Subelliptic Spin$$_\sigma$$ Dirac operators. III: The Atiyah-Weinstein conjecture.(English)Zbl 1169.32008

Summary: We extend the results obtained in part I and II of this paper [the author, Ann. Math. (2) 166, No. 3, 723–777 (2007; Zbl 1154.32017) and ibid., No. 1, 183–214 (2007; Zbl 1154.32016)] to manifolds with 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 $$\overline {\partial}$$-Neumann boundary conditions defined by projection operators, $$\mathcal R_+^{\text{eo}}$$ which give subelliptic Fredholm problems for the Spin$$_{\mathbb C}$$-Dirac operator $$\partial_+^{\text{eo}}$$. We introduce a generalization of Fredholm pairs to the “tame” category. In this context, we show that the index of the graph closure of $$(\partial_+^{eo},\mathcal R_+^{\text{eo}})$$ equals the relative index, on the boundary, between $$\mathcal R^{\text{eo}}_+$$ and the Calderón projector $$P_+^{\text{eo}}$$. Using the relative index formalism, and in particular, the comparison operator $$\mathcal T_+^{\text{eo}}$$ introduced in part I and II, 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 $$\varphi: bX_1 \rightarrow bX_0$$ a contact diffeomorphism. Let $$\mathcal S_0,\mathcal S_1$$ denote generalized Szegő projectors on $$bX_0,bX_1$$, respectively, and $$\mathcal R_0^{\text{eo}},\mathcal R_1^{\text{eo}}$$, the subelliptic boundary conditions they define. If $$\overline {X_1}$$ is the manifold $$X_1$$ with its orientation reversed, then the glued manifold $$X = X_0 \prod_{\varphi} \overline {X_1}$$ has a canonical Spin$$\mathbb C$$-structure and Dirac operator $$\partial_X^{\text{eo}}$$. Applying these results and those of our previous papers, we obtain a formula for the relative index, R-Ind$$(\mathcal S_0,\varphi^{\ast}\mathcal S_1),$$
$\text{R-Ind}(\mathcal S_0,\varphi^{\ast}\mathcal S_1) = \text{Ind}(\partial_X^{\text e}) - \text{Ind}(\partial_{X_0}^{\text e},\mathcal R_0^e) + \text{Ind}(\partial X_1 ^{\text e},\mathcal R_1^e).\tag{1}$
For the special case that $$X_0$$ and $$X_1$$ are strictly pseudoconvex complex manifolds and $$S_0$$ and $$S_1$$ are the classical Szegő projectors defined by the complex structures, this formula implies that
$\text{R-ind}(\mathcal S_0,\varphi^{\ast}\mathcal S_1) = \text{Ind}(\partial_X^{\text e}) - \chi_{\mathcal O}^{\prime}(X_0) + \chi_{\mathcal O}^{\prime}(X 1),\tag{2}$
which is essentially the formula conjectured by Atiyah and Weinstein; see [RIMS Kokyuroku 1014, 1–14 (1997; Zbl 0943.58019)]. 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 the author [Ann. Math. (2) 147, No. 1, 1–59, 61–91 (1998; Zbl 0942.32025, Zbl 0942.32026)] reduces to a conjecture of Stipsicz concerning the Euler numbers and signatures of Stein surfaces with a given contact boundary [A. I. Stipsicz, Mich. Math. J. 51, No. 2, 327–337 (2003; Zbl 1043.53066)].

MSC:

 32W05 $$\overline\partial$$ and $$\overline\partial$$-Neumann operators 19K56 Index theory 32V15 CR manifolds as boundaries of domains 35H20 Subelliptic equations 35J55 Systems of elliptic equations, boundary value problems (MSC2000) 35S30 Fourier integral operators applied to PDEs 32V30 Embeddings of CR manifolds 32Q60 Almost complex manifolds 53C27 Spin and Spin$${}^c$$ geometry
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