Abstract
In this paper we extend the results obtained in [9], [10] to manifolds with $\mathrm{Spin}_{\mathbb{C}}$-structures defined, near the boundary, by an almost complex structure. We show that on such a manifold with a strictly pseudoconvex boundary, there are modified $\bar{\partial}$-Neumann boundary conditions defined by projection operators, $\mathcal{R}^{\mathrm{eo}}_+,$ which give subelliptic Fredholm problems for the $\mathrm{Spin}_{\mathbb{C}}$-Dirac operator, $\eth^{\mathrm{eo}}_{+}$. We introduce a generalization of Fredholm pairs to the “tame” category. In this context, we show that the index of the graph closure of $(\eth^{\mathrm{eo}}_+,\mathcal{R}^{\mathrm{eo}}_+)$ equals the relative index, on the boundary, between $\mathcal{R}^{\mathrm{eo}}_+$ and the Calderón projector, $\mathcal{P}^{\mathrm{eo}}_+.$ Using the relative index formalism, and in particular, the comparison operator, $\mathcal{T}^{\mathrm{eo}}_{+},$ introduced in [9], [10], we prove a trace formula for the relative index that generalizes the classical formula for the index of an elliptic operator. Let $(X_0,J_0)$ and $(X_1,J_1)$ be strictly pseudoconvex, almost complex manifolds, with $\phi:bX_1\to bX_0,$ a contact diffeomorphism. Let $\mathcal{S}_0, \mathcal{S}_1$ denote generalized Szegő projectors on $bX_0, bX_1,$ respectively, and $\mathcal{R}_0^{\mathrm{eo}},$ $\mathcal{R}_1^{\mathrm{eo}},$ the subelliptic boundary conditions they define. If $\overline{X_1}$ is the manifold $X_1$ with its orientation reversed, then the glued manifold $X=X_0\amalg_{\phi}\overline{X_1}$ has a canonical $\mathrm{Spin}_{\mathbb{C}}$-structure and Dirac operator, $\eth_X^{\mathrm{eo}}.$ Applying these results and those of our previous papers we obtain a formula for the relative index, $\mathrm{R-Ind}(\mathcal{S}_0,\phi^*\mathcal{S}_1),$ $$ \mathrm{R-ind}(\mathcal{S}_0,\phi^*\mathcal{S}_1)=\mathrm{Ind}(\eth^{\mathrm{e}}_X)- \mathrm{Ind}(\eth^{\mathrm{e}}_{X_0},\mathcal{R}^{\mathrm{e}}_{0})+\mathrm{Ind}(\eth^{\mathrm{e}}_{X_1},\mathcal{R}^{\mathrm{e}}_{1}). \end{equation} For the special case that $X_0$ and $X_1$ are strictly pseudoconvex complex manifolds and $\mathcalS_0$ and $\mathcalS_1$ are the classical Szegő projectors defined by the complex structures this formula implies that \begin{equation} \mathrm{R-ind}(\mathcal{S}_0,\phi^*\mathcal{S}_1)=\mathrm{Ind}(\eth^{\mathrm{e}}_X)- \chi’_{\mathcal{O}}(X_0)+\chi’_{\mathcal{O}}(X_1), \end{equation} which is essentially the formula conjectured by Atiyah and Weinstein; see [37]. We show that, for the case of embeddable CR-structures on a compact, contact 3-manifold, this formula specializes to show that the boundedness conjecture for relative indices from [7] reduces to a conjecture of Stipsicz concerning the Euler numbers and signatures of Stein surfaces with a given contact boundary; see [35].
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