The odd Chern character and index localization formulae (Q413397)

Let \(H\) be a complex separable Hilbert space. A continuous family of unbounded self-adjoint Fredholm operators \(F=(F_m)_{m \in M}\) on \(H\) defines an element of \(K^1(M)\). In the present paper the author derives formulas for the Chern character of such an element (and their Poincaré dual if \(M\) is a closed oriented manifold). The formulas involve Schubert varieties constructed using a description of the classifying space of \(K^1\) in terms of Lagrangians: Let \(\hat H= H \oplus H\) be endowed with the standard skewhermitian structure and let \({\mathcal Lag}\) be the Grassmannian of Lagrangian subspaces of \(\hat H\). The subset \({\mathcal Lag}^-\) of those Lagrangian subspaces which form a Fredholm pair with the space \(H^-=0 \oplus H\) is a classifying space of \(K^1\). The author studies in detail this space and its relation to other models. Then he fixes a complete decreasing flag \(H^- \supset W_1 \supset W_2 \dots \) and studies the Schulbert varieties \(\overline Z_{i+1}=\{L\in {\mathcal Lag}^- ~|~ \dim L \cap W_i\geq 1\}\). In particular he shows that \(\overline Z_{i+1}\) defines a cohomology class \([\overline Z_{i+1}]\in H^{2i+1}({\mathcal Lag}^-)\). Now one of the main results of this paper is the following formula. Let \(G:M \to {\mathcal Lag}^-\) be the map associating to \(m\) the graph of \(F_m\). Then NEWLINE\[NEWLINE\mathrm{ch}_{2k-1}[F]=\frac{(-1)^{k-1}}{(k-1)!} G^*[\overline Z_k] \;.NEWLINE\]