# Subelliptic $\mathrm{Spin}_{\mathbb{C}}$ Dirac operators, III. The Atiyah–Weinstein conjecture | Annals of Mathematics

Abstract In this paper we extend the results obtained in ,  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, eo+, which give subelliptic Fredholm problems for the Spinℂ-Dirac operator, ðeo+. We introduce a generalization of Fredholm pairs to the “tame” category. In this context, we show that the index of the graph closure of (ðeo+,eo+) equals the relative index, on the boundary, between eo+ and the Calderón projector, eo+. Using the relative index formalism, and in particular, the comparison operator, eo+, introduced in , , we prove a trace formula for the relative 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, with ϕ:bX1→bX0, a contact diffeomorphism. Let 0,1 denote generalized Szegő projectors on bX0,bX1, respectively, and eo0, eo1, the subelliptic boundary conditions they define. If X1⎯⎯⎯⎯ is the manifold X1 with its orientation reversed, then the glued manifold X=X0⨿ϕX1⎯⎯⎯⎯ has a canonical Spinℂ-structure and Dirac operator, ðeoX. Applying these results and those of our previous papers we obtain a formula for the relative index, R−Ind(0,ϕ∗1), R−ind(0,ϕ∗1)=Ind(ðeX)−Ind(ðeX0,e0)+Ind(ðeX1,e1). For the special case that X0 and X1 are strictly pseudoconvex complex manifolds and 0 and 1 are the classical Szegő projectors defined by the complex structures this formula implies that R−ind(0,ϕ∗1)=Ind(ðeX)−χ′(X0)+χ′(X1), which is essentially the formula conjectured by Atiyah and Weinstein; see . 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  reduces to a conjecture of Stipsicz concerning the Euler numbers and signatures of Stein surfaces with a given contact boundary; see .

KEYWORDS

SHARE & LIKE ### 数学年刊（Annals of Mathematics）

0 Following 0 Fans 0 Projects 674 Articles

SIMILAR ARTICLES

Abstract For any nondegenerate, quasi-homogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a correspondin

Abstract For a large class of nonlinear Schrödinger equations with nonzero conditions at infinity and for any speed c less than the sound velocity, we

Abstract Let L2,p(ℝ2) be the Sobolev space of real-valued functions on the plane whose Hessian belongs to Lp. For any finite subset E⊂ℝ2 and p>2, let

Abstract We prove that for any group G in a fairly large class of generalized wreath product groups, the associated von Neumann algebra LG completely

Abstract We study the parity of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. We

Abstract This paper has two main results. Firstly, we complete the parametrisation of all p-blocks of finite quasi-simple groups by finding the so-cal

Abstract We derive sharp Moser-Trudinger inequalities on the CR sphere. The first type is in the Adams form, for powers of the sublaplacian and for ge

Abstract We prove that isoparametric hypersurfaces with (g,m)=(6,2) are homogeneous, which answers Dorfmeister-Neher’s conjecture affirmatively and so

Abstract We prove the periodicity conjecture for pairs of Dynkin diagrams using Fomin-Zelevinsky’s cluster algebras and their (additive) categorificat

Abstract If F(x,y)∈ℤ[x,y] is an irreducible binary form of degree k≥3, then a theorem of Darmon and Granville implies that the generalized superellipt
{"ft_lang_label":"__label__en","ft_lang_prob":0.8056901,"math_prob":0.9573021,"size":3728,"snap":"2019-13-2019-22","text_gpt3_token_len":1050,"char_repetition_ratio":0.11278196,"word_repetition_ratio":0.0035971224,"special_character_ratio":0.21164162,"punctuation_ratio":0.10841655,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9917991,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-03-27T02:31:35Z\",\"WARC-Record-ID\":\"<urn:uuid:3712f7dc-ce4e-42c0-99df-7c2f66cf20e9>\",\"Content-Length\":\"50675\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:5f3b5c8c-dd32-4fc0-b997-e9c9b44db513>\",\"WARC-Concurrent-To\":\"<urn:uuid:59f9fc7b-6117-4337-aacd-c3640feeb583>\",\"WARC-IP-Address\":\"115.159.110.167\",\"WARC-Target-URI\":\"http://www.jijitang.com/article/540faab120296a4535b91bd6/Subelliptic-dollarmathrmSpin_mathbbCdollar-Dirac-operators-III-The-AtiyahWeinstein-conjecture-or-Annals-of-Mathematics\",\"WARC-Payload-Digest\":\"sha1:2JCEW7BX6DOLXSDP763623W6P3R5TPKJ\",\"WARC-Block-Digest\":\"sha1:JTA647PP4DIV4J4Y4I7QSB6FGK7LCDXI\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-13/CC-MAIN-2019-13_segments_1552912207618.95_warc_CC-MAIN-20190327020750-20190327042750-00126.warc.gz\"}"}