The heat kernel Lefschetz fixed point formula for the spin-c dirac operator
Let \(M\) be a compact \(2n\)-dimensional \(C^\infty\) manifold provided with an almost complex structure \(J\), and let \(\pi_L : L\to M\) be a smooth complex vector bundle over \(M\) of (complex) fiber dimension \(l\). In the complex analytic case (i.e., when \(M\) and \(L\) are holomorphic and \(n= \dim_\mathbb{C} M)\), one defines the Riemann-Roch number \(RR (M,L): = \sum^n_{g=0} (-1)^q \dim_\mathbb{C} H^q (M,{\mathcal O} (L))\), where \({\mathcal O}(L)\) is the sheaf of holomorphic sections of \(L\) over \(M\), and, more generally, one defines the holomorphic Lefschetz number of \(\gamma\), a complex analytic automorphism of \(L\) (so it acts on \({\mathcal O} (L))\), by \(\chi_{M,L} (\gamma): = \sum^n_{q=0} (-1)^q \text{trace}_\mathbb{C} (\gamma |H^q (M, {\mathcal O} (L)))\). Hence, \(\chi_{M,L} \text{(id)} = RR(M,L)\). In general, we suppose \(M\) endowed with a Hermitian metric (one has also to fix a Hermitian connection on \(K^*\), the dual of the canonical bundle of \(M\), and on the bundle \(L)\). Denote by \(E^q: = \Lambda^{(0,q)} (\mathbb{C} \otimes T^*M)\) the bundle of \((0,q)\)-forms on \(M\), and by \(\Gamma (M,E^q \otimes L)\) the smooth sections of the latter bundle tensored with \(L\). One constructs an operator (the Dirac operator) \(D: \Gamma (M,E^q \otimes L)\to \Gamma (M,(E^{q+1} \oplus E^{q-1}) \otimes L)\). Write \(E^+\) (resp. \(E^-)\) for the sum of the \(E^q\) over even \(q\)'s (resp. over odd \(q\)'s), and define \(D^\pm: =D |_{\Gamma (M,E^\pm \otimes L)}: \Gamma (M,E^\pm \otimes L) \to\Gamma (M,E^\pm \otimes L)\). Define the virtual character by \(\chi (\gamma) = \text{trace}_\mathbb{C} \gamma |\text{Ker} D^+-\text{trace}_\mathbb{C} \gamma |\text{Ker} D^-\). Notice that \(\chi (\text{id}) = \text{index} D^+ = \dim_\mathbb{C} \text{Ker} D^+ - \dim_\mathbb{C} \text{Ker} (D^+)^*\). Moreover, define \(Q^\pm: = D^\mp \circ D^\pm : \Gamma (M,E^\pm \otimes L) \to \Gamma (M,E^\pm \otimes L)\), and \(Q=Q^+ \oplus Q^-\). Consider (it is the Berline-Getzler-Vergne method) \(\exp (-tQ)\) and denote by \(K(t,x,y)\) the kernel of \(\exp (-tQ)\), for \(t>0\), \(x,y \in M\) (the heat-kernel), and by \(K_\pm (t,x,y)\) the restrictions of \(K(t,x,y)\) to \(E^\pm \otimes L\). Then one has that \(K(t,x,y)\) admits an asymptotic expansion along the diagonal of \(M\) as \(t\to 0+\) and that \((\text{trace}_\mathbb{C} K_+ (t,x,x)- \text{trace}_\mathbb{C} K_- (t,x,x))dx\) converges, as \(t \to 0+\), to a universal polynomial in the geometric data at \(x\) (namely, characteristic classes). Then index \(D^+ = \int_M \lim_{t\to 0+} (\text{trace}_\mathbb{C} K_+(t,x,x) - \text{trace}_\mathbb{C} K_-(t,x,x)) dx\), and, in the complex analytic case, index \(D^+ = RR(M,L)\) (the Hirzebruch-Riemann-Roch formula). When \(\gamma\) is a transformation in \(M\) which is covered by an automorphism \(\gamma_L\) of \(L\) (i.e. \(\pi_L\circ \gamma = \gamma \circ \pi_L) \) which leaves all the given structures invariant, \(\gamma\) induces an action \(\gamma_{op}\) on sections of \(E \otimes L\) then. If \(K^\pm_\gamma\) are the restrictions of the heat-kernel \(K_\gamma\) of \(\gamma_{op} \circ \exp (-tQ)\) to \(E^\pm \otimes L\), and assuming that the fixed-point set \(M^\gamma\) of \(\gamma\) in \(M\) is a smooth (almost complex) submanifold of \(M\) and the action of \(\gamma_L\) on the normal bundle of \(M^\gamma\) is nondegenerate, one has that \(K^\pm_\gamma (t,x,x)\) is rapidly decreasing as \(t\to 0+\) unless \(x\) is a fixed point of \(\gamma\). One then relates the virtual character with the sum of the integrals over the connected components \(\{F_j\}^N_1\) of \(M^\gamma\) of the limit as \(t\to 0+\) of integrals over the diagonal in \(N^j_x \times N^j_x\) of \((\text{trace}_\mathbb{C} K^+_\gamma (t,x,x)- \text{trace}_\mathbb{C} K^-_\gamma (t,x,x))d_{F_j}x\), where \(N^j_x\) is the orthogonal complement of \(T_xF_j\) in \(T_xM\). In the complex analytic case, the virtual character is in turn equal to the holomorphic Lefschetz number, and one gets the holomorphic Lefschetz-fixed-point formula. Because of the fact that (writing \(\sigma (P)\) for the principal symbol of \(P\)) \(\sigma (D^2)(x,\xi) = (\sigma(D) (x,\xi))^2 = g_x(\xi,\xi)I\) (\(g^{-1}\) is the Riemannian metric on \(M\) induced by the Hermitian one), the \(SO(2n)\)-invariance and the \(J\)-invariance, the Dirac operator \(D\) above is associated with a Clifford-algebra bundle over \(M\) (which we denote by \(C\ell (T^* M))\), relative to \(g\) and actually with the complexification of it, \(\mathbb{C} \otimes C \ell(T^*M)\). In fact, in order to take into account the double covering \(\text{Spin} (2n) \to SO(2n)\), and the fact that the elements of \(U(n)\) are defined to be the elements of \(SO(2n)\) which commute with the almost complex structure \(J\), it is necessary to find a group which contains \(U(n)\) and is a double-covering of \(SO(2n) \times U(1)\). The sought for group is \(\text{Spin}^c (2n)\) (it turns out that \(\text{Spin} (2n) \times U(1)/ \{\pm 1\} @>\sim>> \text{Spin}^c (2n))\) and \(\mathbb{C} \otimes C\ell (T^*_x M)\) naturally includes the \(\text{Spin}^c (2n)\) group and the unitary group \(U(n)\). The Dirac operator \(D\) attached to this structure has principal symbol equal to the one of the Dolbeault-Dirac operator \(2(\overline \partial + \overline \partial^*)\). The author's book is a marvelous introduction to such objects and theories. The first 6 chapters of the book are devoted to the detailed construction of the Dirac operator on Clifford bundles. Chapters 7 to 10 concern the asymptotics of the heat kernels considered, while in chapters 11-12-13 one finds the applications to Lefschetz formulas. Chapter 14 contains applications to orbifolds, and chapter 15 applications to symplectic geometry. Chapter 16 is an appendix where one can find all the necessary tools relevant to the equivariant constructions carried out in the main body of the book. In his sharp and clear expository style, the author shows in this book all the beauty of this field of research, linking together geometry, topology, analysis and representation theory. It should be recommended to any student or researcher willing to broaden his/her view in mathematics. Further developments of the theory are the recent Seiberg-Witten invariants of four-manifolds and the very recent results by C. Taubes on the relation between Seiberg-Witten invariants and Gromov invariants of symplectic four-manifolds. The author's book provides, in my opinion, a fundamental background for these matters.
scientific article; zbMATH DE number 833389