A L
E S
E D L ’IN IT ST T U
F O U R
ANNALES
DE
L’INSTITUT FOURIER
Ursula LUDWIG
The geometric complex for algebraic curves with cone-like singularities and admissible Morse functions
Tome 60, no5 (2010), p. 1533-1560.
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THE GEOMETRIC COMPLEX FOR ALGEBRAIC CURVES WITH CONE-LIKE SINGULARITIES AND
ADMISSIBLE MORSE FUNCTIONS
by Ursula LUDWIG
Abstract. — In a previous note the author gave a generalisation of Witten’s proof of the Morse inequalities to the model of a complex singular curveX and a stratified Morse functionf. In this note a geometric interpretation of the complex of eigenforms of the Witten Laplacian corresponding to small eigenvalues is provided in terms of an appropriate subcomplex of the complex of unstable cells of critical points off.
Résumé. — Dans une note précédente, l’auteur a donné une généralisation de la preuve de Witten des inégalités de Morse pour le cas modèle d’une courbe algébrique complexe singulière et d’une fonction de Morse stratifiée. Le but de cette note est de donner une interprétation géométrique du complexe des formes propres du Laplacien de Witten pour des petites valeurs propres à l’aide d’un sous-complexe approprié du complexe des cellules instables.
1. Introduction
Let M be a smooth compact manifold of dimension dim(M) = n. Let f :M →Rbe a Morse function on M, i.e. a function such that for each critical pointp(df(p) = 0) the HessianHessp(f)off inpis non degenerate (as a symmetric bilinear form onTpM). The number of negative eigenvalues of the HessianHessp(f)is called the index off inp. We denote byCriti(f) the set of critical points of f of indexi and by ci(f) := #Criti(f). The celebrated Morse inequalities state that there is a relation between the number of critical points off and the Betti numbers ofM.
A way to prove the Morse inequalities is to show the existence of a complex(C∗, ∂∗) of vector spaces such thatdimCi =ci(f) and such that
Keywords:Morse theory, Witten deformation, Cone-like Singularities.
Math. classification:58Axx, 58Exx.
the homology of the complex is isomorphic to the singular homology ofM. The Morse inequalities follow from the existence of such a complex by a simple algebraic argument. The existence of a complex with the above properties has been shown by geometrical methods by Thom and Smale:
The chain groups of the Thom-Smale complex are generated by the critical points off, the boundary operator is defined by “counting trajectories” of the negative gradient flow (for a generic metric) between critical points of index difference1.
In [16] Witten proposed a different, purely analytical proof of the Morse inequalities. A rigorous account of the analytic proof of the Morse inequal- ities using semi-classical analysis has been done in [8]. The main idea of Witten’s method consists in deforming the de Rham complex(Ω∗(M), d) by means of the Morse function f into a complex (Ω∗(M), dt), where dt = e−tfdetf and t ∈ (0,∞) denotes the deformation parameter. The mapω7→etfω induces an isomorphism of the two complexes and therefore (1.1) H∗ (Ω∗(M), dt)
'H∗ (Ω∗(M), d)
'H∗(M).
The last isomorphism in (1.1) is just the well-known de Rham isomorphism.
Let us denote byδt:=etfδe−tf the adjoint of dtand by
∆t=dtδt+δtdt
the Witten Laplacian. The Hodge theorem for the deformed complex (Ω∗(M), dt)states that
ker(∆t)'H∗ (Ω∗(M), dt) .
The advantage of the deformed complex compared to the initial de Rham complex is that the spectral properties of the Witten Laplacian are “nice”.
In particular one can show that for large deformation parametertthere is a
“gap” in the spectrum of the Witten Laplacian, i.e.spec(∆t)∩(e−ct, Ct) =∅ for somec, C >0. Moreover, for06i6dim(M), the number of eigenvalues (counted with multiplicities) of∆t|Ωi(M)contained in the interval [0,1]is equal to ci(f). We denote by Fit ⊂ Ωi(M) the ci(f)-dimensional vector space generated by the eigenspaces of∆t|Ωi(M)corresponding to eigenvalues in[0,1]. One thus gets a finite dimensional subcomplex(F∗t, dt)of(Ω∗, dt), with
H∗ (F∗t, dt)
'ker(∆t)'H∗(M)
and as indicated above the Morse inequalities (for cohomology) follow.
Witten further suggested in [16] that under some genericity conditions from the complex(F∗t, dt)one can recover the Thom-Smale complex asso- ciated to the Morse functionf by lettingt → ∞. Again a rigorous proof
based on semi-classical analysis can be found in [8]. In [1] Bismut and Zhang gave another proof of this “comparison theorem” using a result of Laudenbach in [9] describing the geometry of the boundary of the unstable cells of the singular points off. (The result is used in the sequel to give an extension of a theorem of Cheeger and Müller on the relation between the Ray-Singer analytic torsion and the Reidemeister torsion.)
In [10] (see also [12]) a generalisation of Witten’s proof of the Morse inequalities to the model of a singular complex algebraic curve and strati- fied Morse functions (in the sense of the theory developed by Goresky and MacPherson in [7]) is given. The model functions considered in [12] were called admissible Morse functions. One can assume that all singularities p ∈ Σ := Sing(X) are unibranched. For p ∈ Σ we denote by m(p) the multiplicity ofX atp. In this situation the Witten method consists in de- forming the complex(C, d)ofL2-integrable forms (instead of the de Rham complex) by means of an admissible Morse functionsf. One can then show that also in this situation the “spectral gap” theorem for the Witten Lapla- cian holds and the vector spaceFit of eigenforms of the Witten Laplacian to small eigenvalues has dimension
dimFit=ci(f) :=
(ci(f|X\Σ) i= 0,2, c1(f|X\Σ) +P
p∈Σ(m(p)−1) i= 1.
The below Morse inequalities for theL2-Betti numbersb(2)i (X)ofX now follow by a simple algebraic argument:
k
X
i=0
(−1)k−ici(f)>
k
X
i=0
(−1)k−ib(2)i (X)fork= 0,1,
2
X
i=0
(−1)ici(f) =
2
X
i=0
(−1)ib(2)i (X).
(1.2)
Since the situation treated in [10] is a model for a singular algebraic curve and certain stratified Morse functions on it as explained in [10], from (1.2) one gets back the Morse inequalities for intersection homology of middle perversity which were already known by [7].
The goal of this note is to generalise the second part of Witten’s pro- gram to the singular situation described above, i.e. to provide a geometric interpretation of the complex(F∗t, dt).
First one has to investigate the structure of the unstable set of points inCrit(f) := Crit(f|X\Σ)∪Σ. Using the structure of the boundary of the unstable sets one can then construct a subcomplex(C∗u0, ∂∗)of the complex
of unstable cells as well as an appropriate basis
{ep1, p∈Crit(f)\Σ} ∪ {epi, p∈Σ, i= 1, . . . , m(p)−1}.
The main result of this article is a comparison theorem between the combinatorial complex (C∗u0, ∂∗) and the complex (F∗t, dt). Let us denote by{Ψp1(t), p∈Crit(f)\Σ} ∪ {Ψpi(t), p∈Σ, i= 1, . . . , m−1}the basis of Ftconstructed in [10] (and recalled in section 2.2). The map
(1.3) R(t) : Hom (C∗u0, ∂∗),R
→(Fet
∗, d), [epi]∗7→etfΨpi(t), is an isomorphism into a subcomplex(Fet, d)of the complex ofL2-integrable forms. One can show that integration yields a well-defined morphism of complexes
(1.4) P∞,t: (eF∗t, d) −→ Hom (C∗u0, ∂∗),R .
We are now ready to state the two main results of the article. The two theorems below generalise Theorem 6.11 and Theorem 6.12 in [3] re- spectively to the singular situation. Denote by F ∈ End(Hom(Ciu0,R)) the homomorphism which acts on[epj]∗ by multiplication with f(p). With I ∈End(Hom(Ciu0,R))we denote multiplication byi.
Theorem 1.1. — The asymptotic behaviour ofP∞,t◦R(t)ast→ ∞is (1.5) P∞,t◦R(t) =etFπ
t
(I−1)/2
(1 +O(e−ct)).
In particular for larget the linear map of vector spacesP∞,t is an isomor- phism.
Theorem 1.2. — There existsc >0 such that fort→ ∞,
R(t)−1◦d◦R(t) = rt
π(1 +O(e−ct))−1e−tF∂∗etF(1 +O(e−ct)).
This note is organised as follows: In Section 2 basic facts on the L2- cohomology of a singular space having cone-like singularities are recalled.
Also, for convenience of the reader the results in [10] are summarised. In particular the construction of the basis
{Ψp1(t), p∈Crit(f)\Σ} ∪ {Ψpi(t), p∈Σ, i= 1, . . . , m−1}
ofFtis explained.
Section 3.1 describes the structure of the boundary of the unstable set for the critical points of the admissible Morse function, thus extending the result in [9] to the singular situation. In Section 3.2 we define the subcomplex (C∗u0, ∂∗)of the complex of unstable cells. One can moreover
show that integration of L2-integrable forms on the cells in (C∗u0, ∂∗) is well-defined and Stokes formula holds.
In Section 4.1 the two comparison theorems Theorem 1.1 and Theo- rem 1.2 are proved by extending the proofs of the corresponding statements for smooth manifolds in [3]. Section 4.2 shortly explains the duality pair- ing for the geometric complexes of the stable and unstable cells and its compatibility with the Poincaré duality.
Some explicit computations are postponed to the appendix. The results contained in this paper were announced in [11].
This article together with [10] is a first attempt to make the Witten method accessible to singular spaces. Note however that the generalisation of both parts to higher dimensional spaces promises to be more involved, one of the problems thereby being that the natural metrics on algebraic varieties, i.e. those induced from a metric on projective space are in general not of cone-type.
Acknowledgements. — The author wishes to thank J.M. Bismut for many helpful discussions and for suggesting work on the subject. The au- thor was supported by a DFG-grant.
2. Witten deformation on a singular curve by means of an admissible Morse function
2.1. L2-cohomology and the Witten deformation
In this note we deal with the following situation: Let(X, g)be the model of an algebraic curve with unibranched singularities, i.e. (X, g) is a Rie- mannian space ofdimX = 2with cone-like singularitiesΣ :={p1, . . . , pN} of multiplicitiesmi=m(pi)∈N,mi >2. More precisely
• X is a topological space, such thatX−Σis a smooth manifold.g is a Riemannian metric onX−Σ.
• There exist open neighbourhoodsU(pi)ofpi inX,i∈ {1, . . . , N}, such thatX−SN
i=1U(pi)is a smooth compact manifold with bound- ary.
• The open set(U(pi)−pi, g|Ui−pi)is isometric to (cone(Sm1i), dr2+r2dϕ2)
for some > 0. Hereby for m ∈ N we denote by Sm1 the circle of length2πmand bycone(Sm1) :=
(r, ϕ)|r∈(0, ), ϕ∈Sm1 .
Definition 2.1. — Letf :X →Rbe a continuous function, which is smooth outside the singularities ofX. The functionf is called an admissible Morse function if the following conditions are satisfied
(1) Each critical pointp∈X−Σoff is a non-degenerate critical point.
(2) Let p ∈ Σ be a singular point of X. Then there exist ap, bp ∈ R,(ap, bp)6= (0,0), such that the functionf has the following form in local coordinates(r, ϕ)nearp:
f(r, ϕ) =f(p) +r(apcos(ϕ) +bpsin(ϕ)).
Remark 2.1. — Note that after change of coordinates on the linkSm1 and rescalingt t0=q
a2p+b2p·tone can always assume that(ap, bp) = (1,0) in the above definition.
We denote by Crit(f) :={p∈X\Σ|df(p) = 0} ∪Σthe set of critical points off. Forp∈X−Σa (smooth) critical point off the indexind(p) off in pis defined as the number of negative eigenvalues of the Hessian off inp. Each singular pointp∈ΣofX is considered to be a critical point off of index ind(p) = 1. Fori = 0,1,2 we denote by Criti(f) the set of critical points off of index i.
A Riemannian singular spaceX ofdimX= 2as above is a metric model for a singular complex projective algebraic curve. An admissible Morse function on X is a model for a stratified Morse function on a complex curve in the sense of the theory developed by Goresky/MacPherson in [7].
Let us explain this in more detail: LetC⊂Pn(C)be a complex projective algebraic curve. Let p ∈ C be a singular point of C and denote by Cj, j = 1, . . . , s, the analytic branches of C at p. Then for each branch Cj
there exist open neighbourhoods Vj ⊂ C of 0 resp. U(p) ⊂ Pn(C) of p, as well as affine coordinates z1, . . . , zn onU(p) and a normalisation map defined by
π:Vj⊂C→U(p)∩Cj
t7→(z1(t), . . . , zn(t)) = (tmj, tqj2fj2(t), . . . , tqjnfjn(t)), such that π|Vj−{0} is a biholomorphic map. Hereby mj < qj2 < qj3 <
. . . < qjn and fjk(0) 6= 0, for k = 2, . . . , n. The multiplicity mj of Cj atpis an analytic invariant, i.e. it does not depend on the choice of local coordinatesz1, . . . , zn.
We denote by eg the Riemannian metric on Cj induced by the Fubini- Study metric onPn(C). Then the metricπ∗egonVj− {0} ⊂Cis isometric to the metric(m2j |t|2(mj−1)+O(|t|2mj−1))dt⊗dt.Moreover the map
Π : (|t|,arg(t))→(|t|mj, mj·arg(t))
induces an isometry from(Vj− {0}, π∗eg)to (cone(Sm1
j),(1 +O(r1/mj))(dr2+r2dϕ2))
(see e.g. [13]). Thus in particular (Cj,eg) is quasi-isometric to a cone-like singularity of multiplicitymj.
The affine line l := {z2 = . . . = zn = 0} is the tangent line to the irreducible branchCj. LetF :Pn(C)∩U(p)→Cbe a holomorphic function such thatf := Re(F)|C : C∩U(p)→ Ris a stratified Morse function in the sense of [7] (Part II). The non-degeneracy condition in [7] (for the branchCj) implies that locally nearpthe functionF has the form
F =F(p) +X
aizi+O(z2), wherea16= 0. One checks easily that
f ◦π◦Π−1:
cone(Sm1)−→R, (r, ϕ)7→r(Re(a) cos(ϕ)−Im(a) sin(ϕ)) +O(r1+δ) for some δ > 0. The leading term is thus of the form given in Defini- tion 2.1 (2).
Let us now recall the main features of theL2-cohomology of(X, g). Let (Ω∗0(X −Σ), d) be the de Rham complex of differential forms acting on smooth forms with compact support. Anideal boundary condition for the elliptic complex(Ω∗0(X−Σ), d)is a choice of closed extensionsDk ofdk in the Hilbert space of square integrablek-forms, such that
Dk(dom(Dk))⊂dom(Dk+1).
We then get a Hilbert complex
0→dom(D0)−−→D0 . . . .−−−→Dn−1 dom(Dn)→0.
(See [4] for the general theory for Hilbert and Fredholm complexes). The minimal and maximal extension ofd
dmin:=d= closure ofd,
dmax:=δ∗= adjoint of the formal adjointδofd are examples ofideal boundary conditions. As shown in [5] in the case of cone-like singularities we haveuniquenessofideal boundary conditions, i.e.
(2.1) dk,min=dk,max for allk.
The equation (2.1) is also called the L2-Stokes theorem. We denote by (C, d, < , >)the unique extension of the de Rham complex
(Ω∗0(X−Σ), d, <, >)
to a Hilbert complex. The cohomology of this complex is the so-calledL2- cohomology ofX
H(2)i (X) := kerdi,max/imdi−1,max= kerdi,min/imdi−1,min. Note that the validity of (2.1) does not imply the essential self-adjointness of the Beltrami-Laplace operator∆|Ω∗
0(X\Σ) =dδ+δd (acting on smooth compactly supported forms). Instead it is equivalent to the self-adjointness of the particular extension∆ =dminδmin+δmindmin.
Moreover the L2-Hodge theorem for Riemannian spaces with cone-like singularities (see [5], Section 1) states that the complex (C, d, < , >) is Fredholm and that the canonical maps
(2.2) ker(∆i)→H(2)i (X), i= 0, . . . , n,
are isomorphisms. (In particularrange(di)is closed for alli and therefore reduced and unreducedL2-cohomology coincide in this case.)
By generalising Witten’s idea to the singular situation described above one can deform the complex ofL2-forms by means of an admissible Morse function. I.e. one starts with the differential complex
(Ω∗0(X−Σ), dt, < , >),
wheredt=e−tfdetf andt∈(0,∞). As shown in [10] the complex (Ω∗0(X−Σ), dt, < , >)
also hasunique ibc. We denote the unique extension of (Ω∗0(X−Σ), dt, < , >)
into a Hilbert complex by(Ct, dt, < , >). It is not difficult to see that there is an isomorphism of Hilbert complexes:
e−tf : (C, d, <, >t)→(Ct, dt, <, >), where by< α, β >t=R
α∧∗βe−2tf we denote the twisted metric. Therefore the complex(Ct, dt, <, >)is also a Fredholm complex whose cohomology is isomorphic to theL2-cohomology ofX, i.e.
(2.3) H∗ (Ct, dt, <, >)
'H(2)∗ (X).
Let us denote by δt the formal adjoint of the operator dt with respect to theL2-metric< , >and by∆t|Ω∗
0(X−Σ)= (dt+δt)2 the corresponding Laplacian (acting on smooth compactly supported forms). Then we have
the following identities (c.f. e.g. [3], Proposition 5.5.) on smooth forms with compact support outside the singularity:
dt=d+tdf∧,
δt=etfδe−tf =δ+t∇f ,
∆t= ∆ +t2k∇fk2+t(L∇f+L∗∇f), (2.4)
where we denote by L∇f = d(∇f ) +∇f d the Lie derivative in the direction of the gradient vector field∇f and byL∗∇f its adjoint. Note that the operatorMf :=L∇f+L∗∇f is a zeroth order operator.
The Hodge theorem holds for the complex(Ct, dt, <, >), i.e.
(2.5) Hi((Ct, dt, <, >))'kerdt,i∩kerδt,i−1'ker ∆t,i, where∆tdenotes the closed extension of∆t|Ω∗
0(X\Σ) with domain:
(2.6) dom(∆t) = Φ
Φ, dtΦ, δtΦ, dtδtΦ, δtdtΦ∈L2 Λ∗(X−Σ) . Note however that dom(∆t) 6= dom(∆) in the presence of singularities and therefor we have to indicate carefully the domain of definition of each operator. The operator ∆t is called the Witten Laplacian. It is a self- adjoint, nonnegative, discrete operator. The main result in [10] states
Theorem 2.2. — (a) (Spectral gap theorem) Let (X, g) be a Riemannian space as above and let f : X → R be an admissible Morse function. Then there exist constantsC1, C2, C3>0andt0>
0depending onX andf such that for anyt > t0 spec(∆t)∩(C1e−C2t, C3t) =∅.
(b) For large parameter t, the subcomplex(Ft, dt, <, >)of the complex (Ct, dt, <, >)generated by the eigenforms of ∆t,i to eigenvalues in [0,1]satisfies
(2.7) rk(Fit) =ci(f) :=
#Criti(f|X−Σ) fori= 0,2,
#Crit1(f|X−Σ) +X
p∈Σ
(m(p)−1) fori= 1.
By (2.3) and (2.5) we know that the cohomology of the complex (F∗t, dt, <, >)
is isomorphic to theL2-cohomology ofX. By a standard algebraic argument one can therefore deduce the Morse inequalities (1.2) from Theorem 2.2.
2.2. Construction of a basis of Ft
In the remaining of this section we recall the construction of a basis of (Ft, dt, <, >) (see [10], [12]). This basis will be needed in Section 4 to construct the comparison morphism. Let us first construct a local model for the Witten Laplacian near a singular pointpof multiplicitym(p)ofX. Let us denote by cone(Sm1) the infinite cone (0,∞)×Sm1 equipped with the metricdr2+r2dϕ2. We considerf : cone(Sm1)→R, f(r, ϕ) = rcosϕ.
By ∆|Ω∗
0(cone(S1m)) we denote the Laplace operator acting on compactly supported smooth forms oncone(S1m). Let us consider the operator (2.8) ∆t,i|Ω∗
0(cone(Sm1)):=dt,i−1δt,i−1+δt,idt,i. In view of the formulas in (2.4) we get
(2.9) ∆t,i|Ω∗
0(cone(Sm1))=∆i|Ω∗
0(cone(S1m))+t2 fori= 0,1,2.
The model Witten Laplacian ∆t is defined to be the self-adjoint exten- sion of∆t|Ω∗
0(cone(Sm1)) with domain dom(∆t) =
Φ
Φ, dtΦ, δtΦ, dtδtΦ, δtdtΦ∈L2 Λ∗(cone(Sm1)) . Theorem 2.3. — (a) spec(∆t,i) =
t2,∞
in casei= 0,2.
(b) spec(∆t,1) = {0} ∪ t2,∞
and dim ker(∆t,1) =m−1. Moreover {ωjp(t)|j= 1, . . . , m−1}(see Appendix) form an ONB ofker(∆t,1).
Proof. — See [10], [12].
We make the additional assumption that in a neighbourhood of p ∈ Critk(f)−Σ, k = 0,1,2 the metric g is the Euclidean flat metric. The local model operator ∆pt,i is closely related to a harmonic oscillator and has been studied in [16]. It is well-known that∆pt is a nonnegative, essen- tially self-adjoint, elliptic operator withker∆pt = ker∆pt,k = span{ωp1(t)}, where ω1p(t) := p
t/πe−tkxk2/2dx1∧. . .∧dxk. (Hereby x1, . . . , xk denote the coordinates in the Morse Lemma, i.e.
f =f(p)−1/2(x21+. . .+x2k) + 1/2(x2k+1+. . .+x22) nearp.)
Now we choose > 0 such that the open neighbourhoods B2(p), p ∈ Crit(f), are pairwise disjoint. We denote byν : [0,∞)→[0,1]a smooth cut-off function, which equals1in the interval [0, /2]and is equal to 0in the interval(,∞). The index setIpis defined by
Ip:=
({1, . . . , m−1} forp∈Σof multiplicitym, {1} forp∈Crit(f)\Σ.
Forp∈Crit(f),j ∈Ip we define
Φpj(t) :=βjp(t)−1ν(r)ωjp(t)
whereβpj(t) =kν(r)ωpj(t)k=kωpj(t)k+O(e−ct). The formsΦpj(t) can be identified with forms in(Ct, dt, <, >).
We denote byP(t,[0,1])the orthogonal projection operator fromCtonFt
(with respect to the metric<, >). Then
{Ψpj(t) :=P(t,[0,1])(Φpj(t)), p∈Crit(f), j∈Ip} is a basis forFt. In section 4 we will need the following proposition:
Proposition 2.4. — The set
{Φpj(t)|p∈Crit(f), j∈Ip}
forms an ONB of span{Φpj(t) | p ∈ Crit(f), j ∈ Ip} with respect to the L2-norm.
Proof. — The proposition follows from Lemma 5.1 in the Appendix, the definition of theβp(t)’s and the fact that, by construction, for p6=q the supports ofΦpj and Φqk are disjoint, for allj∈Ip, k∈Iq.
3. The geometric complex
3.1. Unstable/stable sets of critical points
In this section we will show that the singular spaceXhas a decomposition
(3.1) X = G
p∈Crit(f)
Wu(p),
where for a critical pointp∈Crit(f)we denote byWu(p)its unstable set (see Definition below). The main result of this section is Proposition 3.2, which describes the boundary of eachWu(p), or in other terms “the way the cells are attached to each other”.
Let us denote by −∇gf the negative gradient vector field of f. Note that −∇gf is defined only on X \Σ. We denote by Φ the induced flow.
The flow Φis not defined for all time t ∈ R. However we can define the stable/unstable set for all critical points off (including pointsp∈Σ):
Let p∈ Crit(f). Then the stable (resp. unstable) set of pis defined as follows:
Ws(p) =
x∈X | ∃t+(x)>0, such that lim
t→t+(x)
Φ(x, t) =p ∪ {p}, resp.Wu(p) =
x∈X | ∃t−(x)<0, such that lim
t→t−(x)
Φ(x, t) =p ∪ {p}
.
Thus for a critical pointp∈Crit(f)\Σthe definition above coincides with the usual definition of the stable/unstable set. For p ∈ Σ by the above definition we includedp∈Wu/s(p).
Ifp∈X−Σis a critical point off of indexind(p)it is well-known that the stable (resp. unstable manifold) is a (non closed) manifold of dimension dimWs(p) = 2−ind(p)(resp.dimWu(p) = ind(p)), see e.g. [14].
Proposition 3.1. — Letp∈Σbe a singular point ofX of multiplicity m, thenWu(p)− {p} as well asWs(p)− {p} are manifolds of dimension1 havingmconnected componentsWju/s(p),j∈Z/m(p)Z.
Proof. — The picture below describes the gradient flow in the neigh- bourhood of a singular point (the picture below represents the casem= 3;
note thatϕ∈[0,6π]). By definition of an admissible Morse function there exists a neighbourhoodU(p)ofpin X such that f has the following form in local coordinates(r, ϕ)nearp:
f(r, ϕ) =f(p) +r apcos(ϕ) +bpsin(ϕ) .
As explained in Remark 2.1 we can assume thatap = 1andbp = 0. Then
−∇f(r, ϕ) = −(cos(ϕ),−r−1sin(ϕ)). In particular −∇f(r,(2k+ 1)π) = (1,0) and−∇f(r,2kπ) = (−1,0). Therefore it is easy to see that the local stable (resp. unstable) set ofpare given by
(3.2) Ws(p)∩U(p) = G
i∈Iep
Wis,loc(p)G {p}
(3.3) resp.Wu(p)∩U(p) = G
i∈eIp
Wiu,loc(p)G {p},
where fori∈Iep:=Z/m(p)Zwe denote by Wis,loc(p) :=
(r, ϕ)∈U(p)|r∈R+, ϕ= 2iπ , Wiu,loc(p) :=
(r, ϕ)∈U(p)|r∈R+, ϕ= (2i+ 1)π . (3.4)
p
1
Thus the assumption holds locally nearp. The global statement is shown as usual by “moving the charts” by means of the flow.
An orientation is chosen for allWu(p),p∈Crit(f)−Σ. Forp∈Σ, j∈Iep
the cellsWju(p)are oriented by the negative gradient flow. We denote by
−Wu(p)the cell Wu(p)with its opposite orientation.
It is easy to see that by perturbing the metricg outside of a neighbour- hood ofΣwe can assume that the gradient vector field∇gfis Morse-Smale, i.e. all intersections of stable and unstable manifolds are transversal.
The following proposition is a generalisation of Proposition 2 in [9] to the present situation:
Proposition 3.2. — Letf be an admissible Morse function such that
∇gf satisfies the Morse-Smale condition. Then for each critical point p∈ Crit(f)−Σ the closure Wu(p) is a stratified space. Let p∈ Crit(f)\Σ.
Then the strata ofWu(p)\Wu(p)can be of the following form:
(a) Wu(q), forq∈X−Σ, ind(q)<ind(p),
(b) Wju(q), forq∈Σ,j∈Ieq and1 = ind(q)<ind(p), (c) {q}, forq∈Σ,ind(q)<ind(p).
Moreover the strata of type (b) “come in pairs”, i.e. if there exists j ∈ Ieq such that Wju(q) ⊂ ∂Wu(p) then Wj−1u (q) ⊂ ∂Wu(p) or Wj+1u (q) ⊂
∂Wu(p). Moreover ifWu(p)has2connected components nearWju(q)then
Wju(q)is the boundary of one of these, while−Wju(q)is the boundary of the other one.
Moreover for p∈Σ, i∈Iep we have Wiu(p)'(0,1) and Wiu(p)'[0,1]
where one end of the compactification corresponds topand the other end corresponds to someq∈Crit0(f).
Remark 3.3. — The analogous result holds for the closures of the sta- ble cells.
Proof. — Note first that by the Morse-Smale condition we can always assume that the critical values are pairwise distinct. Note moreover that the statement of the proposition is obvious ifpis a critical point of index0 or1.
Let p be a critical point of index 2. For a ∈ R we denote by Xa :=
Wu(p)∩f−1(a). Ifa < f(p)is such that[a, f(p)]contains no critical value, then
(3.5) Xa 'S1.
Asadecreases this remains true as long as we don’t pass a critical value.
Let nowa1be the first critical value off witha1< f(p). By our assump- tion on the critical values of the Morse function there is a unique critical pointq1∈Crit(f)witha1=f(q1).
Case 1. — IfWu(p)∩Ws(q1) =∅thenXa1−'S1.
Case 2. — Assume that Wu(p)∩Ws(q1) 6= ∅. Then ind(q1) < 2. If ind(q1) = 0. Then obviously Xa1 ' {∗} and Xa1− =∅. (Thus Wu(p) = Wu(p)∪ {q1} in this case.)
Therefore assumeind(q1) = 1. ThenXa1−is no longer a closed manifold.
Ifq16∈Σthis is a consequence of Lemma 4 in [9] and the situation is already well-understood. Ifq1∈Σ, let us denote by(Y1, Y10)the pair of sets:
(3.6) (Y1, Y10) := (Xa1+, Xa1+∩Ws(q1))'
S1, G
i∈eI
0 q1⊂Ieq1
{∗i}
. We have the following easy lemma:
Lemma 3.4. — Each connected component ofY1−Y10 is mapped diffeo- morphically (by means of the flow) to a submanifold off−1(a1−), which is diffeomorphic to an open interval and the closure of which is either home- omorphic toS1 (case (i)) or to[0,1](case (ii)).
We continue the proof of Proposition 3.2. In case (i) we deduce that there exists a j ∈Ieq1 such that Wu(p) has 2 connected components near Wju(q1),Wju(q1)is the oriented boundary of one of them while−Wju(q1) is the oriented boundary of the other one. In case (ii) we deduce that there is aj such thatWju(q1)∪ −Wj−1u (q1)⊂∂Wu(p).
We continue the process by studying the setY2:= Φ(Y1−Y10,R)∩Xa1− when passing the next critical pointq2∈Crit(f). AgainY2stays unchanged ifWu(p)∩Ws(q2) =∅. IfWu(p)∩Ws(q2)6=∅andind(q2) = 0at least one of the connected components of Y2 will be mapped to q2 under the flow.
Let us now assume thatWu(p)∩Ws(q2)6=∅andind(q2) = 1.Denote by (3.7) (Y2, Y20) := (Y2, Y2∩Ws(q2)).
Then by [9] and a generalised version of Lemma 3.4 we deduce that each connected component of Y2−Y20 is mapped under the flow into a submanifold off−1(a2−)which is diffeomorphic to one of the following intervals:
(3.8) (0,1) or[0,1) or(0,1]or [0,1].
Each open end corresponds to a boundary componentWju(q2), j∈Ieq2 of Wu(p). Again this components come in pairs.
Since Crit(f) is finite the process described above finishes after a finite number of steps and we therefore get the result.
3.2. The complex of unstable cells (C∗u, ∂∗)and its subcomplex (C∗u0, ∂∗)
In this section we define the chain complex (C∗u, ∂∗) “generated by the unstable manifolds” of the critical points of f. The chain groups of the complex(C∗u, ∂∗)are defined as follows
C2u:= M
p∈Crit2(f)
R·[Wu(p)],
C1u:= M
p∈Crit1(f) p6∈Σ
R·[Wu(p)]⊕M
p∈Σ j∈Iep
R·[Wju(p)],
C0u:= M
p∈Crit0(f)
R·[Wu(p)]⊕M
p∈Σ
R·[{p}].
(3.9)
Note that sinceCrit(f)is finite the above chain groups are well-defined.
The boundary of a generatorσ∈Ciuis defined by
(3.10) ∂σ=X
n(σ, θ)·θ,
where the sum is taken over all generators ofCi−1 and wheren(σ, θ) = 0 ifθis not in the closure ofσ. Moreover ifθis in the closure ofσwe define n(σ, θ)as follows: Nearθthe cellσhasn=n++n− connected components such thatθis the oriented boundary ofn+ of these and−θ is the oriented boundary of the othern−. Then
(3.11) n(σ, θ) =n+−n−.
It is not difficult to verify that∂2= 0.
In the case of a Morse function on a smooth manifold one can give an interpretation of the coefficientsn(σ, θ)by counting trajectories of the gradient flow between critical points of index difference1. In our situation we can not do so.
Forp∈Σandj∈Ip denote byσju:=Wju(p)∪ −Wj−1u (p)∪ {p} and by [σju] := [Wju(p)]−[Wj−1u (p)]. We denote by
C2u0 :=C2u, C1u0 := M
p∈Crit1(f) p6∈Σ
R·[Wu(p)]M span
[σju(p)]|p∈Σ, j∈Ip ,
C0u0 := M
p∈Crit0(f)
R·[Wu(p)].
(3.12)
Proposition 3.5. — (C∗u0, ∂∗)is a subcomplex of(C∗u, ∂∗).
Proof. — From Proposition 3.2 we deduce that for allσ∈C2u0 we have
∂σ∈C1u0, and for allσ∈C1u0 we have∂σ∈C0u0. Remark 3.6. — The decomposition of X into unstable cells is a CW- decomposition ofX and therefor we have H∗((C∗u, ∂∗))'Hsing(X). Since X has dimension 2 and is (topologically) normal we have moreover that Hsing(X) ' IH∗(X) (see [6], section 4.2 and 4.3). It is not difficult to show that the inclusion of complexes (C∗u0, ∂∗) ,→ (C∗u, ∂∗) is a quasi- isomorphism. Thus the complex (C∗u0, ∂∗) computes the intersection ho- mology ofX. Note however that the cellsσuj are not allowed in the sense of intersection homology.
3.3. Stokes theorem
Denote by (Ω∗(X \Σ), d) the differential complex of smooth forms on X\Σ. Let
(3.13) 0−→ D0−−−→ DdD,0 1−−−→ DdD,1 2−→0 be the differential complex with
(3.14) Di:={α∈L2(Λi(X))∩Ω∗(X\Σ), dα∈L2(Λi+1(X))}.
Fori= 0,1,2define by
(3.15) H(2),Di (X) = kerdD,i/imdD,i−1
thei-th cohomology of the complex (D, d). There is a natural morphism (3.16) i(2):H(2),D∗ (X)→H(2)∗ (X),
which by [5] is an isomorphism.
We can now use the results in Section 3.1 to prove the following propo- sition
Proposition 3.7. — Leti= 1,2andω∈ Di−1. Denote by σ (1) σ:=Wu(p), wherep∈Criti(f)\Σor
(2) σ:=σju(p), wherep∈Σ,j∈Ip(in casei= 1).
Then the Stokes formula holds, i.e. one has (3.17)
Z
σ
dω= Z
∂σ
ω.
In particular both sides of (3.17)are well-defined.
Proof. — As shown in Proposition 6 in [9] the claim holds ifσ∩Σ =∅.
To prove the proposition it is therefore enough to treat the following 2 cases.
Case 1. — σ=Wu(p)wherep∈Crit2(f)andσ∩Σ =q. LetB(q)be an-neighbourhood of qinX. Then the usual Stokes formula gives (3.18)
Z
σ\B(q)
dω= Z
∂(σ\B(q))
ω.
We get the claim by letting → 0: Since ω ∈ L2(Λ1(X)) and dω ∈ L2(Λ2(X))we have ω =O(rβ)dr+O(rγ)dϕ for someβ >−1 and γ >0 anddω=O(rα)drdϕ for someα >0. Therefore we get for the right hand side of (3.18):
(3.19)
Z
σ∩B(q)
dω= Z
σ∩B(q)
O(rα)drdϕ6Cα→0.
Moreover for the left hand side of (3.18) we use that (3.20)
Z
∂(σ∩B(q))
ω= Z
∂(σ∩B(q))
O(rβ)dr+O(rγ)dϕ→0.
Case 2. — σ=σju(p)forp∈Σ,j ∈Ip can be treated similarly.
Remark 3.8. — Note that as a corollary of the above proposition we get a second proof of the fact that(Cu0, ∂∗)is a complex, i.e. that∂2= 0.
4. Relation to the geometric complex 4.1. Proof of the main theorems
For p∈Crit(f)\Σdefineep1 := [Wu(p)], Wu(ep1) :=Wu(p). Forp∈Σ andj∈Ip define
(4.1) epj :=X
l∈Ip
alj[σul(p)], Wu(epj) :=X
l∈Ip
aljσul(p)
whereA= alj
l,j ∈GL(m−1,R)is defined in Lemma 5.3. Let us equip Ciu0 with the unique metric such that {epj | p ∈ Crit(f), j ∈ Ip} is an orthonormal base.
We denote by Ji(t) : Hom(Ciu0,R) −→ Cti the linear map defined by Ji(t)([epj]∗) = Φpj(t).From Proposition 2.4 we deduce thatJi(t)is an isom- etry fromHom(Ciu0,R)into the image ofJi(t).
Denote by (Ft, dt, <, >) the subcomplex of (Ct, dt, <, >) generated by the eigenforms of ∆t to eigenvalues λ ∈ [0,1]. We denote by P(t,[0,1]) the orthogonal projection operator fromCtonFt(with respect to the met- ric<, >).
Proposition 4.1. — There exist a constantc >0and aL2-integrable functionρ:X →Rsuch that for allv∈Hom(Ciu0,R)and allx∈X:
(4.2)
[(Pi(t,[0,1])◦Ji(t)−Ji(t))v](x)
=ρ(x)O(e−ct)kvk.
Proof. — See [10] Proposition 7.1 (see also [12]).
We denote by ∆ftthe Laplacian associated to the complex (C, d, <, >t).
Denote by (Fet, d, <, >t) the subcomplex of (C, d, <, >t) generated by the eigenforms of∆ftto eigenvaluesλ∈[0,1]. Denote byP(t,e [0,1])the orthog- onal projection fromC toFetwith respect to the metric<, >t. Then
