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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].

##### 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