV,τ±ε_{would coincide. But regularity fails because of the wall crossing at} t= 0. So we have to cut out a neighbourhood of the locus with singular action and compute the invariant integral over the newly created boundary component. To describe the situation around the singular locus we need some preparation. SinceWI has codimension one we can fix a primitivee1∈Λ such that

hwi, e1i = 0 for alli∈I, and

hη, e1i > 0.

Denote byT1⊂T the subtorus generated by e1 and byt1 ⊂tits Lie algebra. The quotient group and its Lie algebra are denoted by

T0=T /T1 and t0=t/t1

and as before we identifyt∗₀∼={w∈t∗| hw, e1i= 0}. NowT1acts trivially on

VI := {(v1, . . . , vN)∈V |vν = 0 forν6∈I}.

For a small numberδ >0 we define

Wδ := (V ×[−ε, ε])\Nδ,

whereNδ is theδ-neighbourhood of the set

VI×[−ρ, ρ]⊂V ×[−ε, ε]

with

ρ := π·maxν6∈I|hwν, e1i|

hη, e1i

The reason for this choice will become evident below. We chooseδsmall enough such that Nδ is contained in the interior ofV ×[−ε, ε]. HenceWδ is a smooth

connected oriented manifold with C1_-boundary

∂Wδ = (V × {−ε})t(V × {ε})t∂Nδ.

If we orient ∂Wδ as the boundary ofWδ ⊂V ×[−ε, ε] and ∂Nδ as part of this

boundary then

∂Wδ ∼= (−V)tV t∂Nδ.

On Wδ we have the T-action on theV-factor and we consider the equivariant

map

Φ(v, t) := µ(v)−τt.

Denote byMδ := Φ−1(0) the zero set of Φ. Note that Φ(v, t) = 0 implies that

t = π hη, e1i X ν6∈I |vν|2hwν, e1i. (6.1) Now on∂Nδ we havePν6∈I|vν| 2

≤δand hence the intersection ofMδ with the

boundary component∂Nδ is contained in the cylindrical part

Zδ := {(v, t)∈∂Nδ| |t| ≤ρ}=    v∈V X ν6∈I |vν| 2 =δ    ×[−ρ, ρ].

We introduce the notation

VδI :=    v∈V X ν6∈I |vν| 2 =δ    and Sδ :=    (vν)ν6∈I X ν6∈I |vν|2=δ   

and observe thatZδ =VδI×[−ρ, ρ] andV I δ ∼=V

I_×S δ.

Proposition 6.7. For all η in a dense and open subset of t∗ the value0 ∈t∗ is regular forΦ :Wδ −→t∗ and also forΦ|∂Wδ. TheT-action onΦ

−1_{(0) =M}

δ

is regular.

Proof. Pick (v, t) ∈ Mδ. For t 6= 0 the element τt is a regular value for the

moment map µ : V −→ t∗ and hence dΦ(v,t) is onto. This also holds for the differential of the restriction of Φ to the boundary componentsV × {−ε} and V × {ε}. Fort= 0 we haveµ(v) =τ0. Now the image of dΦ(v,t)is spanned by η and the collection (wj)j∈J withJ :={j∈ {1, . . . , N} |vj 6= 0}. Suppose that

(wj)j∈J does not already span all of t∗. Thenτ0 ∈WJ implies that (wj)j∈J∩I

has rankk−1. Otherwiseτ0 would be contained in two different walls, which we assumed not to be the case. So since η is transversal to WI we also get

surjectivity of dΦ(v,t). So for the first statement of the proposition it remains to show regularity for Φ|∂Nδ and in fact only for Φ|Zδ sinceMδ∩∂Nδ ⊂Zδ as we remarked above.

We define

ϕ := Φ|Zδ :Zδ−→t

∗_.

Let (v, t)∈Zδ. The tangent spaceT(v,t)Zδ is given by pairs (x, s)∈V×Rsuch

thatP ν6∈Ig(vν, xν) = 0. Now dϕ(v,t)(x, s) = 2π N X ν=1 g(vν, xν)·wν−sη.

So withJ :={j∈ {1, . . . , N} |vj6= 0}the image of dϕ(v,t) is given by

I :=    X j∈J cjwj−sη ∈t∗ X j∈J\I cj = 0, cj, s∈R    .

Next we note that (wj)j∈Jspanst∗: Fort6= 0 this follows by regularity ofτt. For

t= 0 we see as above that (wj)j∈J∩I has rank k−1. But nowPν6∈I|vν|

2 =δ implies that there is an index l ∈ J \ I and hence wl spans the remaining

dimension.

Given any τ ∈ t∗ we thus find numbers dj ∈Rwith Pj∈Jdjwj =τ. In case

d:=P

j∈J\Idj = 0 this would already prove τ ∈ I. Else we pickej ∈Rwith

P

j∈Jejwj =η. Then ife:=Pj∈J\Iej 6= 0 we can set

cj := dj−

d

eej , s := − d e

to obtainτ∈ I. So for provingI=t∗it suffices to ensure by a suitable choice of ηthat the equationP

j∈Jejwj =ηhas at least one solution withPj∈J\Iej6= 0.

Set

J := {J0⊂ {1, . . . , N} |(wj)j∈J0 is a basis fort

∗_}.

For every J0∈ J we introduce

E(J0) :=    X j∈J0 ejwj X j∈J0\I ej = 0    ⊂t∗ and we choose η∈t∗\ span(wi)i∈I∪ [ J0∈J E(J0) ! .

This ensures that η is indeed transversal to WI. And to obtain the needed

solution ofP

j∈Jejwj=ηwe pick aJ0∈ J withJ0⊂J. We then get a unique

solution ofP

j∈J0ejwj =η and setting allejforj∈J\J0equal to zero we get the desired solution ofP

j∈Jejwj=η withPj∈J\Iej 6= 0.

This finishes the proof of the first statement in the proposition.

The isotropy subgroups of the T-action at points (v, t) ∈ Mδ with t 6= 0 are

finite by regularity of τt. Now suppose (v,0) ∈ Mδ is fixed by a whole 1-

parameter family {exp(te) ∈ T|t ∈ R} for some e ∈ Λ\ {0}. Again we set J :={j∈ {1, . . . , N} |vj 6= 0} and observe

j∈J ⇐⇒ hwj, ei= 0.

But similarly as above we see that (wj)j∈J spans all oft∗, which would imply

e= 0. This gives the desired contradiction.

So if we pick a generic direction ηfor the wall crossing we get a smooth cobor- dism Mδ with regular T-action. If we fix an orientation oft∗ we get induced

orientations on µ−1_(τ

±ε) and Mδ. We equip the boundary components of Mδ

with the induced boundary orientations and obtain

∂Mδ ∼= (−µ−1(τ−ε))tµ−1(τε)tϕ−1(0).

Proposition 6.8. If we orientϕ−1(0)as part of the boundary of Mδ then

χV,τε_−χV,τ−ε _{= −} Z

ϕ−1_(0)/T

π∗ : H_T∗(V)=∼S(t∗)−→R. (6.2)

where π : Zδ −→ V denotes the projection and integration over ϕ−1(0)/T is

understood asT-invariant integration.

Remark 6.9. This wall crossing formula also holds for non-genericη if we use the cobordism property for Euler classes and interprete the integral on the right hand side as the Euler class of a regular T-moduli problem overZδ.