SubellipticSpinCDirac operators, III The Atiyah-Weinstein conjecture ByCharles L. Epstein* This paper is dedicated to my wife Jane for her enduring love and support. Abstract In this paper we extend the results obtained in [9], [10] to manifolds with SpinC-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 ¯ ∂-Neumann boundary conditions defined by projection operators, Reo +, which give subelliptic Fredholm problems for the SpinC-Dirac operator, ð 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 (ð eo +,Reo +) equals the relative index, on the boundary, betweenReo +and the Calder´ on projector, P eo +. Using the relative index formalism, and in particular, the comparison operator,T eo +, introduced in [9], [10], we prove a trace formula for the rel-ative index that generalizes the classical formula for the index of an elliptic operator. Let (X0,J0) and (X1,J1) be strictly pseudoconvex, almost complex manifolds, withφ: bX1 →bX0, a contact diffeomorphism. LetS0,S1 de-note generalized Szeg˝o projectors onbX0,bX1, respectively, andReo 0 , Reo 1 , the subelliptic boundary conditions they define. IfX1is the manifold X1with its orientation reversed, then the glued manifoldX=X0φX1 has a canonical SpinC-structure and Dirac operator, ð eo X. Applying these results and those of our previous papers we obtain a formula for the relative index, R-Ind(S0,φ ∗ S1), R-Ind(S0,φ ∗ S1) = Ind(ð e X)ưInd(ð e X0 ,Re 0 ) + Ind(ð e X1 ,Re 1 ). (1) For the special case thatX0andX1are strictly pseudoconvex complex mani-folds and S0andS1are the classical Szeg˝o projectors defined by the complex structures this formula implies that R-Ind(S0,φ ∗ S1) = Ind(ð e X)ưχ O(X0)+χ O(X1), (2) *Research partially supported by NSF grant DMS02-03795 and the Francis J. Carey term chair. 300 CHARLES L. EPSTEIN 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]. Introduction LetXbe an even dimensional manifold with a SpinC-structure; see [21]. A compatible choice of metric,g,and connection∇S/ , define a SpinC-Dirac operator,ðwhich acts on sections of the bundle of complex spinors,S/.This bundle splits as a direct sumS/=S/ e⊕S/ o .IfXhas a boundary, then the kernels and cokernels ofð eo are generally infinite dimensional. To obtain a Fredholm operator we need to impose boundary conditions. In this instance, there are no local boundary conditions for ð eo that define elliptic problems. In our earlier papers, [9], [10], we analyzedsubellipticboundary conditions forð eo obtained by modifying the classical¯ ∂-Neumann and dual ¯ ∂-Neumann conditions forX, under the assumption that the SpinC-structure near to the boundary of Xis that defined by an integrable almost complex structure, with the boundary ofXeither strictly pseudoconvex or pseudoconcave. The boundary condi-tions considered in our previous papers have natural generalizations to almost complex manifolds with strictly pseudoconvex or pseudoconcave boundary. A notable feature of our analysis is that, properly understood, we show that the natural generality for Kohn’s classic analysis of the ¯ ∂-Neumann prob-lem is that of an almost complex manifold with a strictly pseudoconvex contact boundary. Indeed it is quite clear that analogous results hold true for almost complex manifolds with contact boundary satisfying the obvious generaliza-tions of the conditionsZ(q), for a qbetween 0 andn; see [14]. The principal difference between the integrable and non-integrable cases is that in the latter case one must consider all form degrees at once because, in general,ð 2 does not preserve form degree. Before going into the details of the geometric setup we briefly describe the philosophy behind our analysis. There are three principles: 1. On an almost complex manifold the SpinC-Dirac operator, ð, is the proper replacement for¯ ∂+¯ ∂ ∗ . 2. Indices can be computed using trace formulæ. 3. The index of a boundary value problem should be expressed as a relative index between projectors on the boundary. The first item is a well known principle that I learned from reading [6]. Tech-nically, the main point here is thatð 2 differs from a metric Laplacian by an SUBELLIPTICSpinCDIRAC OPERATORS, III 301 operator of order zero. As to the second item, this is a basic principle in the analysis of elliptic operators as well. It allows one to take advantage of the remarkable invariance properties of the trace. The last item is not entirely new, but our applications require a substantial generalization of the notion of Fredholm pairs. In an appendix we definetame Fredholm pairs and prove generalizations of many standard results. Using this approach we reduce the Atiyah-Weinstein conjecture to Bojarski’s formula, which expresses the index of a Dirac operator on a compact manifold as a relative index of a pair of Calder´on projectors defined on a separating hypersurface. That Bojarski’s for-mula would be central to the proof of formula (1) was suggested by Weinstein in [37]. The Atiyah-Weinstein conjecture, first enunciated in the 1970s, was a conjectured formula for the index of a class of elliptic Fourier integral opera-tors defined by contact transformations between co-sphere bundles of compact manifolds. We close this introduction with a short summary of the evolution of this conjecture and the prior results. In the original conjecture one began with a contact diffeomorphism between co-sphere bundles:φ:S ∗M1→S ∗M0. This contact transformation defines a class of elliptic Fourier integral opera-tors. There are a variety of ways to describe an operator from this class; we use an approach that makes the closest contact with the analysis in this paper. Let (M, g) be a smooth Riemannian manifold; it is possible to define complex structures on a neighborhood of the zero section inT ∗Mso that the zero section and fibers ofπ:T ∗M→Mare totally real; see [24], [16], [17]. For eachε>0, let B∗ εMdenote the co-ball bundle of radiusε,and let Ω n,0B∗ εM denote the space of holomorphic (n,0)-forms onB∗ εMwith tempered growth at the boundary. For small enoughε>0, the push-forward defines maps Gε:Ω n,0 B ∗ εMư→ C ư∞ (M), (3) such that forms smooth up to the boundary map toC∞(M).Boutet de Monvel and Guillemin conjectured, and Epstein and Melrose proved that there is an ε0 >0 so that, ifε<ε0, thenGε is an isomorphism; see [11]. With S ∗ εM= bB ∗ ε M,we let Ω n,0 b S ∗ εMdenote the distributional boundary values of elements of Ω n,0B∗ ε M.One can again define a push-forward map Gbε:Ω n,0 b S ∗ εMư→ C ư∞ (M). (4) In his thesis, Raul Tataru showed that, for small enoughε,this map is also an isomorphism; see [36]. As the canonical bundle is holomorphically trivial for εsufficiently small, it suffices to work with holomorphic functions (instead of (n,0)-forms). LetM0 andM1 be compact manifolds andφ: S ∗M1→S ∗M0 a contact diffeomorphism. Such a transformation canonically defines a contact diffeo-morphismφε : S ∗ εM1 →S ∗ εM0 for all ε>0. For sufficiently small positiveε,