Lefschetz theory on manifolds with singularities.

*(English)*Zbl 1118.58010
Bojarski, Bogdan (ed.) et al., \(C^*\)-algebras and elliptic theory. Basically formed by contributions from the international conference, Bȩdlewo, Poland, February 2004. In cooperation with Dan Burghelea, Richard Melrose and Victor Nistor. Basel: Birkhäuser (ISBN 3-7643-7686-4/hbk). Trends in Mathematics, 157-186 (2006).

Generalizations of the Lefschetz fixed point theorem of elliptic complexes due to Atiyah-Bott for manifolds with singularities obtained by the authors and their collaboraters are reviewed. Let

\[ 0\to E_0@> D_0>> E_1@> D_1>>\cdots@> D_{m-1}>> E_m\to 0 \] be a complex of vector spaces over \(\mathbb{C}\) with finite-dimensional cohomology, and let

\[ T= \{T_j: E_j\to E_j\}|_{j= 0,\dots, m} \] be an endomorphism of this complex. Then \(T_j\) induces mappings

\[ \widetilde T_j: H^j(E)\to H^j(E),\quad j= 0,\dots, m. \] The Lefschetz number of \(T\) is defined by \({\mathcal L}= \sum^m_{j=0} (-1)^j\text{Trace\,}\widetilde T_j\).

Atiyah-Bott treated the case for each \(E_i= C^\infty(M, F_i)\), \(F_i\) is a vector bundle over a compact smooth manifold \(M\), \(D_i\) is an elliptic differential operator and \(T_i\) is geometric, that is \(T_i\phi(x)= A_i(x)\phi(f(x))\), \(f: M\to M\) is a smooth mapping. The authors say the class of geometric endomorphisms is not a natural framework for the problem of Lefschetz number if \(D_i\) is a pseudodifferential operator. In this case, the class of geometric endomorphisms should be replaced with the class of Fourier integral operators. Then the theory necessarily becomes asymptotic. To obtain meaningful formulas, a small parameter \(h\in(0, 1]\) is introduced and semiclassical (or \(1/h\)) pseudodifferential operators and the Fourier-Maslov integral operator associated with a canonical transformation \(g: T^*M\to T^*M\) are considered. The Lefschetz number depends on \(h\), and under appropriate assumptions about fixed points of \(g\), an expression for the asymptotics of the Lefschetz number as \(h\to 0\) is obtained by applying the stationary phase method to the trace integrals representing this number.

According to this strategy, first the trace of the Fourier-Maslov integral operator \(T(g,\phi)\) with amplitude \(\phi\) associated with the graph of the transformation \(g: T^*M\to T^*M\) is computed assuming \(g\) has finite nondegenerate fixed points \(\alpha_1,\dots, \alpha_N\in T^*M\) (Th. 1.1. V. E. Shatalov and B. Yu. Sternin, Funct. Anal. Appl. 32, No. 4, 247–257 (1998); translation from Funkts. Anal. Prilozh. 32, No. 4, 35–48 (1998; Zbl 0948.58018)], hereafter referred as [1]). Then the Lefschetz number \({\mathcal L}\) is computed. If \(m= 2\), the formula is

\[ {\mathcal L}= \sum^N_{k=1} \exp\Biggl({i\over h} S_k\Biggr) {\text{Trace\,}\phi_1(\alpha_k)- \text{Trace\,}\phi_2(\alpha_k)\over \sqrt{\text{det}(1- g_*(\alpha_k))}}+ O(h), \] where \(\phi_i\) is the amplitude of the endomorphism \(\widehat T_i\), \(i= 1, 2\) and \(S_k\) is the value of the generating function \(g\) at \(\alpha_k\) (Th.1.3, [1]). The case where the set of fixed points is not discrete but a finite union of smooth connected submanifolds \(G_k\), and \(g\) is non-degenerate on \(G_k\) is also treated and \({\mathcal L}\) is expressed by the sum of integrations on \(G_k\) (Th.1.4, [1]). It suggests the underlying geometry of \({\mathcal L}\) is symplectic rather than contact. Explicit computation of the measure on \(G_k\) appear in the asymptotic formula of \({\mathcal L}\) is difficult. But for some extreme cases it is computed (Th.1.5, Th.1.7. of [V. Nazaikinskii, Russ. J. Math. Phys. 6, No. 2, 202–213 (1999; Zbl 1059.58503)]).

Following these arguments, the Atiyah-Bott-Lefschetz theorem for geometric endomorphisms on manifolds with conical singularities and for endomorphisms given by quantized canonical transformations are presented in §2 and §3. In this paper, in both cases \(m\) is assumed to be 2.

In §2, first ellipticity of a differential operator \(\widehat D\) on a manifold \(M\) having \(N\) conical singular points is defined in terms of its symbol and conormal symbol \(\widehat D_0(\alpha_j, p)\), \(\alpha_j\) is a conical point of \(M\). Then equipping \(M\) with metric and measure

\[ d\rho^2+ r^2 d\omega^2,\quad d\mu(x)= {dr\over r}\wedge d\omega, \] the Lefschetz number is shown to be a sum of contributions \({\mathcal L}_{\text{int}}(x^*)\) of interior fixed points \(x^*\) and the contributions \({\mathcal L}_{\text{sing}}(\alpha^*)\) of conical fixed points. \(\sum_{x^*}{\mathcal L}_{\text{int}}(x^*)\) is expressed by the same fixed point formula of Atiyah-Bott. While \(\sum_{\alpha^*}{\mathcal L}_{\text{sing}}(\alpha^*)\) is computed by the localization technique (Th. 2.3). The conormal symbol of \(\widehat B(p)\) is said to be of power type if the orders of poles of \(\widehat B^{-1}(p)\) are boundary and satify some bounded conditions (Def. 2.4. cf. [M. A. Shubin, Pseudodifferential operators and spectral theory. Transl. from the Russian by Stig I. Andersson. Berlin etc.: Springer-Verlag (1987; Zbl 0616.47040)]). If \(\widehat D_0(\alpha^*, p)\) is of power type at some \(\alpha^*\), \({\mathcal L}_{\text{sing}}(\alpha^*)\) is expressed in terms of residue of \(\widehat D^{-1}_0(\alpha^*, p)\) (Th.2.6). As an example,

\[ \widehat D= \Biggl(r{\partial\over\partial r}\Biggr)^2+ {\partial^2\over\partial\omega^2},\quad g(r,\omega)= (\lambda r,\omega), \] on a two-dimensional manifold \(M\) with a conical singularity expressed locally by \((r,\omega)\), \(r\in[0, 1)\), \(\omega\in S^1\) and \(\widehat Tu(x)= u(g(x))\), is considered and it is shown \[ {\mathcal L}_{\text{sing}}= \sum_{k>\gamma} \text{Trace\,Res}_{p= ik}{2\lambda^{-ip}p\over p^2-{\partial^2\over\partial\omega^2}}= \sum_{k>\gamma} a_k\lambda^k, \] where \(\gamma\) is the weight exponent at the conical point and \(a_k= 2\), \(k\neq 0\) and \(a_0= 1\) (§2.5).

In §3, after explaining semiclassical pseudodifferential operators and quantization of canonical transformations [cf. B. W. Nazaikinskii, B. Schulze, B. Sternin and V. Shatalov, A Lefshetz fixed point theorem on manifolds with conical singularities, quantization of symplectic transformations on manifolds with conical singularities, Preprint No.97/20 and 97/23, Univ. Potsdam, Institut für Mathematik, Potsdam (1997), see also Zbl 0981.58016] the asymptotic formula of the Lefschetz number \({\mathcal L}(h)\) is shown to be the sum of contributions of interior fixed points and the sum of conical fixed points and areminder term which is \(O(h)\). It is also shown that the Lefschetz number at a conical fixed point \(\alpha^*\) is expressed by the residue of the conormal symbol (Th.3.11. cf. Th.2.6).

For the entire collection see [Zbl 1097.58001].

\[ 0\to E_0@> D_0>> E_1@> D_1>>\cdots@> D_{m-1}>> E_m\to 0 \] be a complex of vector spaces over \(\mathbb{C}\) with finite-dimensional cohomology, and let

\[ T= \{T_j: E_j\to E_j\}|_{j= 0,\dots, m} \] be an endomorphism of this complex. Then \(T_j\) induces mappings

\[ \widetilde T_j: H^j(E)\to H^j(E),\quad j= 0,\dots, m. \] The Lefschetz number of \(T\) is defined by \({\mathcal L}= \sum^m_{j=0} (-1)^j\text{Trace\,}\widetilde T_j\).

Atiyah-Bott treated the case for each \(E_i= C^\infty(M, F_i)\), \(F_i\) is a vector bundle over a compact smooth manifold \(M\), \(D_i\) is an elliptic differential operator and \(T_i\) is geometric, that is \(T_i\phi(x)= A_i(x)\phi(f(x))\), \(f: M\to M\) is a smooth mapping. The authors say the class of geometric endomorphisms is not a natural framework for the problem of Lefschetz number if \(D_i\) is a pseudodifferential operator. In this case, the class of geometric endomorphisms should be replaced with the class of Fourier integral operators. Then the theory necessarily becomes asymptotic. To obtain meaningful formulas, a small parameter \(h\in(0, 1]\) is introduced and semiclassical (or \(1/h\)) pseudodifferential operators and the Fourier-Maslov integral operator associated with a canonical transformation \(g: T^*M\to T^*M\) are considered. The Lefschetz number depends on \(h\), and under appropriate assumptions about fixed points of \(g\), an expression for the asymptotics of the Lefschetz number as \(h\to 0\) is obtained by applying the stationary phase method to the trace integrals representing this number.

According to this strategy, first the trace of the Fourier-Maslov integral operator \(T(g,\phi)\) with amplitude \(\phi\) associated with the graph of the transformation \(g: T^*M\to T^*M\) is computed assuming \(g\) has finite nondegenerate fixed points \(\alpha_1,\dots, \alpha_N\in T^*M\) (Th. 1.1. V. E. Shatalov and B. Yu. Sternin, Funct. Anal. Appl. 32, No. 4, 247–257 (1998); translation from Funkts. Anal. Prilozh. 32, No. 4, 35–48 (1998; Zbl 0948.58018)], hereafter referred as [1]). Then the Lefschetz number \({\mathcal L}\) is computed. If \(m= 2\), the formula is

\[ {\mathcal L}= \sum^N_{k=1} \exp\Biggl({i\over h} S_k\Biggr) {\text{Trace\,}\phi_1(\alpha_k)- \text{Trace\,}\phi_2(\alpha_k)\over \sqrt{\text{det}(1- g_*(\alpha_k))}}+ O(h), \] where \(\phi_i\) is the amplitude of the endomorphism \(\widehat T_i\), \(i= 1, 2\) and \(S_k\) is the value of the generating function \(g\) at \(\alpha_k\) (Th.1.3, [1]). The case where the set of fixed points is not discrete but a finite union of smooth connected submanifolds \(G_k\), and \(g\) is non-degenerate on \(G_k\) is also treated and \({\mathcal L}\) is expressed by the sum of integrations on \(G_k\) (Th.1.4, [1]). It suggests the underlying geometry of \({\mathcal L}\) is symplectic rather than contact. Explicit computation of the measure on \(G_k\) appear in the asymptotic formula of \({\mathcal L}\) is difficult. But for some extreme cases it is computed (Th.1.5, Th.1.7. of [V. Nazaikinskii, Russ. J. Math. Phys. 6, No. 2, 202–213 (1999; Zbl 1059.58503)]).

Following these arguments, the Atiyah-Bott-Lefschetz theorem for geometric endomorphisms on manifolds with conical singularities and for endomorphisms given by quantized canonical transformations are presented in §2 and §3. In this paper, in both cases \(m\) is assumed to be 2.

In §2, first ellipticity of a differential operator \(\widehat D\) on a manifold \(M\) having \(N\) conical singular points is defined in terms of its symbol and conormal symbol \(\widehat D_0(\alpha_j, p)\), \(\alpha_j\) is a conical point of \(M\). Then equipping \(M\) with metric and measure

\[ d\rho^2+ r^2 d\omega^2,\quad d\mu(x)= {dr\over r}\wedge d\omega, \] the Lefschetz number is shown to be a sum of contributions \({\mathcal L}_{\text{int}}(x^*)\) of interior fixed points \(x^*\) and the contributions \({\mathcal L}_{\text{sing}}(\alpha^*)\) of conical fixed points. \(\sum_{x^*}{\mathcal L}_{\text{int}}(x^*)\) is expressed by the same fixed point formula of Atiyah-Bott. While \(\sum_{\alpha^*}{\mathcal L}_{\text{sing}}(\alpha^*)\) is computed by the localization technique (Th. 2.3). The conormal symbol of \(\widehat B(p)\) is said to be of power type if the orders of poles of \(\widehat B^{-1}(p)\) are boundary and satify some bounded conditions (Def. 2.4. cf. [M. A. Shubin, Pseudodifferential operators and spectral theory. Transl. from the Russian by Stig I. Andersson. Berlin etc.: Springer-Verlag (1987; Zbl 0616.47040)]). If \(\widehat D_0(\alpha^*, p)\) is of power type at some \(\alpha^*\), \({\mathcal L}_{\text{sing}}(\alpha^*)\) is expressed in terms of residue of \(\widehat D^{-1}_0(\alpha^*, p)\) (Th.2.6). As an example,

\[ \widehat D= \Biggl(r{\partial\over\partial r}\Biggr)^2+ {\partial^2\over\partial\omega^2},\quad g(r,\omega)= (\lambda r,\omega), \] on a two-dimensional manifold \(M\) with a conical singularity expressed locally by \((r,\omega)\), \(r\in[0, 1)\), \(\omega\in S^1\) and \(\widehat Tu(x)= u(g(x))\), is considered and it is shown \[ {\mathcal L}_{\text{sing}}= \sum_{k>\gamma} \text{Trace\,Res}_{p= ik}{2\lambda^{-ip}p\over p^2-{\partial^2\over\partial\omega^2}}= \sum_{k>\gamma} a_k\lambda^k, \] where \(\gamma\) is the weight exponent at the conical point and \(a_k= 2\), \(k\neq 0\) and \(a_0= 1\) (§2.5).

In §3, after explaining semiclassical pseudodifferential operators and quantization of canonical transformations [cf. B. W. Nazaikinskii, B. Schulze, B. Sternin and V. Shatalov, A Lefshetz fixed point theorem on manifolds with conical singularities, quantization of symplectic transformations on manifolds with conical singularities, Preprint No.97/20 and 97/23, Univ. Potsdam, Institut für Mathematik, Potsdam (1997), see also Zbl 0981.58016] the asymptotic formula of the Lefschetz number \({\mathcal L}(h)\) is shown to be the sum of contributions of interior fixed points and the sum of conical fixed points and areminder term which is \(O(h)\). It is also shown that the Lefschetz number at a conical fixed point \(\alpha^*\) is expressed by the residue of the conormal symbol (Th.3.11. cf. Th.2.6).

For the entire collection see [Zbl 1097.58001].

Reviewer: Akira Asada (Takarazuka)

##### MSC:

58J10 | Differential complexes |

58J40 | Pseudodifferential and Fourier integral operators on manifolds |

58J20 | Index theory and related fixed-point theorems on manifolds |

32S50 | Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants |

##### Keywords:

Lefschetz number; singular manifold; elliptic operator; Fourier integral operator; semiclassical method
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\textit{V. Nazaikinskii} and \textit{B. Sternin}, in: \(C^*\)-algebras and elliptic theory. Basically formed by contributions from the international conference, Bȩdlewo, Poland, February 2004. In cooperation with Dan Burghelea, Richard Melrose and Victor Nistor. Basel: Birkhäuser. 157--186 (2006; Zbl 1118.58010)