Witten’s perturbation on strata with general adapted metrics

WITTEN’S PERTURBATION ON STRATA WITH GENERAL ADAPTED METRICS JES´ US A. ´ ALVAREZ L ´ OPEZ, MANUEL CALAZA, AND CARLOS FRANCO Abstract. Let Mbe a stratum of a compact stratified space A. It is equipped with a general adapted metric g, which is slightly more general than the adapted metrics of Nagase and Brasselet-Hector-Saralegi. In particular, g has a general type, which is an extension of the type of an adapted metric. A restriction on this general type is assumed, and then gis called good. We consider the maximum/minimum ideal boundary condition, dmax/min, of the compactly supported de Rham complex on M, in the sense of Br¨uning-Lesch. Let H∗ max/min(M) and ∆max/min denote the cohomology and Laplacian of dmax/min. The first main theorem states that ∆max/min has a discrete spectrum satisfying a weak form of the Weyl’s asymptotic formula. The second main theorem is a version of Morse inequalities using H∗ max/min(M) and what we call rel-Morse functions. An ingredient of the proofs of both theorems is a version for dmax/min of the Witten’s perturbation of the de Rham complex. Another ingredient is certain perturbation of the Dunkl harmonic oscillator previously studied by the authors using classical perturbation theory. The condition on gto be good is general enough in the following sense. Assume that Ais a stratified pseudomanifold, and consider its intersection homology I¯pH∗(A) with perversity ¯p; in particular, the lower and upper middle perversities are denoted by ¯mand ¯n, respectively. Then, for any perversity ¯p≤¯m, there is an associated good adapted metric on Msatisfying the Nagase isomorphism Hr max(M)∼ =I¯pHr(A)∗(r∈N). If Mis oriented and ¯p≥¯n, we also get Hr min(M)∼ =I¯pHr(A). Thus our version of the Morse inequalities can be described in terms of I¯pH∗(A). Contents 1. Introduction 2 2. Preliminaries 11 3. A perturbation of the Dunkl harmonic oscillator 15 4. Two simple types of elliptic complexes 19 5. Witten’s perturbation on a cone 32 6. Splitting of the Witten’s complex on a cone 34 7. Relatively local model of the Witten’s perturbation 40 8. Proof of Theorem 1.1 41 9. Functional calculus 41 1991 Mathematics Subject Classification. 58A14, 32S60. Key words and phrases. Morse inequalities, ideal boundary condition, stratification, general adapted metric, Witten’s perturbation. The first author is partially supported by MICINN, Grant MTM2011-25656, and by MEC, Grant MTM2014-56950-P. The third author has received financial support from the Xunta de Galicia and the European Union (European Social Fund - ESF). 1 2 J.A. ´ ALVAREZ L ´ OPEZ, M. CALAZA, AND C. FRANCO 10. The wave operator 42 11. Proof of Theorem 1.2 42 References 44 1. Introduction 1.1. Ideal boundary conditions of the de Rham complex. The following usual notation is used for a densely defined linear operator Tin a Hilbert space. Its domain and range are denoted by D(T) and R(T). If Tis essentially self-adjoint, its closure is denoted by T. If Tis self-adjoint, its smooth core is D∞(T) := T∞ m=1 D(Tm), and its spectrum is denoted by σ(T). AHilbert complex (D,d) is a differential complex of finite length given by a densely defined closed operator din a graded separable Hilbert space H[9]. Then the operator D=d+d∗, with D(D) = D(d)∩D(d∗), is self-adjoint in H, and therefore the Laplacian ∆=D2=dd∗+d∗dis also self-adjoint. Moreover D∞(∆) is a subcomplex of (D,d) with the same homology [9, Theorem 2.12]; it may be also said that D∞(∆) is the smooth core of d. The above notion is applied here in the following case. For a Riemannian manifold M, let Ω0(M) be the space of compactly supported differential forms, and L2Ω(M) the graded Hilbert space of square integrable differential forms. Let dand δbe the de Rham derivative and coderivative acting on Ω0(M), and let D=d+δ and ∆ = D2=dδ +δd (the Laplacian). Every Hilbert complex extension d of din L2Ω(M) is called an ideal boundary condition (i.b.c.) [9], giving rise to self-adjoint extensions Dand ∆of Dand ∆ in L2Ω(M). There exists a minimum/maximum i.b.c., dmin =dand dmax =δ∗, inducing self-adjoint extensions Dmax/min and ∆max/min of Dand ∆. If Mis oriented, then ∆max corresponds to ∆min by the Hodge star operator. The corresponding cohomologies, Hmax/min(M), are quasi-isometric invariants of M; for instance, Hmax(M) is the usual L2co- homology H(2)(M) [12]. They give rise to versions of Betti numbers and Euler characteristic, βr max/min =βr max/min(M) and χmax/min =χmax/min(M). These concepts can indeed be defined for arbitrary elliptic complexes [9]. It is well known that dmin =dmax if Mis complete. Thus considering an i.b.c. becomes interesting when Mis not complete. For example, if Mis the interior of a compact Riemannian manifold Nwith with ∂N 6=∅, then dmax/min is defined by taking absolute/relative boundary conditions. With more generality, we will assume that Mis a stratum of a compact stratified space A[41, 31, 32, 42], equipped with a generalization of the adapted metrics considered in [33, 34, 8]. As we will see, we can assume M=Aif desired (it can be said that Mis the regular strum in this case). 1.2. Stratified spaces. Roughly speaking, a (Thom-Mather)stratified space (or stratification) is a Hausdorff, locally compact and second countable space Aequipped with a partition into C∞manifolds (the strata), satisfying certain conditions [41, 31]. In particular, an order relation on the family of strata is defined by declaring X≤Ywhen X⊂Y. With respect to this ordering, the maximum length of chains of strata less or equal than a stratum Xis called the depth of X. The supremum of the strata depth is called the depth of A, denoted depth A. The precise definition and needed preliminaries were collected in [4, Section 3], where we WITTEN’S PERTURBATION ON STRATA 3 have mainly followed [42]. Instead of recalling it, let us describe how the strata of Afit together, describing also morphisms/isomorphisms of stratifications, and, in particular, the group of automorphisms, Aut(A). We proceed by induction on its depth. If depth A= 0, then Ais just a C∞manifold, and Aut(A) consists of its diffeomorphisms. Now, given any k∈Z+, assume that any stratified space Lis described if depth L < k, as well as Aut(L). If Lis compact, the cone with link Lis c(L) = (L×[0,∞))/(L×{0}), whose vertex is the point ∗=L×{0} ∈ c(L). Let L0be another compact stratification of depth < k, and φ:L→L0a morphism. Then let c(φ) : c(L)→c(L0) be the map induced by φ×id : L×[0,∞)→L0×[0,∞); in particular, we get the group c(Aut(L)) = {c(φ)|φ∈Aut(L)}. It is also declared that c(∅) = {∗}, for the empty stratification, and c(∅) = id, for the empty map. The cone c(L) is used as a model stratified space of depth kif Lis of depth k−1, whose strata are {∗} and the manifolds Y×R+for strata Yof L. The second factor projection L×[0,∞)→[0,∞) defines a c(Aut(L))-invariant function ρ:c(L)→[0,∞), called the radial function. The restrictions of ρto the strata are C∞. A conic bundle is a fiber bundle Tover a manifold Xwith typical fiber c(L) and structural group c(Aut(L)). Then ρinduces a radial function on T, also denoted by ρ, and the vertex of c(L) defines the vertex section of T, whose image is identified with X. Moreover the stratified structure defined on c(L) can be used to define a stratified structure on T, where Xbecomes the vertex stratum. For any stratification Aof depth k, every stratum Xhas an open neighborhood (a tube representative) that is isomorphic to an open neighborhood of Xin some conic bundle TXover X(with the obvious restrictions of stratified structures to open subsets). The typical fiber of TXis of the form c(LX) for some compact stratification LX(the link of X) with depth LX<depth A. The vertex and radial function of c(LX) are denoted by ∗Xand ρX. Two such neighborhoods of X represent the same tube if their structure is equal on some smaller neighborhood of X. Note that Xis open in Aif and only if LX=∅. Finally, a morphism between two stratifications is a continuous map sending every stratum to another stratum, whose restrictions to the strata are C∞, and whose restrictions to small enough tube representatives are restrictions of conic bundle morphisms. Then isomorphisms and automorphisms of stratifications have the obvious meaning. This completes the description because the depth is locally finite by the local compactness. The (topological) dimension of a stratification Aequals the supremum of the dimensions of its strata. It may be infinite, but it is locally finite. The codimension of every stratum Xis dim A−dim X. Our main results will assume that the stratification is compact, but non-compact stratifications will be also used in the proofs. In any case, we will only consider stratifications of finite dimension. If the above description of Ais modified by requiring that, at every inductive step, only stratifications with no strata of codimension 1 are used, then Ais called a stratified pseudomanifold. A locally closed subset B⊂Ais called a substratification of Aif the restrictions of the strata and tubes of Ato Bdefine a stratified structure on B. For instance, Acan be restricted to any open subset, to any locally closed union of strata, and to the closure of any stratum. If moreover there are tube representatives of Awhose restrictions to Bhave the same fibers over points of B, then Bis called saturated. 4 J.A. ´ ALVAREZ L ´ OPEZ, M. CALAZA, AND C. FRANCO Let xbe a point of a stratum Xof dimension mXin a stratification A. A local trivialization of TXon some open neighborhood Uof xdefines a chart O≡O0 of Afor some open O0⊂RmX×c(LX). We can assume O0=U0×c(LX), where U0is some open neighborhood of 0 in RmXand c(LX) is the subset of c(LX) defined by the condition ρX< , for some  > 0. This chart is said to be centered at xif x≡(0,∗X)∈O0. The corresponding concept of atlas has the obvious meaning. These concepts can be generalized as follows. Any finite product of stratifications has a non-canonical stratified structure [4, Section 3.1.2]; in particular, any finite product of cones is isomorphic to a cone [4, Lemma 3.8]. Moreover Aut(P)×Aut(Q) is canonically injected in Aut(P×Q) for stratifications Pand Q. Thus it makes sense to consider a decomposition c(LX)∼ =QaX i=1 c(LX,i) (aX∈N), for compact stratifications LX,i. The vertex and radial function of every c(LX,i) are denoted by ∗X,i and ρX,i. Then we can also consider general tube representatives given by bundles TXwith typical fibers QaX i=1 c(LX,i) and structural groups QaX i=1 c(Aut(LX,i)). This gives rise to a general chart O≡O0 around xfor some open O0⊂RmX×QaX i=1 c(LX,i), which is centered at xif x≡(0,∗X,1,...,∗X,aX)∈O0. As above, we can assume O0=U0×QaX i=1 c(LX,i) for some  > 0. Let ρX,0denote the norm function on RmX. The function ρ= (ρ2 X,0+··· +ρ2 X,aX)1/2is called the radial function of RmX×QaX i=1 c(LX,i), even though, when mX= 0, ρis not the radial function of any cone structure on QaX i=1 c(LX,i) [4, Example 3.6 and Proof of Lemma 3.8]. A collection of general charts covering Ais called a general atlas. We can suppose that the strata of Aare connected [4, Remark 1 (v)]. Fix a stratum Mof dimension nin A. Since the stratified structure of Acan be restricted to M[4, Section 3.1.1], we can also assume without loss of generality that M=A(any other stratum is < M); in particular, depth A= depth Mand dim A=n. With the above notation, for a chart O≡O0centered at x, we get M∩O≡M0∩O0, where M0=RmX×N×R+for some dense stratum Non LX. In the case of a general chart O≡O0centered at x, we have M∩O≡M0∩O0for M0=RmX×QaX i=1(Ni×R+), where every Niis some dense stratum of LX,i. We will use the notation kX,i = dim Ni+ 1. 1.3. General adapted metrics. Ageneral adapted metric gon Mis defined by induction on the depth of M. It is any (Riemannian) metric if depth M= 0. Now, assume that depth M > 0 and general adapted metrics are defined for lower depth. Given any general chart O≡O0as above, take any general adapted metric ˜gion every Ni(depth Ni<depth M), and let gi=ρ2uX,i X,i ˜gi+ (dρX,i)2on Ni×R+for some uX,i >0. Let also g0be the Euclidean metric on RmX. Then gis a general adapted metric if, via any such general chart, g|Ois quasi-isometric to (PaX i=0 gi)|O0. In this case, the mapping X7→ uX:= (uX,1, . . . , uX,aX)∈RaX +(X < M) is called the general type of g. Such a general chart is called compatible with g, or with its general type. Let us point out that a general metric does not completely determine its general type. For instance, suppose uX,i =uX,j = 1 for indices i6=j. Write c(LX,i)× c(LX,j)≡c(L), with radial function ρ, for some stratification L. Then Ni×R+× Nj×R+≡N×R+for some dense stratum Nof L. Moreover there is a general adapted metric ˜gon Nsuch that gi+gjis quasi-isometric to ρ2˜g+ (dρ)2via the above identity. Therefore we can omit uX,i or uX,j in uX, obtaining a different type of g. This cannot be done if uX,i =uX,j 6= 1 (Proposition 2.1). WITTEN’S PERTURBATION ON STRATA 5 If the above definition of general adapted metric is modified by requiring that, at every inductive step, the general type satisfies uX,i ≤1 for all X < M and i= 1, . . . , aX, then the general adapted metric is called good for the scope of this paper. On the other hand, if the definition is modified by requiring at every inductive step that aX= 1 and uXdepends only on k:= kX,1= codim Xfor all X < M, then we get the adapted metrics considered in [33, 34, 8]. In this case, the general charts compatible with the general type are indeed charts. Writing uk=uX≡uX,1∈R+, the condition on an adapted metric gto be good becomes uk≤1 for all k, at every inductive step of its definition. In [33, 34, 8], it is assumed that Ais a stratified pseudomanifold, and then ˆu= (u2, . . . , un) stands for the type of g. This ˆuis determined by g. In particular, if the definition is modified by taking uk= 1 for all kat every inductive step, we get the adapted metrics of conic type considered in [12, 13, 14]. Be alerted about the three slightly different terms used for the scope of this paper: adapted metrics of conic type, adapted metrics and general adapted metrics. The class of (good) general adapted metrics is preserved by products, as well as the class of adapted metrics of conic type, but the class of adapted metrics does not have this property. The existence of general adapted metrics with any possible general type can be shown like in the case of adapted metrics [33, Lemma 4.3], [8, Appendix]. Like in [4], the term “relative(ly)” (or simply “rel-”) usually means that some condition is required in the intersection of Mwith small neighborhoods of the points in M, or that some concept can be described using those intersections. Let Mbe equipped with a general adapted metric g, with a general type X7→ uX as above. The rel-local metric completion c Mof Mconsists of the points in the metric completion represented by Cauchy sequences that converge in M(c Mis the metric completion of Mif Mis compact). Figure 1 illustrates this concept. The limits of Cauchy sequences define a continuous map lim : c M→M. The following properties can be proved like in the case of conic metrics [4, Proposition 3.20 (i),(ii)]. c Mhas a unique stratified structure with connected strata so that lim : c M→Mis a morphism whose restrictions to the strata are local diffeomorphisms. Moreover g is also a general adapted metric with respect to c M. (a) M(b) c M Figure 1. The stratified space c M. 1.4. Relatively Morse functions. A smooth function fon Mis called reladmissible when the functions f,|df|and |Hess f|are rel-bounded. In this case, 6 J.A. ´ ALVAREZ L ´ OPEZ, M. CALAZA, AND C. FRANCO fmay not have any continuous extension to M, but it has a continuous extension to c M. So it makes sense to say that x∈c Mis a rel-critical point of fwhen lim inf |df(y)|= 0 as y→xin c Mwith y∈M. The set of rel-critical points of fis denoted by Critrel(f). It is said that fis a rel-Morse function if it is rel-admissible and has the following description around every x∈Critrel(f): •there is a general chart O≡O0of c M, centered at xand compatible with g, such that M∩O≡M0∩O0for M0=RmX×QaX i=1(Ni×R+), where X is the stratum of c Mcontaining x; and •f|M∩O≡f(x) + 1 2(ρ2 +−ρ2 −)|M0∩O0, where ρ±is the radial function of Rm±×Qi∈I±c(LX,i) for some expression mX=m++m−(m±∈N) and some partition of {1, . . . , aX}into sets I±. This local condition is used instead of requiring that Hess fis “rel-non-degenerate” at the rel-critical points because a “rel-Morse lemma” is missing. Moreover, for every r∈ {0, . . . , n}, let νr x,max/min =X (r1,...,raX) aX Y i=1 βri max/min(Ni),(1) where (r1, . . . , raX) runs in the subset of NaXdetermined by r=m−+PaX i=1 ri+|I−|, ri<kX,i−1 2+1 2uX,i if i∈I+ ri≥kX,i−1 2+1 2uX,i if i∈I−)for νr x,max , ri≤kX,i−1 2−1 2uX,i if i∈I+ ri>kX,i−1 2−1 2uX,i if i∈I−)for νr x,min .                  (2) When aX= 0 in (1), the singleton N0consists of the empty sequence, obtaining1 νr x,max/min =δr,m−with the convention that the value of empty products is 1. Finally, let νr max/min =Pxνr x,max/min with xrunning in Critrel(f). The notation νr x,max/min(f) and νr max/min(f) may be used if necessary. The existence of rel-Morse functions for general adapted metrics holds like in the case of adapted metrics [4, Proposition 4.9]. 1.5. Main theorems. The following is our first main theorem, where property (ii) is a weak version of the Weyl’s asymptotic formula. Theorem 1.1. The following properties hold on any stratum of a compact stratification with a good general adapted metric: (i) ∆max/min has a discrete spectrum, 0≤λmax/min,0≤λmax/min,1≤ ···, where every eigenvalue is repeated according to its multiplicity. (ii) lim infkλmax/min,k k−θ>0for some θ > 0. Our second main result is the following version of Morse inequalities for rel-Morse functions. 1Kronecker’s delta symbol is used. WITTEN’S PERTURBATION ON STRATA 7 Theorem 1.2. For any rel-Morse function on a stratum of dimension nof a compact stratification, equipped with a good general adapted metric, we have k X r=0 (−1)k−rβr max/min ≤ k X r=0 (−1)k−rνr max/min (0 ≤k < n), χmax/min = n X r=0 (−1)rνr max/min . In the case of adapted metrics of conic type, Theorem 1.1 (i) is essentially due to Cheeger [12, 13] (see also [1, 2, 4]), Theorem 1.1–(ii) was proved by the authors [4], and Theorem 1.2 was proved by the authors [4] and Ludwig [30] (with more restrictive conditions but stronger consequences). Other developments of elliptic theory on strata were made in [10, 25, 23, 39, 16, 2, 1], all of them using adapted metrics of conic type. The main novelty of our paper is the extension of the elliptic theory on strata to the wider class of good general adapted metrics, including good adapted metrics. 1.6. Applications to intersection homology. Consider now the case where A is a stratified pseudomanifold, and therefore Mis its regular stratum. Let I¯pH∗(A) denote its intersection homology with perversity ¯p[19, 20], taking real coefficients. Let β¯p r=β¯p r(A) and χ¯p=χ¯p(A) denote the versions of Betti numbers and Euler characteristic for I¯pH∗(A). Every perversity can be considered as a sequence ¯p= (p2, p3, . . . ) in Nsatisfying p2= 0 and pk≤pk+1 ≤pk+ 1. For example, the zero perversity is ¯ 0 = (0,0, . . . ), the top perversity is ¯ t= (0,1,2, . . . ) (tk=k−2), the lower middle perversity is ¯m= (0,0,1,1,2,2,3, . . . ) (mk=bk 2c−1), and the upper middle perversity is ¯n= (0,1,1,2,2,3,3, . . . ) (nk=dk 2e−1). Recall also that two perversities ¯pand ¯qare called complementary if ¯p+ ¯q=¯ t. Write ¯p≤¯qif pk≤qkfor all k. Let gbe an adapted metric on Mof type ˆu= (u2, . . . , un). If ˆuis associated with a perversity ¯p≤¯min the sense 1 k−1−2pk≤uk<1 k−3−2pkif 2pk≤k−3, 1≤uk<∞if 2pk=k−2,)(3) then Hr (2)(M)∼ =I¯pHr(A)∗[33, 34, 8], and therefore β¯p r=βr max. In particular, Hr (2)(M)∼ =I¯mHr(A)∗if gis an adapted metric of conic type [14]. Thus the incompatibility of adapted metrics with products is related to the subtleties of the versions of the K¨unneth theorem for intersection homology [15, 17]. For instance, the isomorphism I¯pH∗(P×Q)∼ =I¯pH∗(P)⊗I¯pH∗(Q), for arbitrary pseudomanifolds Pand Q, only holds with some special perversities ¯p, including ¯p= ¯m. According to (3), there exist good adapted metrics on Mwhose type is associated with any given perversity ≤¯m. In (3), only the choices 2pk=k−2, k −4, . . . are possible if kis even, and only the choices 2pk=k−3, k −5, . . . are possible if kis odd. Thus, for every k, (3) establishes a bijection between the possibilities for pkand a partition of [ 1 k−1,∞) into semi-open intervals, where ukis taken. Let fbe a rel-Morse function on M, let x∈Critrel(f), let Xbe the stratum of c Mcontaining x, and let k= codim X. With the above notation for a chart O≡O0of c Mcentered at x, there is an adapted metric ˜gon Nso that, via the chart, g|Ois quasi-isometric to the restriction of g0+ρ2uk X˜g+ (dρX)2to M0∩O0. 8 J.A. ´ ALVAREZ L ´ OPEZ, M. CALAZA, AND C. FRANCO Then the type of ˜gis also associated with ¯p. Moreover there is some expression, mX=m++m−(m±∈N), and some decomposition, c(LX)≡c(L+)×c(L−), so that M0≡Rm+×N+×R+×Rm−×N−×R+for dense strata N±of L±, and f|O≡f(x) + 1 2(ρ2 +−ρ2 −)|O0, where ρ±is the radial function of Rm±×c(L±). Let k±= dim N±+ 1; thus k=k++k−. Here, some of the stratifications L±may be empty; in fact, L+6=∅ 6=L−only can happen if uk= 1 (Section 1.3). From (1) and (2), it follows that the numbers νr x,max are independent of the choice of ˆu associated with ¯p, and therefore the notation ν¯p x,r =ν¯p x,r(f) will be used. Precisely, they have the following expressions: •If L+6=∅ 6=L−(only if uk= 1), then ν¯p x,r =X (r+,r−) β¯p r+(L+)β¯p r−(L−), where (r+, r−) runs in the subset of N2determined by the conditions r=m−+r++r−+ 1 , r+<k+ 2, r−≥k− 2. •If LX=L+6=∅(L−=∅), then ν¯p x,r =X r+ β¯p r+(LX), where r+runs in the subset of Ndetermined by the conditions r=m−+r+, r+<(k−1−pkif uk<1 k 2if uk= 1 . •If LX=L−6=∅(L+=∅), then ν¯p x,r =X r− β¯p r−(LX), where r−runs in the subset of Ndetermined by the conditions r=m−+r−+ 1 , r−≥(k−1−pkif uk<1 k 2if uk= 1 . •If LX=∅, then ν¯p x,r =δr,m−. Finally, let ν¯p r=ν¯p r(f) = Pxν¯p x,r (x∈Critrel(f)), which equals νr max. Suppose now that Ais oriented (Mis oriented) and compact. We have βr min = βn−r max for all rbecause ∆min corresponds to ∆max by the Hodge star operator. On the other hand, for any perversity ¯q≥¯n, if ¯p≤¯mis complementary of ¯q, then I¯qHr(A)∼ =I¯pHn−r(A)∗[19, 20], and therefore β¯q r=β¯p n−r, obtaining β¯q r=βr min. As before, it follows from (1) and (2) that the numbers νr x,min are independent of the choice of ˆuassociated with ¯p. Precisely, with the notation ν¯q x,r =ν¯q x,r(f) = νr x,min, they have the following expressions: •If L+6=∅ 6=L−(only if uk= 1), then ν¯q x,r =X (r+,r−) β¯q r+(L+)β¯q r−(L−), where (r+, r−) runs in the subset of N2determined by the conditions r=m−+r++r−+ 1 , r+≤k+ 2−1, r−>k− 2−1. WITTEN’S PERTURBATION ON STRATA 9 •If LX=L+6=∅(L−=∅), then ν¯q x,r =X r+ β¯q r+(LX), where r+runs in the subset of Ndetermined by the conditions r=m−+r+, r+≤(k−2−qkif uk<1 k 2−1 if uk= 1 . •If LX=L−6=∅(L+=∅), then ν¯q x,r =X r− β¯q r−(LX), where r−runs in the subset of Ndetermined by the conditions r=m−+r−+ 1 , r−>(k−2−qkif uk<1 k 2−1 if uk= 1 . •If LX=∅, then ν¯q x,r =δr,m−. Like ν¯p r, we also define ν¯q r=ν¯q r(f) = Pxν¯q x,r (x∈Critrel(f)), which equals νr min. Theorem 1.2 has the following direct consequence. Corollary 1.3. Let Abe a compact pseudomanifold of dimension n, let Mbe its regular stratum, and let ¯pbe a perversity. If ¯p≤¯m, or if Ais oriented and ¯p≥¯n, then, for any rel-Morse function on M(with respect to any good adapted metric), we have k X r=0 (−1)k−rβ¯p r≤ k X r=0 (−1)k−rν¯p r(0 ≤k < n), χ¯p= n X r=0 (−1)rν¯p r. Stratified Morse theory was introduced by Goresky and MacPherson [21], and has a great wealth of applications. In particular, Goresky and MacPherson have proved Morse inequalities on complex analytic varieties with Whitney stratifications, involving the intersection homology with perversity ¯m[21, Chapter 6, Section 6.12]. Ludwig also gave an analytic interpretation of Morse theory in the spirit of Goresky and MacPherson for conformally conic manifolds [26, 27, 28, 29]. Our version of Morse functions, critical points and associated numbers is different from those used in [21], even in the case of perversity ¯m. To the authors’ knowledge, Corollary 1.3 is the first version of Morse inequalities for intersection homology with perversity 6= ¯m. 1.7. Ideas of the proofs. In the proofs of Theorems 1.1 and 1.2, several steps are like in the case of adapted metrics of conic type [4]. Only brief indications of those steps are given in this paper, whereas the parts with new ideas are explained with detail. We adapt the well-known analytic method of Witten [43]; specially, as described in [36, Chapters 9 and 14]. Thus, given a rel-Morse function fon M, we consider the Witten’s perturbation ds=e−sf desf =d+s df∧on Ω0(M) (s > 0). Let ds,max/min denote its maximum/minimum i.b.c., with corresponding Laplacian ∆s,max/min. Since ∆s,max/min −∆max/min is bounded, it is enough to 16 J.A. ´ ALVAREZ L ´ OPEZ, M. CALAZA, AND C. FRANCO and k kc,d, respectively. The simpler notation L2 +,h,iand k k is used when c= 0. Recall that the harmonic oscillator on C∞ +is the operator H=−d2 dρ2+s2ρ2(s > 0). For c1, c2, d1, d2∈R, let P0=H−2c1ρ−1d dρ +c2ρ−2, Q0=H−2d1d dρ ρ−1+d2ρ−2.(10) Proposition 3.1 ([3, Theorem 1.4]).If a∈Rsatisfies a2+ (2c1−1)a−c2= 0 ,(11) σ:= a+c1>−1 2,(12) then the following holds: (i)P0, with D(P0) = ρaSev,+, is essentially self-adjoint in L2 c1,+. (ii)The spectrum of P0:= P0consists of the eigenvalues λk= (2k+ 1 + 2σ)s , (13) for k∈2N, with multiplicity one and corresponding normalized eigenfunctions χk=χs,σ,a,k := √2ρaφs,σ,k,+. (iii)D∞(P0) = ρaSev,+. Proposition 3.2 (See [3, Section 5]).If b∈Rsatisfies b2+ (2d1+ 1)b−d2= 0 ,(14) τ:= b+d1>−3 2,(15) then the following holds: (i)Q0, with D(Q0) = ρbSodd,+, is essentially self-adjoint in L2 d1,+. (ii)The spectrum of Q0:= Q0consists of the eigenvalues given by the expression (13), for k∈2N+ 1 and using τinstead of σ, with multiplicity one and corresponding normalized eigenfunctions χk=χs,τ,b,k := √2ρbφs,τ,k,+. (iii)D∞(Q0) = ρbSodd,+. Proposition 3.3 ([5, Corollary 8.1]).Let ξ > 0and 0< u < 1.(16) If a∈Rsatisfies (11) and σ:= a+c1> u −1 2,(17) then there is a positive self-adjoint operator Pin L2 c1,+satisfying the following: (i)ρaSev,+is a core of P1/2and, for all φ, ψ ∈ρaSev,+, hP1/2φ, P1/2ψic1=hP0φ, ψic1+ξhρ−uφ, ρ−uψic1.(18) (ii)Phas a discrete spectrum. Let λ0≤λ2≤ ··· be its eigenvalues, repeated according to their multiplicity. There is some D=D(σ, u)>0and, for any  > 0, there is some C=C(, σ, u)>0so that, for all k∈2N, λk≥(2k+ 1 + 2σ)s+ξDsu(k+ 1)−u,(19) λk≤(2k+ 1 + 2σ)(s+ξsu) + ξCsu.(20) Proposition 3.4 ([5, Corollary 8.2]).For ξand ulike in Proposition 3.3, if b∈R satisfies (14) and τ:= b+d1> u −3 2,(21) then there is a positive self-adjoint operator Qin L2 d1,+satisfying the following: WITTEN’S PERTURBATION ON STRATA 17 (i)ρbSodd,+is a core of Q1/2and, for all φ, ψ ∈ρbSodd,+, hQ1/2φ, Q1/2ψid1=hQ0φ, ψid1+ξhρ−uφ, ρ−uψid1.(22) (ii)Qhas a discrete spectrum. Let λ1≤λ3≤ ··· be its eigenvalues, repeated according to their multiplicity. There is some D=D(τ, u)>0and, for any  > 0, there is some C=C(, τ, u)>0so that (19) and (20) are satisfied, for k∈2N+ 1 and with τinstead of σ. Proposition 3.5 ([5, Corollary 8.3]).Consider the notation and conditions of Propositions 3.3 and 3.4. Fix also some η∈R, and let θ > −1 2.(23) Moreover suppose that the following properties hold: (a)If σ=θ6=τand τ−σ6∈ −N, then σ−1< τ < σ + 1,2σ+1 2.(24) (b)If σ6=θ=τand σ−τ6∈ −N, then −τ, τ −1< σ < 3τ+ 1,11τ+ 2, τ + 1 .(25) (c)If σ6=θ=τ+ 1 and σ−τ−16∈ −N, then τ+ 1 < σ < τ + 3,2τ+7 2.(26) (d)If σ6=θ6=τand σ−θ, τ −θ6∈ −N, then σ−τ 2−1,τ−σ 2,σ+τ−1 4,σ+3τ−2 14 ,3σ+τ−4 14 ,σ+τ−1 2< θ < σ+τ+1 2, τ−1< σ < τ + 3 .(27) Then there is a positive self-adjoint operator Win L2 c1,d1,+satisfying the following: (i)ρaSev,+⊕ρbSodd,+is a core of W1/2, and, for φ= (φ1, φ2)and ψ= (ψ1, ψ2) in ρaSev,+⊕ρbSodd,+, hW1/2φ, W1/2ψic1,d1=h(P0⊕Q0)φ, ψic1,d1+ξhρ−uφ, ρ−uψic1,d1 +ηhρ−a−b−1φ2, ψ1iθ+hφ1, ρ−a−b−1ψ2iθ.(28) (ii)Whas a discrete spectrum. Its eigenvalues form two groups, λ0≤λ2≤ ··· and λ1≤λ3≤ ···, repeated according to their multiplicity, such that there is some D=D(σ, τ, u)>0and, for every  > 0, there are some C= C(, σ, τ, u)>0and E=E(, σ, τ, θ)>0so that, for all k∈N, λk≥(2k+ 1 + 2ςk)s−2|η|sv+1 2+ξDsu(k+ 1)−u−2|η|Esv+1 2,(29) λk≤(2k+ 1 + 2ςk)s+ξsu+ 2|η|sv+1 2+ξCsu+ 2|η|Esv+1 2,(30) where v=σ+τ−2θ,ςk=σif kis even, and ςk=τif kis odd. (iii)Let ˜u∈Rsuch that 0, v, τ −2θ+1 2, σ −2θ−1 2<˜u < 1, v + 1, σ +1 2, τ +3 2,(31) and let ˆu= max{˜u, v + 1 −˜u}. There is some D=D(σ, τ, u)>0and, for any  > 0, there is some e C=e C(, σ, τ, u)>0so that, for all k∈N, λk≥(2k+ 1 + 2ςk)s−|η|sˆu+ξDsu(k+ 1)−u−|η|e Csˆu.(32) 18 J.A. ´ ALVAREZ L ´ OPEZ, M. CALAZA, AND C. FRANCO (iv)If u=v+1 2and ξ≥ |η|, then there is some e D=e D(σ, τ, u)>0so that, for all k∈N, λk≥(2k+ 1 + 2ςk)s+ (ξ−|η|)e Dsu(k+ 1)−u.(33) (v)If we add the term ξ0hφ1, ψ1ic1+ξ00hφ2, ψ2id1to the right-hand side of (28), for some ξ0, ξ00 ∈R, then the result holds as well with the additional term max{ξ0, ξ00}in the right-hand side of (30), and the additional term, ξ0for k∈2Nand ξ00 for k∈2N+ 1, in the right-hand sides of (29),(32) and (33). Remark 7.(i) If his a bounded measurable function on R+with h(ρ)→1 as ρ→0, then hhχ0, χ0ic1→1 as s→ ∞ [4, Lemma 7.3]. (ii) The existence of a∈Rsatisfying (11) is characterized by the condition (2c1− 1)2+ 4c2≥0, which holds if c2≥min{0,2c1}. If c2= 0, then (11) means that a∈ {0,1−2c1}. If c2= 2c1, then (11) means that a∈ {1,−2c1}. (iii) The existence of b∈Rsatisfying (14) is characterized by the condition (2d1+ 1)2+ 4d2≥0, which holds if d2≥min{0,−2d1}. If d2= 0, then (14) means that b∈ {0,−1−2d1}. If d2=−2d1, then (14) means that b∈ {−1,−2d1}. (iv) Propositions 3.1 and 3.2 are indeed equivalent, as well as Propositions 3.3 and 3.4, because, if c1=d1+ 1 and c2=d2, then Q0=ρP0ρ−1by (9), and ρ:L2 c1,+→L2 d1,+is a unitary isomorphism. (v) We have P=P,Q=Qand W=W, where P=P0+ξρ−2u, Q =Q0+ξρ−2u,(34) W=P ηρ2(θ−c1)−a−b−1 ηρ2(θ−d1)−a−b−1Q,(35) with D(P) = D∞(P), D(Q) = D∞(Q) and D(W) = D∞(W) [5, Remark 1.4 (i) and Section 8]. (vi) We have D(P1/2) = D(P1/2 0),D(Q1/2) = D(Q1/2 0),D(W1/2) = D((P0⊕Q0)1/2). Thus the expressions (18), (22) and (28) can be extended to φand ψin D(P1/2), D(Q1/2) and D(W1/2), respectively, using hP1/2 0φ, P1/2 0ψic1,hQ1/2 0φ, Q1/2 0ψid1,h(P0⊕Q0)1/2φ, (P0⊕Q0)1/2ψic1,d1 instead of hP0φ, ψic1,hQ0φ, ψid1,h(P0⊕Q0)φ, ψic1,d1, respectively [5, Remark 3.21 and Section 8]. (vii) In Proposition 3.5 (iii), the condition (31) means that (16), (17) and (21) also hold with ˜uand v+ 1 −˜uinstead of u. There exists ˜usatisfying (31) just when 0, v, τ −2θ+1 2, σ −2θ−1 2<1, v + 1, σ +1 2, τ +3 2.(36) This property is satisfied in the cases (b) and (d) by (16), (17), (21), (23), (25) and (27); in particular, we can take ˜u=v+1 2. By (16), (17), (21), (23) and (24) (respectively, (26)), in the case (a) (respectively, in the case (c)), we have (36) if and only if τ < 3σ(respectively, σ < 3τ+ 4). Consider the conditions and notation of Proposition 3.3, and the notation of Proposition 3.1. Take a complete orthonormal system {ˆχk= ˆχP,k |k∈2N}of L2 c1,+so that every ˆχkis a λk-eigenfunction of P. Let ˆχ0 k= ˆχ0 P,k and ˆχ00 k= ˆχ00 P,k WITTEN’S PERTURBATION ON STRATA 19 denote the orthogonal projections of every ˆχkto the subspaces spanned by χkand {χi|k > i ∈2N}, respectively; in particular, ˆχ00 0= 0. Let also ˆχ000 k= ˆχ000 P,k = ˆχk−ˆχ0 k−ˆχ00 k. Lemma 3.6. kˆχ0 P,kkc1→1as s→ ∞ for every k∈2N. Proof. We proceed by induction on k. For k= 0, take some  > 0 and C > 0 satisfying (20). By Propositions 3.1 (ii) and 3.3 (ii), and Remark 7 (vi), (1 + 2σ)(s+ξsu) + ξCsu≥λ0=hP1/2ˆχ0,P1/2ˆχ0ic1>hP1/2 0ˆχ0,P1/2 0ˆχ0ic1 =hP1/2 0ˆχ0 0,P1/2 0ˆχ0 0ic1+hP1/2 0ˆχ000 0,P1/2 0ˆχ000 0ic1 ≥(1 + 2σ)skˆχ0 0k2 c1+ (5 + 2σ)skˆχ000 0k2 c1= (1 + 2σ)s+ 4skˆχ000 0k2 c1, giving kˆχ000 0k2 c1<((1 + 2σ)+C)ξ 4s1−u→0 as s→ ∞, and therefore kˆχ0 0k2 c1→1. Now, take any even integer k > 0 and suppose that the result holds for all even indices < k. This yields kˆχ00 kkc1→0 as s→ ∞. Thus, given any δ > 0, we have kˆχ00 kk2 c1< δ/k for slarge enough. Take some  > 0 and C > 0 satisfying (20). By Propositions 3.1 (ii) and 3.3 (ii), and Remark 7 (vi), (2k+ 1 + 2σ)(s+ξsu) + ξCsu≥λk=hP1/2ˆχk,P1/2ˆχkic1>hP1/2 0ˆχk,P1/2 0ˆχkic1 =hP1/2 0ˆχ0 k,P1/2 0ˆχ0 kic1+hP1/2 0ˆχ00 k,P1/2 0ˆχ00 kic1+hP1/2 0ˆχ000 k,P1/2 0ˆχ000 kic1 ≥(2k+ 1 + 2σ)skˆχ0 kk2 c1+ (1 + 2σ)skˆχ00 kk2 c1+ (2k+ 5 + 2σ)skˆχ000 kk2 c1 = (1 + 2σ)s+ 2ks(kˆχ0 kk2 c1+kˆχ000 kk2 c1)+4skˆχ000 kk2 c1 >(1 + 2σ)s+ 2ks(1 −δ/k)+4skˆχ000 kk2 c1, giving kˆχ000 kk2 c1<((2k+ 1 + 2σ)+C)ξ 4s1−u+δ 2< δ for slarge enough. Thus kˆχ000 kk2 c1→0 as s→ ∞, and the result follows.  Corollary 3.7. If his a bounded measurable function on R+such that h(ρ)→1 as ρ→0, then hhˆχP,0,ˆχP,0ic1→1as s→ ∞. Proof. This follows from Lemma 3.6 and Remark 7 (i).  Similar results hold for Qand W, but they are omitted because they are not used. 4. Two simple types of elliptic complexes Here, we study two simple elliptic complexes on R+, which will show up in a direct sum splitting of the rel-local model of Witten’s perturbation (Section 6). 20 J.A. ´ ALVAREZ L ´ OPEZ, M. CALAZA, AND C. FRANCO 4.1. An elliptic complex of length one. Consider the standard metric on R+. Let Ebe the graded Riemannian/Hermitian vector bundle over R+whose nonzero terms are E0and E1, which are real/complex trivial line bundles equipped with the standard Riemannian/Hemitian metrics. Thus C∞(E0)≡C∞ +≡C∞(E1), L2(E0)≡L2 +≡L2(E1), where real-/complex-valued functions are considered in C∞ +and L2 +. For any fixed s > 0 and κ∈R, let C∞(E0)C∞(E1) d δ -  be the differential operators defined by d=d dρ −κρ−1±sρ , δ =−d dρ −κρ−1±sρ . It is easy to check that (E, d) is an elliptic complex, and that4δ=d†. 4.1.1. Self-adjoint operators defined by the Laplacian. By (9), the homogeneous components of ∆ (or ∆±) are: ∆0=H+κ(κ−1)ρ−2∓s(1 + 2κ),(37) ∆1=H+κ(κ+ 1)ρ−2±s(1 −2κ),(38) where His the harmonic oscillator on C∞ +defined with the constant s. Then ∆0 and ∆1are like P0and Q0in (10), with c1=0=d1, plus a constant. Then, by Propositions 3.1 and 3.2, ∆0and ∆1define the self-adjoint operators Aiand Bi in L2 +indicated in Table 1, where the conditions come from (12) and (15). The notation A± iand B± imay be used as well to specify that these operators are defined by ∆± 0and ∆± 1. In these cases, we have c1=d1= 0, and therefore σ=aand τ=b, which are given by (11) and (14). σ τ Condition ∆0A1κ κ > −1 2 A21−κ κ < 3 2 ∆1B1κ κ > −3 2 B2−1−κ κ < 1 2 Table 1. Self-adjoint operators defined by ∆0and ∆1 There are the following overlaps in Table 1: •Both A1and A2are defined if −1 2< κ < 3 2, and they are equal just when κ=1 2. •Both B1and B2are defined if −3 2<κ<1 2, and they are equal just when κ=−1 2. The cores of Aiand Bi, given by Propositions 3.1 and 3.2, will be denoted by E0 iand E1 i, respectively. Note that the graded subspace Ei=E0 i⊕E1 iof C∞(E)∩L2(E), whenever defined, is preserved by D=d+δ. Propositions 3.1 and 3.2 also describe the spectra of Aiand Bi: 4The superindex †is used to denote the formal adjoint. WITTEN’S PERTURBATION ON STRATA 21 •The spectrum of A1consists of the eigenvalues (2k+ (1 ∓1)(1 + 2κ))s(k∈2N) (39) of multiplicity one. •The spectrum of A2consists of the eigenvalues (2k+ 4 −(1 ±1)(1 + 2κ))s(k∈2N) (40) of multiplicity one. •The spectrum of B1consists of the eigenvalues (2k+ 2 + (1 ∓1)(−1+2κ))s(k∈2N+ 1) (41) of multiplicity one. •The spectrum of B2consists of the eigenvalues (2k−2−(1 ±1)(−1+2κ))s(k∈2N+ 1) (42) of multiplicity one. These eigenvalues have normalized eigenfunctions χk, defined for the corresponding values of a=σand b=τ. For A+ 1, (39) becomes 2ks. For A− 1, (39) is 2(k+1+2κ)s. For A+ 2, (40) becomes 2(k+ 1 −2κ)s. For A− 2, (40) is 2(k+ 2)s. For B+ 1, (41) is 2(k+ 1)s. For B− 1, (41) becomes 2(k+ 2κ)s. For B+ 2, (42) is 2(k−2κ)s. For B− 2, (42) becomes 2(k−1)s. Using this, we get the information about the sign of the eigenvalues of Aiand Bigiven in Table 2. In the tables, grey color is used for cases that will be disregarded later (for instance, if there may exist some negative eigenvalue), and a question mark is used for unknown information. Sign of eigenvalues Sign of eigenvalues A+ 1 0 if k= 0 B+ 1+∀k∈2N+ 1 + if k≥2 even B− 1 κ > −1 2+∀k∈2N+ 1 A− 1+∀k∈2Nκ=−1 2 0 if k= 1 A+ 2 κ > 1 2 −if k < 2κ−1+ if k≥3 odd 0 if k= 2κ−1 κ < −1 2 −if k < −2κ + if k > 2κ−1 0 if k=−2κ κ=1 2 0 if k= 0 + if k > −2κ + if k≥2 even B+ 2+∀k∈2N+ 1 κ < 1 2+∀k∈2NB− 2 0 if k= 1 A− 2+∀k∈2N+ if k≥3 odd Table 2. Sign of the eigenvalues of Aiand Bi 4.1.2. Laplacians of the maximum/minimum i.b.c. Proposition 4.1 ([4, Proposition 8.4]).Table 3 describes ∆max/min. 22 J.A. ´ ALVAREZ L ´ OPEZ, M. CALAZA, AND C. FRANCO ∆max,0∆min,0∆max,1∆min,1 κ≥1 2A1B1 |κ|<1 2A1A2B1B2 κ≤ −1 2A2B2 Table 3. Description of ∆max/min Remark 8.(i) In [4], the proof of Proposition 4.1 uses the following property [4, Lemma 8.5]. Suppose that either θ > 1 2, or θ=1 2=κ(respectively, θ=1 2=−κ). Then, for every ξ∈ρθSev,+, considered as subspace of C∞(E0) (respectively, C∞(E1)), there is a sequence (ξn) in C∞ 0(E0) (respectively, C∞ 0(E1)), independent of κ, such that limnξn=ξin L2(E0) (respectively, L2(E1)) and limndξn=dξ in L2(E1) (respectively, limnδξn=δξ in L2(E0)). In particular, ρθSev,+is contained in D(dmin) (respectively, D(δmin)). Moreover, according to the proof of [4, Lemma 8.5], given 0 < a < b, we can take ξn=αnξfor some αn∈C∞ +satisfying χ[b n,na]≤αn≤χ[a n,nb], where χS denotes the characteristic function of every subset S⊂R+. (ii) E0 i(respectively, E1 i) is also a core of dmax/min (respectively, δmin/max) when ∆max/min,0=Ai(respectively, ∆max/min,1=Bi). 4.2. An elliptic complex of length two. Consider again the standard metric on R+. Let Fbe the graded Riemannian/Hermitian vector bundle over R+whose nonzero terms are F0,F1and F2, which are trivial real/complex vector bundles of ranks 1, 2 and 1, respectively, equipped with the standard Riemannian/Hermitian metrics. Thus C∞(F0)≡C∞ +≡C∞(F2), C∞(F1)≡C∞ +⊕C∞ +, L2(F0)≡L2 +≡L2(F2), L2(F1)≡L2 +⊕L2 +, where real-/complex-valued functions are considered in C∞ +and L2 +. Fix s, µ > 0, 0<u<1 and κ∈R. Let C∞(F0)C∞(F1)C∞(F2) d0≡d0,1 d0,2 δ0≡δ0,1δ0,2 d1≡d1,1d1,2 δ1≡δ1,1 δ1,2 - -  be the differential operators defined by d0,1=µρ−u, d0,2=d dρ −(κ+u)ρ−1±sρ , d1,1=d dρ −κρ−1±sρ , d1,2=−µρ−u, δ0,1=µρ−u, δ0,2=−d dρ −(κ+u)ρ−1±sρ , δ1,1=−d dρ −κρ−1±sρ , δ1,2=−µρ−u. Observe that δ0=d† 0and δ1=d† 1. We may also use the more explicit notation d± r,δ± r,d± r,i and δ± r,i. A direct computation shows that d0and d1define an elliptic complex (F, d) of length two. Note that, by (9), d1,1=ρ−ud0,2ρu, δ0,2=ρ−uδ1,1ρu.(43) WITTEN’S PERTURBATION ON STRATA 23 4.2.1. Self-adjoint operators defined by the Laplacian. By (9), the homogeneous components of the corresponding Laplacian ∆ (or ∆±) are given by ∆0=H+ (κ+u)(κ+u−1)ρ−2+µ2ρ−2u∓s(1 + 2(κ+u)) , ∆2=H+κ(κ+ 1)ρ−2+µ2ρ−2u±s(1 −2κ), ∆1=∆1,1−2µuρ−u−1 −2µuρ−u−1∆1,2, ∆1,1=H+κ(κ−1)ρ−2+µ2ρ−2u∓s(1 + 2κ), ∆1,2=H+ (κ+u)(κ+u+ 1)ρ−2+µ2ρ−2u±s(1 −2(κ+u)) . (We may also use (37) and (38) to compute easily some parts of the above components of ∆.) The operators ∆0, ∆2, ∆1,1and ∆1,2are like Pand Qin (34), with c1= 0 = d1, plus a constant term. Write ∆1=U∓sV , where V=1+2κ0 0−1 + 2(κ+u).(44) Then, by Propositions 3.3, 3.4 and 3.5, and Remark 7 (v), ∆0, ∆2and ∆1define the self-adjoint operators Piand Qjin L2 +, and Wi,j in L2 +⊕L2 +, indicated in Table 4, where the conditions come from (17), (21), (23), (24), (25), (26) and (27). The notation P± i,Q± jand W± i,j may be used as well to specify that these operators are defined by ∆± 0, ∆± 2and ∆± 1. Note that v=ufor all Wi,j. The cores of P1/2 i, Q1/2 jand W1/2 i,j , given by Propositions 3.3, 3.4 and 3.5, will be denoted by F0 i,F2 j and F1 i,j =F1,1 i⊕F1,2 j, respectively. Remark 9.In contrast to Eiin Section 4.1.1, note that the graded subspace F0 i⊕ F1 i,j ⊕F2 jof C∞(F)∩L2(F), whenever defined, is not preserved by D=d+δ. For instance, it is preserved by dbut not by δwhen i=j= 1, and it is preserved by δ but not by dwhen i=j= 2. σ τ θ Condition ∆0P1κ+u κ > −1 2 P21−κ−u κ < 3 2−2u ∆2Q1κ κ > u −3 2 Q2−1−κ κ < 1 2−u ∆1 W1,1κ κ +u κ κ > u −1 2 W2,21−κ−1−κ−u−κ−u κ < 1 2−2u 6 ∃ W1,2κ−1−κ−u−1 2−uImpossible W2,11−κ κ +u1 2−1−u 2< κ < 1−u 2 Table 4. Self-adjoint operators defined by ∆0, ∆2and ∆1 Let us explain the contents of Table 4. Since c1=d1= 0, we have σ=aand τ=b, which are given by (11) and (14). Moreover σ,τand udetermine θin Table 4 so that Uis of the form (35) because 2θ−σ−τ=−u. Let us check the conditions written in this table, which are given by the hypothesis of Propositions 3.3–3.5. For 24 J.A. ´ ALVAREZ L ´ OPEZ, M. CALAZA, AND C. FRANCO Piand Qj, only (17) and (21) are required. For Wi,j, we also require (23), and the hypothesis (a)–(d) of Proposition 3.5, obtaining the following: •For W1,1, we have σ=θ6=τand τ−σ=u6∈ −N. Thus (a) applies in this case. Note that (17), (21) and (23) mean κ>u−1 2. Then (24) holds because 0 < u < 1 and κ > u −1 2. So (a) is satisfied. •For W2,2, we have σ6=θ=τ+ 1 and σ−τ−1 = 1 + u6∈ −N. Thus (c) applies in this case. Now, (17), (21) and (23) mean κ < 1 2−2u. Then (26) holds because 0 < u < 1 and κ < 1 2−2u. So (c) is satisfied. •There is no W1,2because θ < −1 2in that case. •For W2,1, (17), (21) and (23) mean −3 2< κ < 3 2−u, and we have the following possibilities: –The case σ=θ=τis not possible because u6= 0. –The case σ=θ6=τhappens when κ=1 2. Then σ=1 2and τ=1 2+u, obtaining τ−σ=u6∈ −N. Thus (a) applies in this case. Moreover (24) holds because 0 <u<1. So (a) is satisfied. –The case σ6=θ=τhappens when κ=1 2−u. Then σ=1 2+uand τ=1 2, obtaining σ−τ=u6∈ −N. Thus (b) applies in this case. Moreover (25) holds because 0 < u < 1. Hence (b) is satisfied. –The case σ6=θ=τ+ 1 happens when κ=−1 2−u. Then σ=3 2+u and τ=−1 2, obtaining σ−τ−1 = 1+u6∈ −N. Thus (c) applies in this case. Moreover (26) holds because 0 < u < 1. Hence (c) is satisfied. –Finally, assume that σ6=θ6=τ. The condition σ−θ, τ −θ6∈ −N means that κ6∈ (1 2+N)∪(1 2−u−N), which in turn means that κ6=1 2,1 2−u, −1 2−ubecause −3 2<κ<3 2−u. But σ=θif κ=1 2, τ=θif κ=1 2−u, and θ=τ+ 1 if κ=−1 2−u, as we have seen in the previous cases. So σ−θ, τ −θ6∈ −N, and (d) applies in this case. Moreover, since 0 <u<1, (27) holds just when −1−u 2<κ<1−u 2. Thus (d) is satisfied assuming the stated conditions on κ. Therefore W2,1is defined in one of the above ways if −1−u 2<κ<1−u 2. There are the following overlaps of the conditions in Table 4: •Both P1and P2are defined for −1 2< κ < 3 2−2u, and P1=P2just when κ=1 2−u. •Both Q1and Q2are defined for u−3 2< κ < 1 2−u, and Q1=Q2just when κ=−1 2. •Both W1,1and W2,2are defined for u−1 2< κ < 1 2−2u(if u < 1 3), but W1,16=W2,2for all such κ. •Both W1,1and W2,1are defined for u−1 2<κ<1−u 2, and W1,1=W2,1 just when κ=1 2. •Both W2,2and W2,1are defined for −1−u 2< κ < 1 2−2u, and W2,2=W2,1 just when κ=−1 2−u. Propositions 3.3, 3.4 and 3.5 also give the following spectral estimates, for all  > 0: •The spectrum of P1consists of eigenvalues λ0≤λ2≤ ···, taking multiplicity into account, such that there are some D=D(κ, u)>0 and WITTEN’S PERTURBATION ON STRATA 25 C=C(, κ, u)>0 so that, for all k∈2N, λk≥(2k+ (1 ∓1)(1 + 2(κ+u)))s+µ2Dsu(k+ 1)−u,(45) λk≤(2k+ (1 ∓1)(1 + 2(κ+u)))s + (2k+ 1 + 2(κ+u))µ2su+µ2Csu.(46) The first term of the right-hand side of (45) and (46) for P+ 1and P− 1is 2ks and 2(k+ 1 + 2(κ+u))s, respectively. •The spectrum of P2consists of eigenvalues λ0≤λ2≤ ···, taking multiplicity into account, such that there are some D=D(κ, u)>0 and C=C(, κ, u)>0 so that, for all k∈2N, λk≥(2k+ 4 −(1 ±1)(1 + 2(κ+u)))s+µ2Dsu(k+ 1)−u,(47) λk≤(2k+ 4 −(1 ±1)(1 + 2(κ+u)))s + (2k+ 3 −2(κ+u))µ2su+µ2Csu.(48) The first term of the right-hand side of (47) and (48) for P+ 2and P− 2 becomes 2(k+ 1 −2(κ+u))sand 2(k+ 2)s, respectively. •The spectrum of Q1consists of eigenvalues λ1≤λ3≤ ···, taking multiplicity into account, such that there are some D=D(κ, u)>0 and C=C(, κ, u)>0 so that, for all k∈2N+ 1, λk≥(2k+ 2 −(1 ∓1)(1 −2κ))s+µ2Dsu(k+ 1)−u,(49) λk≤(2k+ 2 −(1 ∓1)(1 −2κ))s+ (2k+ 1 + 2κ)µ2su+µ2Csu.(50) The first term of the right-hand side of (49) and (50) for Q+ 1and Q− 1is 2(k+ 1)sand 2(k+ 2κ)s, respectively. •The spectrum of Q2consists of eigenvalues λ1≤λ3≤ ···, taking multiplicity into account, such that there are some D=D(κ, u)>0 and C=C(, κ, u)>0 so that, for all k∈2N+ 1, λk≥(2k−2 + (1 ±1)(1 −2κ))s+µ2Dsu(k+ 1)−u,(51) λk≤(2k−2 + (1 ±1)(1 −2κ))s+ (2k−1−2κ)µ2su+µ2Csu.(52) The first term of the right-hand side of (51) and (52) for Q+ 2and Q− 2is 2(k−2κ)sand 2(k−1)s, respectively. •For W2,1, we can take ˜u=u+1 2satisfying (31). Moreover the maximum eigenvalue of ∓sV is s(1∓(2κ+u)−u). Thus the spectrum of W2,1consists of two groups of eigenvalues, λ0≤λ2≤ ··· and λ1≤λ3≤ ···, repeated according to multiplicity, such that there are some D=D(κ, u)>0, C= C(, κ, u)>0, e C=e C(, κ, u)>0 and E=E(, κ)>0 so that, for all k∈2N, λk≥1−2µusu−1 2(2k+ 3 −2κ)s +µ2Dsu(k+ 1)−u−2µu e Csu+1 2∓(1 + 2κ)s , (53) λk≤(2k+ 4 −(1 ±1)(2κ+u))s + (2k+ 3 −2κ)(µ2su+ 4µusu+1 2) + µ2Csu+ 4µuEsu+1 2,(54) 32 J.A. ´ ALVAREZ L ´ OPEZ, M. CALAZA, AND C. FRANCO Since t7→ φtdefines a differentiable map with values in L2(F), it follows that there is a sequence a<ci↑ ∞ such that φt(ci)→0, and d dt Z∞ a+t|φt(ρ)|2dρ = lim i d dt Zci a+t|φt(ρ)|2dρ ≤lim i|φt(ci)|2= 0 . So Z∞ a+t|φt(ρ)|2dρ ≤Z∞ a|φ0(ρ)|2dρ =Z∞ a|φ(ρ)|2dρ = 0 . 5. Witten’s perturbation on a cone For rel-Morse functions, the rel-local analysis of the Witten’s perturbed Laplacian will be reduced to the case of the functions ±1 2ρ2on a stratum of a cone with a model adapted metric, where ρdenotes the radial function. This kind of rel-local analysis begins in this section. 5.1. Witten’s perturbation. To begin with, recall the following generalities about the Witten’s perturbation. Let M≡(M, g) be a Riemannian n-manifold. For all x∈Mand α∈TxM∗, let αy= (−1)nr+n+1 ? α∧?=−ια]on r ^TxM∗, involving the Hodge star operator ?on VTxM∗defined by any choice of orientation of TxM. For any f∈C∞(M), E. Witten [43] has introduced the following perturbations of d,δ,Dand ∆, depending on s≥0: ds=e−sf d esf =d+s df∧,(71) δs=esf δ e−sf =δ−s dfy,(72) Ds=ds+δs=D+sR , ∆s=D2 s=dsδs+δsds= ∆ + s(RD +DR) + s2R2,(73) where R=df∧−dfy. Notice that δs=d† s; thus Dsand ∆sare formally self-adjoint. By analyzing the terms RD +DR and R2, the expression (73) becomes ∆s= ∆ + sHessf+s2|df|2,(74) where Hessfis an endomorphism defined by Hess f[36, Lemma 9.17], satisfying |Hessf|=|Hess f|[4, Section 9]. 5.2. De Rham operators on a cone. Let Lbe a non-empty compact stratification. Consider a stratum Nof L, and the corresponding stratum M=N×R+of c(L). We use the notation ˜n= dim Nand n= dim M= ˜n+ 1. Let π:M→N be the first factor projection, and ρthe radial function on c(L). From VTM∗= VTN∗VTR∗ +, we get a canonical identity r ^TM∗≡π∗ r ^TN∗⊕dρ ∧π∗ r−1 ^TN∗≡π∗ r ^TN∗⊕π∗ r−1 ^TN∗(75) for every degree r. So Ωr(M)≡C∞(R+,Ωr(N)) ⊕dρ ∧C∞(R+,Ωr−1(N)) (76) ≡C∞(R+,Ωr(N)) ⊕C∞(R+,Ωr−1(N)) .(77) Here, smooth functions R+→Ω(N) are defined by considering Ω(N) as Fr´echet space with the weak C∞topology. In this section, all matrix expressions of vector WITTEN’S PERTURBATION ON STRATA 33 bundle homomorphisms on VrTM∗or differential operators on Ωr(M) will be considered with respect to the decompositions (75) and (77). Let dand ˜ ddenote the exterior derivatives on Ω(M) and Ω(N), respectively. We have [4, Lemma 10.1] d≡ ˜ d0 d dρ −˜ d!.(78) Fix a general adapted metric ˜gon N. For u > 0, the metric g=ρ2u˜g+dρ2is a general adapted metric on M. The induced metrics on VTM∗and VTN∗are also denoted by gand ˜g, respectively. Fix some degree r∈ {0,1, . . . , n}, and, to simplify the expressions, let κ= (n−2r−1)u 2.(79) According to (75), g≡ρ−2ru ˜g⊕ρ−2(r−1)u˜g(80) on VrTM∗. Choose an orientation on an open subset W⊂N, and let ˜ωdenote the corresponding ˜g-volume form on W. Consider the orientation on W×R+⊂M so that the corresponding g-volume form is ω=ρ(n−1)udρ ∧˜ω . (81) The corresponding Hodge star operators on VT(W×R+)∗and VTW∗will be denoted by ?and ˜ ?, respectively. Like in [4, Lemma 10.2], from (80) and (81), it follows that ?≡0ρ2(κ+u)˜ ? (−1)rρ2κ˜ ?0(82) on VrT(W×R+)∗. Let L2Ωr(M) = L2Ωr(M, g) and L2Ωr(N) = L2Ωr(N, ˜g). From (80) and (81), we also get that (77) induces the identity of Hilbert spaces6 L2Ωr(M)≡L2 κ,+b ⊗L2Ωr(N)⊕L2 κ+u,+b ⊗L2Ωr−1(N).(83) Let δand ˜ δdenote the exterior coderivatives on Ω(M) and Ω(N), respectively. Like in [4, Lemma 10.3], using (78), (82) and (9), we get δ≡ ρ−2u˜ δ−d dρ −2(κ+u)ρ−1 0−ρ−2u˜ δ!(84) on Ωr(M). Let ∆ and e ∆ denote the Laplacians on Ω(M) and Ω(N), respectively. Like in [4, Corollary 10.4], from (78), (84) and (9), it follows that ∆≡P−2uρ−1˜ d −2uρ−2u−1˜ δ Q (85) on Ωr(M), where P=ρ−2ue ∆−d2 dρ2−2κρ−1d dρ ,(86) Q=ρ−2ue ∆−d2 dρ2−2(κ+u)d dρ ρ−1.(87) 6Recall that, for Hilbert spaces H0and H00, with scalar products h,i0and h,i00, the notation H0b ⊗H00 is used for the Hilbert space tensor product. This is the Hilbert space completion of the algebraic tensor product H0⊗H00 with respect to the scalar product defined by hu0⊗u00, v0⊗v00i= hu0, v0i0hu00, v00i00. 34 J.A. ´ ALVAREZ L ´ OPEZ, M. CALAZA, AND C. FRANCO 5.3. Witten’s perturbation on a cone. Let ds,δs,Dsand ∆s(s≥0) denote the Witten’s perturbations of d,δ,Dand ∆ induced by the function f=±1 2ρ2on M. The more explicit notation d± s,δ± s,D± sand ∆± smay be used if needed. In this case, df =±ρ dρ. According to (77), ρ dρ∧ ≡ 0 0 ρ0,−ρ dρy≡0ρ 0 0. So, by (78), (84), (71) and (72), ds≡ ˜ d0 d dρ ±sρ −˜ d!,(88) δs≡ ρ−2u˜ δ−d dρ −2(κ+u)ρ−1±sρ 0−ρ−2u˜ δ!,(89) on Ωr(M). Now, R=±ρ(dρ∧−dρy)≡ ±0ρ ρ0, and therefore R2≡ρ20 0ρ2≡ρ2.(90) Like in [4, Lemma 10.6], we get RD +DR =∓V(91) on Ωr(M), where Vis given by (44). As a consequence of (73), (85) and (91), we obtain ∆s≡Ps−2uρ−1˜ d −2uρ−2u−1˜ δ Qs(92) on Ωr(M), where Ps=ρ−2ue ∆ + H−2κρ−1d dρ ∓s(1 + 2κ),(93) Qs=ρ−2ue ∆ + H−2(κ+u)d dρ ρ−1∓s(−1 + 2(κ+u)) .(94) 6. Splitting of the Witten’s complex on a cone 6.1. Spectral decomposition on the link of the cone. Theorem 1.1 is proved by induction on the depth. Thus, with the notation of Section 5, suppose that ˜g is good, and e ∆max/min satisfies the statement of Theorem 1.1. Moreover suppose that gis also good; that is, u≤1. Let e Hmax/min = ker e Dmax/min = ker e ∆max/min, which is a graded subspace of Ω(N)∩L2Ω(N). For every degree r, let e Rmax/min,r−1,e R∗ max/min,r ⊂L2Ωr(N) be the images of ˜ dmax/min,r−1and ˜ δmax/min,r, respectively, which are closed subspaces. By restriction, e ∆max/min defines self-adjoint operators in e Rmax/min,r−1and e R∗ max/min,r−1, with the same eigenvalues [4, Section 5.1]. For any eigenvalue ˜ λof WITTEN’S PERTURBATION ON STRATA 35 the restriction of e ∆max/min to e Rmax/min,r−1, let e Rmax/min,r−1,˜ λand e R∗ max/min,r−1,˜ λ denote the corresponding ˜ λ-eigenspaces. We have7 L2Ωr(N) = e Hr max/min ⊕[ M ˜ λ,˜ λ0e Rmax/min,r−1,˜ λ⊕e R∗ max/min,r,˜ λ0,(95) where ˜ λand ˜ λ0run in the spectrum of the restrictions of e ∆max/min to e Rmax/min,r−1 and e R∗ max/min,r, respectively. 6.2. Subcomplexes of length one. Given 0 6=γ∈e Hr max/min, consider the canonical identities C∞ +≡C∞ +γ⊂Ωr(M), C∞ +≡C∞ +dρ ∧γ⊂Ωr+1(M).(96) The following result follows from (88) and (89). Lemma 6.1. For s≥0,dsand δsdefine maps 0C∞ +γ C∞ +dρ ∧γ0.. ds,r−1 δs,r−1 ds,r δs,r ds,r+1 δs,r+1 - - -  Moreover, using (96), ds,r =d dρ ±sρ , δs,r =−d dρ −2κρ−1±sρ . Let Eγ,0denote the subcomplex of length one of (Ω(M), ds) defined by Er γ,0=C∞ +,0γ≡C∞ +,0,Er+1 γ,0=C∞ +,0dρ ∧γ≡C∞ +,0. The closure of Eγ,0in L2Ω(M) is denoted by L2Eγ. By (83), L2Er γ=L2 κ,+γ≡L2 κ,+, L2Er+1 γ=L2 κ,+dρ ∧γ≡L2 κ,+. Assume now that s > 0. With the notation of Section 4.1, consider the real version of the elliptic complex (E, d) determined by sand κ(given by (79)). Using Lemma 6.1 and (9), like in [4, Proposition 12.3], we get the following. Proposition 6.2. The operator ρκ:L2 κ,+→L2 +defines a unitary isomorphism L2Eγ→L2(E), which restricts to an isomorphism of complexes, (Eγ,0, ds)→ (C∞ 0(E), d), up to a shift of degree. By Proposition 6.2, (Eγ,0, ds) has a maximum/minimum Hilbert complex extension in L2Eγ. Let (Dγ,ds,γ) be the maximum/minimum Hilbert complex extension of (Eγ,0, ds) if γ∈e Hr max/min, and ∆s,γ the corresponding Laplacian. Let Hs,γ =Hr s,γ ⊕ Hr+1 s,γ = ker ∆s,γ , with the induced grading. The more explicit notation d± s,γ,∆± s,γ and H± s,γ =H±,r s,γ ⊕H±,r+1 s,γ may be also used. Corollary 6.3. (i)∆s,γ has a discrete spectrum. (ii)The dimensions of H±,r s,γ and H±,r+1 s,γ are given in Table 11. (iii)If es∈ Hs,γ with norm one for every s, and his a bounded measurable function on R+with h(ρ)→1as ρ→0, then hhes, esi → 1as s→ ∞. (iv)All nonzero eigenvalues of ∆s,γ are positive and in O(s)as s→ ∞. 36 J.A. ´ ALVAREZ L ´ OPEZ, M. CALAZA, AND C. FRANCO γ∈e Hr max γ∈e Hr min H+,r s,γ H+,r+1 s,γ H−,r s,γ H−,r+1 s,γ H+,r s,γ H+,r+1 s,γ H−,r s,γ H−,r+1 s,γ κ≥1 21001 0 0 |κ|<1 20 1 κ≤ −1 20 1 Table 11. Dimensions of H±,r s,γ and H±,r+1 s,γ Proof. This follows from Propositions 6.2 and 4.1, Corollary 3.7, Section 4.1.1, and the choice made to define ds,γ. 6.3. Subomplexes of length two. Let µ=p˜ λfor an eigenvalue ˜ λof the restriction of e ∆max/min to e Rmax/min,r−1. According to [4, Section 5.1], there are nonzero differential forms, α∈e Rmax/min,r−1,˜ λ⊂Ωr(N), β ∈e R∗ max/min,r−1,˜ λ⊂Ωr−1(N), such that ˜ dβ =µα and ˜ δα =µβ. Consider the canonical identities C∞ +≡C∞ +β⊂Ωr−1(M), C∞ +≡C∞ +dρ ∧α⊂Ωr+1(M),(97) C∞ +⊕C∞ +≡C∞ +α+C∞ +dρ ∧β⊂Ωr(M).(98) The following result follows from (88) and (89). Lemma 6.4. For s≥0,dsand δsdefine maps 0C∞ +β C∞ +α+C∞ +dρ ∧β C∞ +dρ ∧α0. ds,r−2 δs,r−2 ds,r−1 δs,r−1 ds,r δs,r ds,r+1 δs,r+1 - -  - -  Moreover, according to (97) and (98), ds,r−1=µ d dρ ±sρ, δs,r−1=µρ−2u−d dρ −2(κ+u)ρ−1±sρ, ds,r =d dρ ±sρ −µ, δs,r =−d dρ −2κρ−1±sρ −µρ−2u. 7Consider a family of Hilbert spaces, Hawith scalar product h,ia. Recall that the Hilbert space direct sum, c LaHa, is the Hilbert space completion of the algebraic direct sum, LaHa, with respect to the scalar product h(ua),(va)i=Pahua, vaia. Thus c LaHa=LaHaif and only if the family is finite. WITTEN’S PERTURBATION ON STRATA 37 Let Fα,β,0=Fr−1 α,β,0⊕ Fr α,β,0⊕ Fr+1 α,β,0denote the subcomplex of length two of (Ω(M), ds) defined by Fr−1 α,β,0=C∞ +,0β≡C∞ +,0,Fr+1 α,β,0=C∞ +,0dρ ∧α≡C∞ +,0, Fr α,β,0=C∞ +,0α+C∞ +,0dρ ∧β≡C∞ +,0⊕C∞ +,0. The closure of Fα,β,0in L2Ω(M) is denoted by L2Fα,β. By (83), L2Fr−1 α,β =L2 κ+u,+β≡L2 κ+u,+, L2Fr+1 α,β =L2 κ,+dρ ∧α≡L2 κ,+, L2Fr α,β =L2 κ,+α+L2 κ+u,+dρ ∧β≡L2 κ,+⊕L2 κ+u,+. Assume now that s > 0. With the notation of Section 4.2, consider the real version of the elliptic complex (F, d) determined by sand κ(given by (79)). Using Lemma 6.4 and (9), we get the following (cf. [4, Proposition 12.9]). Proposition 6.5. If u < 1, then ρκ:L2 κ,+→L2 +and ρκ+u:L2 κ+u,+→L2 + define a unitary isomorphism L2Fα,β →L2(F), which restricts to an isomorphism of complexes, (Fα,β,0, ds)→(C∞ 0(F), d), up to a shift of degree. By Proposition 6.5, (Fα,β,0, ds) has a maximum/minimum Hilbert complex extension in L2Fα,β. Let (Dα,β,ds,α,β) be the maximum/minimum Hilbert complex extension of (Fα,β,0, ds) if α∈e Rmax/min,r−1,˜ λand β∈e R∗ max/min,r−1,˜ λ. Let ∆s,α,β denote the corresponding Laplacian. The more explicit notation d± s,α,β and ∆± s,α,β may be used. Corollary 6.6. (i)∆s,α,β has a discrete spectrum. (ii)The eigenvalues of ∆s,α,β are positive and in O(s)as s→ ∞. Proof. In the case u < 1, this follows from Proposition 6.5 and Corollary 4.5. In the case u= 1, this is the content of [4, Proposition 12.11].  Remark 11.According to (91)–(94), we have ∆s≡H−2κρ−1d dρ ∓s(1 + 2κ) on C∞ +≡C∞ +γ , ∆s≡H−2κd dρ ρ−1∓s(−1+2κ) on C∞ +≡C∞ +dρ ∧γ , ∆s≡H−2(κ+u)ρ−1d dρ +µ2ρ−2u∓s(1 + 2(κ+u)) on C∞ +≡C∞ +β , ∆s≡H−2κd dρ ρ−1+µ2ρ−2u∓s(−1+2κ) on C∞ +≡C∞ +dρ ∧α , and ∆s≡Pµ,s −2µuρ−1 −2µuρ−2u−1Qµ,s  on C∞ +⊕C∞ +≡C∞ +α+C∞ +dρ ∧β, where Pµ,s =H−2κρ−1d dρ +µ2ρ−2u∓s(1 + 2κ), Qµ,s =H−2(κ+u)d dρ ρ−1+µ2ρ−2u∓s(−1 + 2(κ+u)) . So the results of Section 3 could be applied to these expressions. We opted for analyzing first the complexes of Section 4 for the sake of simplicity because we have a=b= 0, L2 +is used instead of L2 κ,+or L2 κ+u,+, and Remark 8 is directly applied. 38 J.A. ´ ALVAREZ L ´ OPEZ, M. CALAZA, AND C. FRANCO 6.4. Splitting into subcomplexes. Let Cmax/min,0denote an orthonormal frame of e Hmax/min consisting of homogeneous differential forms. For every positive eigenvalue µof e Dmax/min, let Cmax/min,µ be an orthonormal frame of the µ-eigenspace of e Dmax/min consisting of differential forms α+βlike in Section 6.3. Then let ds,max/min =M γ ds,γ ⊕d M µM α+β ds,α,β , where γruns in Cmax/min,0,µruns in the positive spectrum of e Dmax/min, and α+β runs in Cmax/min,µ. The notation d± s,max/min may be also used when d± s,γ and d± s,α,β are considered. Proposition 6.7. We have ds,max/min =ds,max/min. Proof. This follows like [4, Proposition 12.12], using [4, Lemma 5.2], [9, Lemma 3.6 and (2.38b)], (76) and (95).  Let Hs,max/min =LrHr s,max/min = ker ∆s,max/min, with the induced grading. The superindex “±” may be added to this notation to indicate that we are referring to ∆± s,max/min. Corollary 6.8. (i) ∆s,max/min has a discrete spectrum. (ii)Table 12 describes the isomorphism class of H±,∗ s,max/min. (iii)If es∈ Hs,max/min has norm one for every s, and his a bounded measurable function on R+with h(ρ)→1as ρ→0, then hhes, esi → 1as s→ ∞. (iv)Let 0≤λs,max/min,0≤λs,max/min,1≤ ··· be the eigenvalues of ∆s,max/min, repeated according to their multiplicities. Given k∈N, if λs,max/min,k >0for some s, then λs,max/min,k >0for all s, and λs,max/min,k ∈O(s)as s→ ∞. (v)There is some θ > 0such that lim infkλs,max/min,kk−θ>0. H+,r s,max H−,r+1 s,max H+,r s,min H−,r+1 s,min κ≥1 2Hr max(N) 0 Hr min(N) 0 |κ|<1 20Hr min(N) κ≤ −1 20Hr max(N) Table 12. Spaces isomorphic to H±,∗ s,max/min Proof. In the case u= 1, this result was already shown in [4, Corollary 12.13]. So we consider only the case 0 <u<1. For all γ,µand α+βas above, ∆s,γ and ∆s,α,β have a discrete spectrum by Corollaries 6.3 (i) and 6.6 (i). Moreover the union of their spectra has no accumulation points according to Section 4 and since e ∆max/min is discrete. Then (i) follows by Proposition 6.7. Now, properties (ii)–(iv) follow directly from Corollaries 6.3 and 6.6, and Proposition 6.7. To prove (v), let 0 ≤˜ λmax/min,0≤˜ λmax/min,1≤ ··· denote the eigenvalues of e ∆max/min, repeated according to their multiplicities. Since Nsatisfies Theorem 1.1 (ii) with ˜g, there is some C0, θ0>0 such that ˜ λmax/min,` ≥C0`θ0(99) WITTEN’S PERTURBATION ON STRATA 39 for all `large enough. Consider the counting function N± s,max/min(λ)=#nk∈N|λ± s,max/min,k < λ o(λ > 0) . From Proposition 4.3, Corollary 4.5, (39)–(42), (45), (47), (49), (51), (53), (55) and (99), and the choices made to define dγand dα,β (Sections 6.2 and 6.3), it follows that there are some C1, C2>0 and C3, C0 3∈Rsuch that N± s,max/min(λ) ≤#n(k, `)∈N2|C1k+C2˜ λmax/min,`(k+ 1)−u+C0 3≤λo ≤#{(k, `)∈N2|C1k+C2C0`θ0(k+ 1)−u+C3≤λ} ≤#((k, `)∈N20≤λ−C3 C1 , ` ≤λ−C3−C1k C2C01 θ0(k+ 1) u θ0). Consider the function f:−1, a := λ−C3 C1→[0,∞), f(x) = λ−C3−C1x C2C01 θ0(x+ 1) u θ0. Elementary calculus shows that fvanishes at x=−1, a, it reaches its maximum at x=b:= λu −C3u−C1 C1(1 + u), and it is strictly increasing (respectively, decreasing) on [−1, b] (respectively, [b, a]). It follows that8 N± s,max/min(λ)≤Za 0 f(x)dx + 2f(b) + a+ 1 . But f(b) = λ−C3+C1 (1 + u)C2C01 θ0u(λ−C3+C1) (1 + u)C1u θ0, and Za 0 f(x)dx ≤ Zλ−C3 C1 0λ−C3−C1x C2C02 θ0dx!1 2 Zλ−C3 C1 0 (x+ 1)2u θ0dx!1 2 ≤ θ0(λ−C3)2 θ0+1 (2 + θ0)(C2C0)2 θ0C1!1 2 θ0(λ−C3+C1)2u θ0+1 (2u+θ0)C 2u θ0+1 1   1 2 =θ0(λ−C3)1 θ0+1 2(λ−C3+C1)u θ0+1 2 (2 + θ0)1 2(2u+θ0)1 2(C2C0)1 θ0C1+ u θ0 1 . 8A similar argument is made in the proof of [4, Corollary 12.13-(viii)]. In that case, the authors use a strictly decreasing function f: (−∞, a]→[0,∞). The resulting estimate should be N± s,max/min(λ)≤Za 0 f(x)dx +f(0) + a+ 1 , but the terms f(0) + a+ 1 were missing in that publication. This correction does not affect the final estimate of N± s,max/min(λ) obtained there. 40 J.A. ´ ALVAREZ L ´ OPEZ, M. CALAZA, AND C. FRANCO So N± s,max/min(λ)≤Cλ1+u θ0+1 for some C > 0 and all large enough λ, giving (v) with θ=1+u θ0+ 1.  Table 13 describes the above conditions on κin terms of r. κ≥1 2r≤n−1 2−1 2u |κ|<1 2|r−n−1 2|<1 2u κ≤ −1 2r≥n−1 2+1 2u Table 13. Correspondence between conditions on κand r 7. Relatively local model of the Witten’s perturbation Let m∈N, and let L1, . . . , Labe compact stratifications. For each i= 1, . . . , a, let Nibe a dense stratum of Li, let ki= dim Ni+1, and let ∗iand ρibe the vertex and radial function of c(Li). Then M:= Rm×Qa i=1(Ni×R+) is a dense stratum of A:= Rm×Qa i=1 c(Li). For any relatively compact open neighborhood Oof x:= (0,∗1,...,∗a), all general adapted metrics on Mare quasi-isometric on M∩O to a metric of the form g=g0+Pa i=1 ρ2ui i˜gi+ (dρi)2, where g0is the Euclidean metric on Rm, every ˜giis a general adapted metric on Ni, and ui>0. Suppose that gis good; i.e., the metrics ˜giare good, and ui≤1. We can assume that every Niis connected, which means that the fiber of lim : c M→Mover xconsists of a unique point, which can be identified to x(see [4, Proof of Proposition 3.20]). According to Section 1.4, the rel-local model of a rel-Morse function around a relcritical point is of the form f=1 2(ρ2 +−ρ2 −), where ρ±is the radial function of Rm±×Qi∈I±c(Li), for some decomposition m=m++m−(m±∈N), and some partition of {1, . . . , a}into sets I±. The rel-critical set of fconsists only of x. Let ds,δs,Dsand ∆sbe the Witten’s perturbations of d,δ,Dand ∆ on Ω(M) induced by f. Let Hs,max/min =LrHr s,max/min = ker ∆s,max/min, with the induced grading. The following result is a direct consequence of Corollary 6.8 and [4, Example 9.1 and Lemma 5.1], taking also into account Table 13. Corollary 7.1. (i) ∆s,max/min has a discrete spectrum. (ii)We have Hr s,max/min ∼ =M (r1,...,ra) a O i=1 Hri max/min(Ni), where (r1, . . . , ra)runs in the subset of Nadefined by the conditions r=m−+ a X i=1 ri+|I−|, ri<ki−1 2+1 2uiif i∈I+ ri≥ki−1 2+1 2uiif i∈I−)for Hr s,max , ri≤ki−1 2−1 2uiif i∈I+ ri>ki−1 2−1 2uiif i∈I−)for Hr s,min . WITTEN’S PERTURBATION ON STRATA 41 (iii)If es∈ Hs,max/min with norm one for every s, and his a bounded measurable function on R+with h(ρ)→1as ρ→0, then hhes, esi → 1as s→ ∞. (iv)Let 0≤λs,max/min,0≤λs,max/min,1≤ ··· be the eigenvalues of ∆s,max/min, repeated according to their multiplicities. Given k∈N, if λs,max/min,k >0for some s, then λs,max/min,k >0for all sand λs,max/min,k ∈O(s)as s→ ∞. (v)There is some θ > 0such that lim infkλs,max/min,k k−θ>0. For every ρ > 0, let Bρbe the open ball of center 0 and radius ρin Rm, and let Ux,ρ =Bρ× a Y i=1 (Ni×(0, ρ)) ⊂M . Taking complex coefficients, by Propositions 6.2, 6.5 and 6.7, the following result clearly boils down to the case of Proposition 4.6. Proposition 7.2. For α∈L2Ω(M), let αt= exp(itDs,max/min)α. If supp α⊂Ux,a for some a > 0, then supp αt⊂Ux,a+|t|for all t∈R. 8. Proof of Theorem 1.1 This theorem follows from Corollary 7.1 (i),(v) with the same arguments as [4, Theorem 1.1]. More precisely, [4, Propositions 14.2 and 14.3] are used to globalize the properties of the rel-local model, the min-max principle (see e.g. [35, Theorem XIII.1]) is used to show that the properties of the statement are invariant by taking Witten’s perturbation defined by rel-admissible functions, and Remark 6 (iii),(iv) is used to produce rel-admissible cutoff functions and partitions of unity with bounded differential. These functions are needed for the Witten’s perturbation and to apply [4, Propositions 14.2 and 14.3]. 9. Functional calculus Let Mbe a stratum of a compact stratification, equipped with a good general adapted metric g. Let fbe any rel-admissible function on M, and let ds,δs,Dsand ∆sbe the corresponding Witten’s perturbations of d,δ,Dand ∆. Since fis reladmissible, for every s, ∆s−∆ is a homomorphism with uniformly bounded norm by (74). From (74) and the min-max principle (see e.g. [35, Theorem XIII.1]), it also follows that D(∆s,max/min) = D(∆max/min), D∞(∆s,max/min) = D∞(∆max/min), and that the properties stated in Theorem 1.1 can be extended to the perturbation ∆s,max/min. For any rapidly decaying function φon R,φ(∆s,max/min) is a Hilbert-Schmidt operator on L2Ω(M) by the version of Theorem 1.1 (ii) for ∆s,max/min. In fact, φ(∆s,max/min) is a trace class operator because φcan be given as the product of two rapidly decaying functions, |φ|1/2and sign(φ)|φ|1/2, where sign(φ)(x) = sign(φ(x)) ∈ {±1}if φ(x)6= 0. Like in the case of closed manifolds (see e.g. [36, Chapters 5 and 8]), φ(∆s,max/min) is given by a Schwartz kernel Ks, and Tr φ(∆s,max/min) equals the integral of the pointwise trace of Kson the diagonal. But we do not know whether Ksis uniformly bounded because a “rel-Sobolev embedding theorem” is missing [4, Section 19]. Theorem 1.1 (ii) becomes important in our arguments to make up for this lack.