The Witten deformation of the Dolbeault complex

The Witten deformation of the Dolbeault complex J. ´ Alvarez L´opez∗and P. Gilkey Abstract. We introduce a Witten-Novikov type perturbation ¯ ∂¯ωof the Dolbeault complex of any complex K¨ahler manifold, defined by a form ωof type (1,0) with ∂ω = 0. We give an explicit description of the associated index density which shows that it exhibits a nontrivial dependence on ω. The heat invariants of lower order are shown to be zero. Mathematics Subject Classification (2010). 58J20. Keywords. Witten deformation, local index density, Dolbeault complex, K¨ahler manifold. 1. Introduction 1.1. Historial Summary and motivation Let Mbe a closed manifold of dimension m, and let hbe a smooth function on M. Witten [25] introduced a perturbation of the de Rham differential of the form dh=d+ ext(dh), using exterior multiplication by dh. Since dhis gauge equivalent to d, the Betti numbers are unchanged. Given a Riemannian metric gon M, the perturbed de Rham codifferential is δh=δ+ int(dh) where int denotes interior multiplication. In general, the perturbed Laplacian, ∆h=dhδh+δhdh, is not gauge equivalent to ∆ and thus can have a different spectrum. In fact, when his a Morse function, Witten gave a beautiful analytical proof of the Morse inequalities by analyzing the spectrum of ∆sh as s→ ∞. This program of Witten was continued by Helffer and Sj¨ostrand [14], and by Bismut and Zhang [4]. With more generality, Novikov [18, 19] defined similar perturbed operators dι,διand ∆ι, replacing dh with a real closed 1-form ιon M. He used Witten’s procedure to estimate the zeros of ιif ιis of Morse type. Since ∗Corresponding author. Research partially supported by Projects MTM2016-75897-P and MTM2017-89686- P (AEI/FEDER, UE). 2 J. ´ Alvarez L´opez and P. Gilkey dιneed not be gauge equivalent to d, the new twisted Betti numbers can be different. However, one can show that the twisted Betti numbers of dsι (s∈R) are constant except for a finite number of values of s, where the dimensions may jump. Those ground values of twisted Betti numbers are called the Novikov numbers of the cohomology class [ι]; they are used in the Novikov version of the Morse inequalities. We refer to related work of Braverman and Farber [5] and of Pazhitnov [22], and, more recently, to the work of many other authors [6, 7, 13, 16, 17]. In previous work [1], we used methods of invariance theory to prove that the local index density for the Witten-Novikov Laplacian ∆ιis the Euler form if mis even and, in particular, does not depend on ι. If mis odd, the local index density vanishes. The heat trace invariants of smaller order are also trivial, but the heat trace invariants of higher order exhibit a nontrivial dependence on ι. A different proof of the invariance of the twisted index density was also given by the first author, Kordyukov, and Leichtnam [2], where it was applied to study certain trace formulas for foliated flows (our original motivation). In [1], we also extended the invariance of the twisted index density to the setting of manifolds with boundary, and gave an equivariant version of that invariance for maps. The situation in the complex setting is quite different. We proved that the local index density for a Witten-Novikov type perturbation of the Dolbeault complex exhibits non-trivial dependence on the twisting 1-form in the case of Riemann surfaces. In the present paper, we extend the study of the Witten-Novikov type perturbation of the Dolbeault complex to the case of an arbitrary complex K¨ahler manifold (M, g, J) of dimension m= 2m. We consider the Dolbeault complex ¯ ∂with coefficients in an auxiliary holomorphic vector bundle E over Mequipped with a Hermitian metric h. The Hirzebruch-Riemann-Roch Theorem states that its index is given by the integral on Mof {Td(M, g, J)∧ ch(E, h)}m(the homogeneous component of degree mof the product of the Todd genus of (M, g, J) and the Chern character of (E, h)). This theorem was refined by the second author [9, 10], and by Atiyah, Bott, and Patodi [3], showing that {Td(M, g, J)∧ch(E, h)}mis indeed the index density that shows up in the asymptotic expansion of the heat kernel. In this complex setting, the Witten-Novikov deformation of the Dolbeault complex is ¯ ∂¯ω=¯ ∂+ ext(¯ω), where ωis a form of type (1,0) on M with ∂ω = 0. We use methods of invariance theory to give an explicit description of its index density, which is a perturbation of {Td(M, g, J)∧ch(E, h)}m with a non trivial dependence on ω. The other heat invariants of lower order are shown to be zero. As a possible application, this description might be a step in a version for the leafwise Dolbeault complex of the trace formula for foliated flows given in [2]. 1.2. Operators of Laplace Type Henceforth, let dvol be the measure defined by a Riemannian metric gon a closed manifold Mof dimension mand let hbe a Hermitian fiber metric on a vector bundle Eover M. A second order partial differential operator D Witten Deformation 3 on C∞(E) is said to be of Laplace type if the leading symbol is given by the metric tensor, i.e. if D=−   m X i,j=1 gij ∂2 ∂xi∂xjid + m X k=1 Ak∂ ∂xk+B   relative to a system of local coordinates (x1, . . . , xm) for Mand relative to a local frame for Ewhere gij =g(dxi, dxj) and where Akand Bare endomorphisms of E. The following result follows from work of Seeley [23] and others. Theorem 1.1. Let Dbe an operator of Laplace type. 1. There exists a smooth kernel K(t, x, y, D)for t > 0so that e−tDφ(x) = ZM K(t, x, y, D)φ(y) dvol(y). 2. There exist local invariants am,2n(D)(x)so that as t↓0, TrExK(t, x, x, D)∼∞ X n=0 t(2n−m)/2am,2n(D)(x), TrL2{e−tD} ∼ ∞ X n=0 t(2n−m)/2ZM am,2n(D)(x) dvol(x). 1.3. The local index density Let E={di:C∞(Ei)→C∞(Ei+1)}where (Ei, hi) is a finite collection of Hermitian vector bundles and where the diare first order partial differential operators. We shall say that Eis an elliptic complex of Dirac type if di+1di= 0 and if the associated self-adjoint second order operators Di:= d∗ idi+di−1d∗ i−1 are of Laplace type. The cohomology groups of Eare given by Hi(E) := kernel(di:C∞(Ei)→C∞(Ei+1)) image(di−1:C∞(Ei−1)→C∞(Ei)) . The Hodge Decomposition Theorem permits us to identify Hi(E) with ker(Di); these groups are finite dimensional and we take the super-trace to define am,2n(E) := X i (−1)iam,2n(Di) and index(E) := X i (−1)idim Hi(E). If mis odd, then index(E) = 0 so we assume meven henceforth. The invariant am,2n(E) for 2n=mis called the local index density as a cancellation argument due to Bott shows that ZM am,2n(E)(x) dvol(x) = index(E) if 2n=m 0 if 2n6=m. 4 J. ´ Alvarez L´opez and P. Gilkey 1.4. The de Rham complex Let Λi(M) be the vector bundle of i-forms and let d:C∞(Λi(M)) →C∞(Λi+1(M)) for 0 ≤i≤m−1, be exterior differentiation. This defines an elliptic complex of Dirac type we shall denote by EdeR(M, g). Let χ(M) be the Euler-Poincar´e characteristic of M. The Hodge–de Rham theorem permits us to identify Hp(EdeR(M, g)) with the topological cohomology groups Hp(M;C) and shows that index(EdeR(M, g)) = χ(M). 1.5. The Dolbeault complex Let Jbe an integrable almost complex structure on a smooth manifold Mof complex dimension mand corresponding real dimension m= 2m. Let gbe a Jinvariant Riemannian metric on M; (M, g, J) is a Hermitian holomorphic manifold. Let Ω(X, Y ) := g(X, JY ) be the K¨ahler form; we say (M, g, J) is K¨ahler if dΩ = 0. Let Ebe an auxiliary holomorphic vector bundle over M equipped with a Hermitian metric h. Let √2¯ ∂be the normalized Dolbeault operator; the normalizing constant of √2 is present to ensure that this is of Dirac type. The Dolbeault complex EDol(M, g, J, E, h) is defined by √2¯ ∂:C∞(Λ0,i(M)⊗E)→C∞(Λ0,i+1(M)⊗E) for 0 ≤i≤m−1. Let Hp(M;O(E)) be the cohomology groups of Mwith coefficients in the sheaf of holomorphic sections to E. Identify Hp(EDol(M, g, J, E, h)) with Hp(M;O(E)); if Eis the trivial line bundle, then index{EDol(M, g, J, E, h)} is the arithmetic genus of M. 1.6. The Chern-Gauss-Bonnet and Hirzebruch-Riemann-Roch Theorems Let m= 2m. Let Rijkl denote the components of the curvature tensor relative to a local orthonormal frame for the tangent bundle of M. We follow the discussion in Chern [8] and define the Pfaffian or Euler form by setting: Pfm(x, g) := m X i1,...,im,j1,...,jm=1 (−1)m 8mπmm!g(ei1∧···∧eim, ej1∧···∧ejm) Ri1i2j1j2. . . Rim−1imjm−1jm(x). In the complex setting, let Td(M, g, J) be the total Todd genus of the complex tangent bundle of (M, g, J) and let ch(E, h) be the total Chern character; we refer to Hirzebruch [15] for details. We set {Td(M, g, J)∧ch(E, h)}m=X i+j=m Tdi(M, g, J)∧chj(E, h). We use the Hodge ?operator to identify top dimensional forms with scalar functions. We refer to Chern [8] for the proof of Assertion (1) and to Hirzebruch [15] for the proof of Assertion (2) in the following result. Theorem 1.2. Witten Deformation 5 1. index(EdeR(M, g)) = ZM Pfm(M, g) dvol. 2. index(EDol(M, g, J, E, h)) = ZM ?{Td(M, g, J)∧ch(E, h)}mdvol. By identifying the local index densities of the de Rham and Dolbeault complexes with the integrands of Theorem 1.2, Patodi [20, 21] gave a heat equation proof of Theorem 1.2 (1) in the real setting and of Theorem 1.2 (2) in the complex K¨ahler setting by showing: Theorem 1.3. 1. am,2n(EdeR(M, g)) = 0if 2n<m Pfm(M, g)if 2n=m. 2. If (M, g, J)is K¨ahler, then am,2n(EDol(M, g, J, E, h)) =0if 2n<m ?{Td(M, g, J)∧ch(E, h)}mif 2n=m. Shortly thereafter, other proofs of Theorem 1.3 were given. Gilkey [9, 10] used invariance theory directly and Atiyah, Bott, and Patodi [3] combined invariance theory with a study of the twisted signature complex and the twisted spin-c complex to prove Theorem 1.3. The subject has an extensive history and we refer to [11] for further details. As noted, the twisted signature complex and twisted spin complex can be treated using heat equation methods [3, 9] and a heat equation proof of the full Atiyah-Singer index theorem given thereby. We note that the local index density for the Dolbeault complex does not agree in general with the Hirzebruch-Riemann-Roch integrand of Theorem 1.2 (2) in the non-K¨ahler setting as was shown in later work by Gilkey, Nikˇcevi´c, and Pohjanpelto [12]; Theorem 1.3 (2) can fail if (M, g, J) is not assumed K¨ahler. 1.7. The Witten deformation If ιis a closed 1-form on a Riemannian manifold (M, g), one can define the deformed de Rham complex EdeR(M, g, ι) by setting dι:= d+ ext(ι) : C∞(ΛiM)→C∞(Λi+1(M)) . The assumption that ιis closed ensures that d2 ι= 0; since ιintroduces a 0th perturbation, the leading symbol of the associated second order operators is unchanged so EdeR(M, g, ι) is an elliptic complex of Dirac type. The authors [1] showed previously that the deformation ιdoes not enter in the index density in this setting am,2n(EdeR(M, g, ι)) = 0 if 2n<m Pfm(M, g) if 2n=m. This result is sharp; if 2n > m, then am,2n(EdeR(M, g, ι)) does depend upon ιin general. In the complex setting, let ωbe a form of type (1,0) on Mwith ∂ω = 0 and let M:= (M, g, J, ω, E, h). Set ¯ ∂¯ω:= ¯ ∂+ext(¯ω). The assumption ∂ω = 0 6 J. ´ Alvarez L´opez and P. Gilkey implies ¯ ∂¯ω= 0 and ensures ¯ ∂2 ¯ω= 0. Let EDol(M) be the Witten perturbation of the Dolbeault complex with coefficients in Edefined by taking √2¯ ∂¯ω:C∞(Λ(0,i)⊗E)→C∞(Λ(0,i+1) ⊗E) for 0 ≤i≤m−1. This is an elliptic complex of Dirac type. Let =(ω) = 1 2√−1(ω−¯ω) be the imaginary part of ω. Let Θ := X k 1 k!πk{d=(ω)}k, {Td ∧ch ∧Θ}m:= X i+j+k=m Td(M, g, J)i∧ch(E, h)j∧Θk. The following is the main new result of this paper. Theorem 1.4. am,2n(EDol(M)) = 0if 2n<m ?{Td ∧ch ∧Θ}mif 2n=m. 1.8. The signature and spin complexes Let Mbe an oriented manifold of dimension 4k. Let d+δ:C∞(Λ±(M)) → C∞(Λ∓(M)) be the Hirzebruch signature complex. We then have dι+δι= d+δ+ (ext + int)(ι). Now (ext −int)(ι):Λ±→Λ∓but (ext + int)(ι) does not have this property if ι6= 0. So dι+διdoes not induce a map on the signature complex; it is not possible to deform the signature complex in this fashion. Similarly the spin complex can not be deformed in this fashion. The de Rham and Dolbeault complexes are Zgraded and this seems to be crucial in studying the Witten deformation; the signature and spin complexes, on the other hand, are Z2graded and this makes all the difference. For this reason, we shall not follow the approach of Atiyah, Bott, and Patodi [3] to study the Witten deformation of the Dolbeault complex by passing to the spin-c complex. Instead, we shall return to the original treatment of Gilkey [10] and apply invariance theory directly. 1.9. Brief guide to the paper In Section 2, we discuss product formulas for the heat trace asymptotics. In Section 3, we normalize the systems of coordinates and vector bundle frames to be considered up to arbitrarily high, but finite, order; this in effect reduces the structure group to the unitary group. In Section 4, we introduce the requisite spaces of invariants; the precise notion of what is meant by a “local invariant” or a “local formula” is crucial to our study. In Section 5, we discuss the restriction map. In Section 6, we use invariance under the action of the unitary group U(m) to establish certain technical results. In Section 7, we complete the proof of Theorem 1.4. 2. Product formulas The following observations are well known – see, for example, the discussion in Gilkey [11]. Let M=M1×M2where Miare closed manifolds of dimension Witten Deformation 7 mi. Let πi:M→Mibe projection on the ith factor. Let gibe Riemannian metrics on Miand let g=π∗ 1g1+π∗ 2g2be the associated Riemannian metric on M. Let (Ei, hi) be Hermitian vector bundles over Miand let E:= π∗ 1E1⊗ π∗ 2E2and h:= π∗ 1h1⊗π∗ 2h2define the associated vector bundle and Hermitian inner product over M. Let D=D1⊗id + id ⊗D2be an operator of Laplace type on C∞(E) over Mwhere Diare operators of Laplace type on C∞(Ei) over Mi. We have e−tD =e−tD1⊗e−tD2and the associated kernel function is given by K(t, (x1, x2),(y1, y2), D) = K(t, x1, y1, D1)⊗K(t, x2, y2, D2). We multiply the resulting asymptotic expansions for the heat kernels and equate coefficients of tto obtain corresponding local expressions am,2n(D)(x1, x2) = X n1+n2=n am1,2n1(D1)(x1)·am2,2n2(D2)(x2).(2.a) Let E1and E2be elliptic complexes of Dirac type over M1and M2, respectively. The elliptic complex of Dirac type E:= E1⊗E2over Mis defined by setting Ek=⊕i+j=kπ∗ 1(E1,i)⊗π∗ 2(E2,j), dk=⊕i+j=kd1,i ⊗id +(−1)iid ⊗d2,j ; the factor of (−1)iis present to ensure d2= 0. The associated operators of Laplace type then take the form Dk=⊕i+j=kD1,i ⊗id + id ⊗D2,j and consequently taking the super trace and applying Equation (2.a) yields am,2n(E)(x1, x2) = X n1+n2=n am1,2n1(E1)(x1)·am2,2n2(E2)(x2).(2.b) We now turn to the Dolbeault complex. Let M1= (M1, g1, J1, E1, h1, ω1) and M2= (M2, g2, J2, E2, h2, ω2) be given. Let M:= M1×M2, g := π∗ 1g1+π∗ 2g2, E := π∗ 1E1⊗π∗ 2E2, h := π∗ 1h1⊗π∗ 2h2 be as given above. Let ω:= π∗ 1ω1+π∗ 2ω2. We have that J:= π∗ 1J1⊕π∗ 2J2on T(M) = π∗ 1T(M1)⊕π∗ 2T(M2) is an integrable almost complex structure on M; the auxiliary bundle Eis then holomorphic. We set M=M1× M2= (M, g, J, E, h, ω) and obtain that EDol(M) = EDol(M1)⊗EDol(M2). Equation (2.b) then yields a corresponding decomposition of the local heat trace invariants am,2n(EDol(M)) = X n1+n2=n am1,2n1(EDol(M1)) am2,2n2(EDol(M2)) (2.c) 8 J. ´ Alvarez L´opez and P. Gilkey 3. Normalizing the coordinates and the local frame Let ~z = (z1,...zm) where zα=xα+√−1yαis a system of local holomorphic coordinates on M. Let ∂ ∂zα := 1 2∂ ∂xα−√−1∂ ∂yαand ∂ ∂¯zα := 1 2∂ ∂xα+√−1∂ ∂yα. We extend gto a symmetric bilinear form on the complex tangent bundle; the condition that gis Jinvariant, then yields g∂ ∂zα,∂ ∂zβ=g∂ ∂¯zα,∂ ∂¯zβ= 0 so gα¯ β:= g∂ ∂zα,∂ ∂¯zβ defines a positive definite Hermitian form. Introduce formal variables gα0¯ β0/α1...αj¯ β1... ¯ βk:= ∂ ∂zα1 . . . ∂ ∂zαj ∂ ∂¯zβ1 . . . ∂ ∂¯zβk g∂ ∂zα0 ,∂ ∂¯zβ0 for the holomorphic and anti-holomorphic derivatives of the components of g where there are no holomorphic derivatives if j= 0 and no anti-holomorphic derivatives if k= 0. We may express the K¨ahler form Ω(X, Y ) = g(X, JY ) as Ω = 1 2√−1 m X α0=1 m X β0=1 gα0¯ β0dzα0∧d¯zβ0. We say (M, g, J) is K¨ahler if dΩ = 0 and we impose this condition henceforth. This condition is equivalent to the symmetries: gα0¯ β0/α1=gα1¯ β0/α0and gα0¯ β0/¯ β1=gα0¯ β1/¯ β0. We can differentiate these relations to see the variables gα0¯ β0/α1...αj¯ β1... ¯ βkare symmetric in {α0. . . αj}and in {β0. . . βk}. Set g(α0...αj;¯ β0... ¯ βk):= gα0¯ β0/α1...αj¯ β1... ¯ βk.(3.a) If σ1and σ2are permutations of j+ 1 and k+ 1 indices, respectively, then g(α0...αj;¯ β0... ¯ βk)=g(ασ1(0)...ασ1(j);¯ βσ2(0)... ¯ βσ2(k)).(3.b) Similarly introduce formal variables h(p¯q;α1...αj;¯ β1... ¯ βk):= hp¯q/α1...αj¯ β1... ¯ βk(3.c) for the derivatives of the components of the Hermitian metric on E. If σ3and σ4are permutations of jand kindices, respectively, then h(p¯q;α1...αj;¯ β1... ¯ βk)=h(p¯q;ασ3(1)...ασ3(j);¯ βσ4(1)... ¯ βσ4(k)).(3.d) Since ∂ω = 0, ωα0/α1=ωα1/α0and similarly we set ω(α0...αj;¯ β1... ¯ βk)=ωα0/α1...αj¯ β1... ¯ βk, ¯ω(α1...αj;¯ β0... ¯ βk)= ¯ω¯ β0/α1...αj¯ β1... ¯ βk.(3.e) Witten Deformation 9 If σ5is a permutation of j+ 1 indices, σ6is a permutation of kindices, σ7is a permutation of jindices, and σ8is a permutation of k+ 1 indices, then ω(α0...αj;¯ β1... ¯ βk)=ω(ασ5(0)...ασ5(j);¯ βσ6(1)... ¯ βσ6(k)), ¯ω(α1...αj;¯ β0... ¯ βk)= ¯ω(ασ7(1)...ασ7(j);¯ βσ8(0)... ¯ βσ8(k)).(3.f) Lemma 2 of [10] yields the following result. Lemma 3.1. Let M= (M, g, J, ω, E, h). Fix a point z0∈Mand a positive integer N. There is a holomorphic coordinate system ~z = (z1, . . . , zm) centered at z0and a holomorphic frame ~e for Edefined near z0so that 1. gα¯ β(z0) = δαβ and hp¯q(z0) = δp,q. 2. g(α0...αj;¯ β0)(z0) = g(α0;¯ β0... ¯ βk)(z0) = 0 for 1≤j, k ≤N. 3. h(p¯q;α1...αj)(z0) = h(p¯q;¯ β1... ¯ βk)(z0)=0for 1≤j, k ≤N. In other words, these variables vanish at the basepoint if either there are no holomorphic or there are no anti-holomorpic derivatives. With these normalizations, the variables g(·;·),h(p¯q;·;·),ω(·;·), and ¯ω(·;·)are tensorial; we have reduced the structure group to the unitary groups U(m) and U(dim(E)) modulo transformations of order O(|z|N+1). 4. Spaces of invariants We must be rather precise in what is meant by a “local invariant” or a “local formula”. We do this as follows. 4.1. The algebra A We introduce the polynomial algebra Amin the variables of Equations (3.a), (3.c), and (3.e) and impose the relations of Equations (3.b), (3.d), (3.f), and Lemma 3.1: Am:= C[g(α0...αj1;¯ β0... ¯ βk1), h(p¯q;α1...αj2;¯ β1... ¯ βk2), ω(α0...αj3;¯ β1... ¯ βk3), ¯ω(α1...αj4;¯ β0... ¯ βk4)] for j1≥1, k1≥1, j2≥1, k2≥1, j3≥0, k3≥0, j4≥0, and k4≥0. If k3= 0, there are no anti-holomorphic derivatives of ωand if j4= 0, there are no anti-holomorphic derivatives of ¯ω. We introduce the complex dimension minto the notation as it plays an important role; we suppress the fiber dimension of Ein the interests of notational simplicity. If P∈Amand if Ais a monomial, we let c(A, P ) be the coefficient of A in Pand express P=PAc(A, P)A. We say Ais a monomial of Por that Aappears in Pif c(A, P)6= 0. The maps P→c(A, P) are linear maps from Amto C. 16 J. ´ Alvarez L´opez and P. Gilkey estimate m≥2n= weight(A)≥2a+ 2b+ 2c+ 2d+e+f ≥2a+ 2b+ 2c+ 2d+ 2 ≥2m=m . Thus all the inequalities are in fact equalities. This implies 2n=m,e=f= 1, and the remaining variables comprising Aall have weight 2. If a ¯ω∗variable of weight 2 does not contain a holomorphic index, Lemma 6.6 shows degν(A) = 0 for some holomorphic index which is false since r(A) = 0. Since Lemma 6.6 also holds for the anti-holomorphic indices, if any of the ω∗variables of weight 2 does not contain an anti-holomorphic index, then deg¯ν(A) = 0 for some anti-holomorphic index which is false. Thus all the variables of weight 2 which divide Abelong to Bmand Assertion (2) holds.  Let <(·) and =(·) be the real and imaginary parts of a 1-form. Lemma 7.2. If Mis a Riemann surface, then a2,2(M) = ?Todd2(M) + d=ω π. Proof. ´ Alvarez L´opez and Gilkey [1] showed that a2,2(M) = τ 8π−1 πδ(<(ω)). We have τ 8π=?Todd2(M) by Hirzebruch [15]. Let ω= (u+√−1v)(dx+√−1dy)=(udx −vdy)+√−1(udy +vdx); −δ(<(ω)) = −δ(udx −vdy) = ux−vy, and d(=(ω)) = d(udy +vdx)=(ux−vy)dx ∧dy. We must improve Lemma 6.6. Lemma 7.3. Let 06=P∈ker(r)∩Pm,m. There exists a monomial Aof P so A=g(11; ¯ β1¯ β2). . . g(aa;¯ β2a−1¯ β2a)h(pa+1 ¯qa+1;a+1;a+1) . . . h(pb¯qb;a+b;a+b) ω(a+b+1;a+b+1) . . . ω(a+b+c;a+b+c)¯ω(a+b+c+1;a+b+c+1) . . . ¯ω(a+b+c+d;a+b+c+d)Ξ where Ξ=1or Ξ=Ξ(m;¯ m)and βν≤afor 1≤ν≤2a . Proof. Apply Lemma 6.6 and Lemma 7.1 to choose a monomial Aof Pof the form A=g(11; ¯ β1¯ β2). . . g(aa;¯ β2a−1¯ β2a)h(pa+1 ¯qa+1;a+1; ¯ βa+1). . . h(pb¯qb;a+b;¯ βa+b) ω(a+b+1; ¯ βa+b+1). . . ω(a+b+c;¯ βa+b+c)¯ω(a+b+c+1; ¯ βa+b+c+1) . . . ¯ω(a+b+c+d;¯ βa+b+c+d)Ξ where Ξ = 1 or Ξ(m;¯ βm). (7.a) We use Lemma 6.1 and Equation (7.a) to see a<β ⇔degβ(A)=1 ⇔deg¯ β(A) = 1 . We say that an anti-holomorphic index ¯ βtouches an anti-holomorphic index ¯γin Aif Ais divisible by g(αα;¯ β¯γ)for some α; we say that ¯ βtouches itself in Aif we can take ¯ β= ¯γ. Witten Deformation 17 Let a<βso degβ(A) = deg¯ β(A) = 1. Let γ6=β. We construct a monomial Bby replacing ¯ βby ¯γ; deg¯ βB= 0 and Ais obtained from Bby changing ¯γ→¯ β. We apply Lemma 6.5 to find A1∈ B(B) which appears in Pwith A6=A1. Since deg¯ βB= 0 and deg¯ β(A1)6= 0, A1is obtained from Bby changing ¯γ→¯ βor, equivalently, A1arises from Aby interchanging a ¯ βwith a ¯γindex. Thus, in particular, since A16=A, two anti-holomorphic indices of degree 1 in Acan not touch in A. Choose Aof the form given in Equation (7.a) so the number of antiholomorphic indices which touch themselves in Ais maximal. Suppose deg¯ β(A) = 1 and ¯ βtouches another anti-holomorphic index ¯γin A. Then there is a monomial A1of Pdifferent from Adefined by interchanging ¯ βand ¯γ. This is not possible since ¯γwould touch itself in A1which contradicts the maximality of A. Thus {¯ V1,..., ¯ Va}consists solely of anti-holomorphic indices of degree 2 in Aand hence must comprise all the anti-holomorphic indices of degree 2 in A. We say that a holomorphic index αtouches an anti-holomorphic index ¯ βin Aif Ais divisible by ω(α;¯ β), by ¯ω(α;¯ β)or by Ξ(α;¯ β). If Ais as constructed above, then every holomorphic index αof degree 1 touches an anti-holomorphic index ¯ βof degree 1 in A. Among all the monomials Aconstructed above, choose Aso the number of holomorphic indices αwhich touch ¯αin Ais maximal. Let a+b < α. If αtouches ¯ βin Awith α6=β, we can interchange ¯ βand ¯αto construct a monomial A1of Pwhere there is one more holomorphic - anti-holomorphic touching which is impossible. Therefore A has the form given in the Lemma.  We use Lemma 7.1 to improve Equation (2.c). Let M=M1× M2. Since ami,2ni= 0 for 2ni< mi, taking 2n=min Equation (2.c) yields am,m(EDol(M))(z1, z2) = am1,m1(EDol(M1))(z1)·am2,m2(EDol(M2))(z2). (7.b) Let Nk(0) = (N2k, JN, gN, EN, hN,0) be a structure of complex dimension kwith trivial twisting (1,0)–form ω0= 0. Let T2(ωi)=(S1×S1, dx2+dy2, J2,1, h0, ωi) be the torus S1×S1with the flat metric dx2+dy2, usual complex structure J2:∂ ∂x →∂ ∂y and J2:∂ ∂y → − ∂ ∂x , flat line bundle 1= (S1×S1)×C, trivial Hermitian metric h0, and (possibly) non-trivial twisting (1,0)-form ωiwith ∂ωi= 0. Let ~ω = (ω1, . . . , ωm−k). Set M(k;~ω) := Nk(0) ×T(ω1)×···×T(ωm−k). By Equation (7.b), am,m(EDol(M(k;~ω))) =a2k,2k(EDol(Nk))a2,2(EDol(T2(ω1))) . . . a2,2(EDol(T2(ωm−k))) . By Lemma 7.3, if 0 6=P∈ker(r)∩Pm,m, then P(M(k;~ω)) 6= 0 for some k and some ~ω. On the other hand, we may use Equation (7.b), Theorem 1.3, 18 J. ´ Alvarez L´opez and P. Gilkey and Lemma 7.2 to see that {am,m −{Td ∧ch ∧Θ}m}(M(k;~ω)) = 0 for all kand ~ω . Theorem 1.4 now follows.  7.1. The kernel of r It seems useful to identify ker(r:Pm,2m→ Pm−1,2m) in a bit more detail. Let chkbe the kth component of the Chern character (see [15]). We decompose the graded ring of characteristic forms into homogeneous components Cm:= C[chk(T M, J, g),chk(E, h)] = ⊕kC2k m. We also consider the graded ring Dm:= C[chk(T M, J, g),chk(E, h), dω, d¯ω, ω, ¯ω] = ⊕kD2k m. Let Pg,E m,2mbe the subspace of invariants which are independent of ω. Lemma 7.4. 1. ker(r:Pg,E m,m → Pg,E m−1,m) = ?Cm m. 2. ker(r:Pm,m → Pm−1,m) = ?Dm m. Proof. Assertion (1) follows from Theorem 1 of Gilkey [10]. Assertion (2) is a scholium to the arguments we have given above. We use Assertion (1) to control the metric terms. We express P∈ker(r) on M(k;~ω) as a metric invariant on Mktimes invariants on the T2(ωi). The metric invariant is itself in the kernel of rand hence is a characteristic class; the remaining invariants only involve Θ. Thus there is an element of Dwhich hits this invariant and hence by Lemma 7.3, Pcan be decomposed appropriately.  Conflict of interest On behalf of all authors, the corresponding author states that there is no conflict of interest. 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