The Atiyah-Bott-Lefschetz fixed-point theorem in symplectic geometry.

*(English. Russian original)*Zbl 0903.58052
Dokl. Math. 53, No. 3, 358-361 (1996); translation from Dokl. Akad. Nauk, Ross. Akad Nauk 348, No. 2, 165-168 (1996).

Let \(M\) be a smooth compact manifold without boundary and \(g:T^*M\rightarrow T^*M\) a symplectic transformation that satisfies some nondegeneracy condition in terms of the components of the fixed point set of \(g\). Assume that \(E_1\rightarrow M\), \(E_2\rightarrow M\) are vector bundles over \(M\) and \(\widehat{a}:L_2(M,E_1)\rightarrow L_2(M,E_2)\) is an elliptic \(1/h\)-pseudodifferential operator with a symbol from the class \(\Sigma^{m,0}\) that acts in sections of those bundles. Classes \(\Sigma^{m,0}\) are modification of the classes \(\Sigma_{1,0}^{m,n}\) which were introduced by M. A. Shubin [‘Pseudodifferential operators and spectral theory’ (Moscow, Nauka) (1978; Zbl 0451.47064)] for the case of compact manifolds with boundary. Let \(T(g,\phi)\) denote the Fourier integral operator that corresponds to the symplectic transformation \(g\) with symbol \(\phi\). Assume that \(\widehat{U}_1=T(g,\phi_1)\), \(\widehat{U}_2=T(g,\phi_2)\) define an endomorphism of this complex. The Lefschetz number is defined as
\[
{\mathcal{L}}={\mathcal{L}}(\widehat{a},\widehat{U}_1, \widehat{U}_2)= \text{Tr}\{\widehat{U}_1\mid_{Ker\;\widehat{a}}\}- \text{Tr}\{\widehat{U}_2\mid_{\text{Coker }\widehat{a}}\}.
\]
The following generalization of the Atiyah-Bott-Lefschetz fixed-point theorem in the context of symplectic geometry is given.

Theorem. Assume that \(\widehat{a}\), \(\widehat{U}_1\), and \(\widehat{U}_2\) satisfy the above conditions. Then the formula \[ {\mathcal{L}}={\mathcal{L}}(h) =\sum_h\exp\{{{i}\over{h}}S_k\}\text{Tr} \int_{F_k}{{(\phi_1-\phi_2)\;dm}\over{\sqrt{\Pi_j\lambda_j}}} \pmod {O(h)} \] is valid, where \(S_k\) is the volume of the nonsingular action on the Lagrangian manifold \(L=\text{graph }g\) at points of \(F_k\), \(\lambda_j\) are the nonzero eigenvalues of the operator \(1-g_*\). The arguments of the eigenvalues \(\lambda_j\) are chosen in the interval \((-3\pi/2,\pi/2)\). The proof is based on the asymptotic behaviour of the trace of the Fourier \(1/h\)-integral operator for \(h\rightarrow 0\). In the paper, the general formula is given in terms of fixed points of the corresponding symplectic transformation \(g\) in the situation when the set of fixed points of \(g\) is a disjoint union of a finite number of compact smooth submanifolds without boundary, which may have various dimensions. Thus for a symplectic transformation defined by a Hamiltonian vector field, it is possible to consider not only fixed points defined by zeros of the corresponding Hamiltonian vector field, but also the fixed points that are defined by its closed trajectories.

Theorem. Assume that \(\widehat{a}\), \(\widehat{U}_1\), and \(\widehat{U}_2\) satisfy the above conditions. Then the formula \[ {\mathcal{L}}={\mathcal{L}}(h) =\sum_h\exp\{{{i}\over{h}}S_k\}\text{Tr} \int_{F_k}{{(\phi_1-\phi_2)\;dm}\over{\sqrt{\Pi_j\lambda_j}}} \pmod {O(h)} \] is valid, where \(S_k\) is the volume of the nonsingular action on the Lagrangian manifold \(L=\text{graph }g\) at points of \(F_k\), \(\lambda_j\) are the nonzero eigenvalues of the operator \(1-g_*\). The arguments of the eigenvalues \(\lambda_j\) are chosen in the interval \((-3\pi/2,\pi/2)\). The proof is based on the asymptotic behaviour of the trace of the Fourier \(1/h\)-integral operator for \(h\rightarrow 0\). In the paper, the general formula is given in terms of fixed points of the corresponding symplectic transformation \(g\) in the situation when the set of fixed points of \(g\) is a disjoint union of a finite number of compact smooth submanifolds without boundary, which may have various dimensions. Thus for a symplectic transformation defined by a Hamiltonian vector field, it is possible to consider not only fixed points defined by zeros of the corresponding Hamiltonian vector field, but also the fixed points that are defined by its closed trajectories.

Reviewer: N.Blažić (Beograd)

##### MSC:

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