Cobordism, relative indices and Stein fillings (Q941852)

This paper consists of two parts. In the first part the author extends his results [Ann. Math. (2) 166, No.~1, 183--214 (2007; Zbl 1154.32016), ibid. 166, No.~3, 723--777 (2007; Zbl 1154.32017), ibid. 168, 299--365] concerning Fredholm boundary value problems for the Spin-Dirac operator. More precisely, let \(X\) be a \(2n\)-dimensional Spin-manifold with contact boundary \(Y\); it was assumed that \(Y\) is connected. In the present work, the author assumes that \(Y\) can have several connected components, some pseudoconvex and some pseudoconcave. The Spin-structure must be defined in a neighborhood of the boundary by an almost complex structure \(J\) and the contact structure in \(Y\) is assumed to be compatible with this structure \(J\), which defines a splitting of \(TX\otimes\mathbb{C}\), i.e., \(TX\otimes\mathbb{C}=T^{1,0}X\oplus T^{0,1}X\); and this gives a dual splitting of \(T^{\ast}X\otimes\mathbb{C}=\Lambda^{1,0}X\oplus \Lambda^{0,1}X\), which leads to the definition of the \(\overline{\partial}\)-operator. If \(\Lambda^{p,q}\) denotes the bundle of forms of type \((p,q)\) the bundle \(\underline{\mathbb{S}}\) of complex spinors can be identified (in a neighborhood \(U\) of \(Y\)) as \(\bigoplus^n_{q=0}\Lambda^{0,q}X\). Selecting a Hermitian metric g on \(T^{0,1}X\) one can define a formal adjoint \(\overline{\partial}^{\ast}\). The Spin-Dirac operator, \(\underline{\partial}\) can be expressed (over \(U\)) as \(\underline{\partial}=\overline{\partial}+\overline{\partial}^* +\varepsilon^{\mathbb C}\) where \(\varepsilon:\underline{\mathbb S}\rightarrow\underline{\mathbb S}\) is a bundle endomorphism. If \(\rho\) is a defining \(C^\infty\) function for the boundary of \(X\), the compatibility of the contact structure with \(J\) is that the hyperplane field \(H\) on \(Y\) is the null-space of the real 1-form \(\theta=i\overline{\partial}p|_{TY}\). The Levi-form \({\mathcal L}_{y}(X,Y)\) is \(\frac{1}{2}[d\theta_{y}(X,JY)+ d\theta_{y}(Y,JX)]\) for \(X\in H_{y}\), and pseudoconvexity (resp., pseudoconcativity) is defined with respect to this Levi-form. In order to obtain a Fredholm operator in the integrable case, one has to change the boundary condition in degree 0, i.e., one requires that \(S(\sigma^{0,0}|_{b_X})=0\) where \(S\) is the (classical) Szegö projector, and in order to obtain a formally self adjoint operator, the boundary condition in degree 1 is modified by requiring that \((Id-S)[\overline{\partial}\rho_{\rfloor}\sigma^{0,1}|_{b_X}]=0\). Along with the \(\overline{\partial}\)-Neumann conditions in higher degrees \((\overline{\partial}\rho_{\rfloor}\sigma^{\rho,q}|_{b_X}=0)\) it defines a projector \({\mathcal R}_+\) acting on sections of \(\underline{\mathbb S}|_{b_X}\). Summing up, the modified \(\overline{\partial}\)-Neumann condition can be expressed as \({\mathcal R}_+ [\sigma|_{b_X}]=0\) and the pair (\(\underline{\partial},{\mathcal R}\)) denotes the operator defined by \(\underline{\partial}\) acting on the domain defined by the above condition. It was proved in (loc. cit.) that this operator is essentially self adjoint and its graph closure is Fredholm. The spin-bundles and operators split into even and odd parts and in the integrable case \(\text{Ind}(\underline{\partial}^e,{\mathcal R}^e)\) computes the renormalized meromorphic Euler characteristic. In the non integrable case, one needs to introduce the generalized Szegő projectors. This generalized Szegő projection can be defined because the contact structure in \(Y\) defines an algebra of pseudodifferential operators \(\Psi_H^*(Y)\) (the Heisenberg algebra). The almost complex structure \(J\) in the fiber of \(H\) is positive if the induced Levi form is positive definite and this defines a function \(s_J\) on the cotangent bundle \(T^*Y\) which is the principal symbol of an \(S\in\Psi^0_H(Y)\). By definition \(S\in\Psi^0_H(Y)\) is a generalized Szegő projector if it \(\rho\) is a projector and if there is a positive almost complex structure \(J\) in \(H\) so that the principal symbol of \(S\) satisfies \(\sigma_0^H(S)=s_J\). It is important that the generalized Szegő projector exist in any contact manifold with positive almost complex structure and if \(S_1\), \(S_2\) are two such generalized Szegő projectors on \((Y,H)\), the restriction \(S_1:\operatorname{range} S_2\to\operatorname{range}S_1\) is Fredholm [\textit{C. Epstein} and \textit{R. Melrose}, Math. Res. Lett. 5, No. 3, 363--381 (1998; Zbl 0929.58012)]. Its index is denoted \(R\text{-Ind}(S_1,S_2)\). The modified pseudoconvex \(\overline{\partial}\)-Neumann condition can be defined, using generalized Szegő projectors, in any strictly pseudoconvex Spin-manifold \(X\) satisfying the condition described previously. Namely, let \(S\in\Psi^0_H(Y)\) be such an operator. Then the modified pseudoconvex \(\overline{\partial}\)-Neumann condition defined by \(S\) is given by: {\parindent=7mm \begin{itemize}\item[(i)] \(S[\sigma^{0,0}|_{b_X}]=0\), \item[(ii)] \((\text{Id}-S)[\overline{\partial}_\rho\rfloor\sigma^{0,1}|_{b_X}]=0\), \item[(iii)] \([\underline{\overline{\partial}}_\rho\rfloor\sigma^{0,q}|_{b_X}]=0\) for \(\rho >a\). \end{itemize}} In [loc. cit.], it is shown that \((\underline{\partial},R_+)\) is essentialy self adjoint and its closure is a Fredholm operator. Denoting with \((\underline{\partial}^e,R_+^e)\), respectively, \((\underline{\partial}^0,R_+^0)\), the even and odd parts, we have that \((\underline{\partial}^{e,0},R_+^{e,0})*=\overline{(\underline{\partial}^{e,0}, R_+^{e,0})}\) and if \(X\) is strictly pseudoconcave, then some result holds with \(\text{Id}-R_+\) instead of \(R_+\). Let \({\mathcal P}\) be the projection operator acting on sections of \(\underline{\mathbb S}|_{b_X}\) with range equal to the boundary values of elements of \(\operatorname{Ker}\underline{\partial}\) (the Calderon projector). As any fundamental solution of \(\underline{\partial}\) leads to the construction of a Calderon projector, it is important that two such Calderon projectors are smoothly homotopic. The author studies the Calderon projector in the case of multiple components of the boundary, and shows that the Calderon projector can be deformed through projectors to a block diagonal matrix. It is shown (relying on the results and methods of [loc. cit.]) that if \(X\) is a Spin-manifold such that the Spin-structure is defined in a neighborhood of \(bX\) by an almost complex structure, making each boundary component of \(X\) either strictly pseudoconvex or strictly pseudoconcave, then one has an inequality \(\|\sigma\|_{H^{\frac12}(X)}\leq C[\|\underline{\partial}\sigma\|_{L^2(X)}+\|\sigma\|_{L^2(X)}]\) for any \(\sigma\) beloging to the domain of \({\mathcal R}\), and the chiral restrictions \(\underline{\partial}^{e,0}\) are Fredholm. The main point that there is, under the same hypothesis about \(X\) as above, a relation between \(\text{Ind}(\underline{\partial}^{e,0},{\mathcal R}^{e,0})= {\mathcal R}\text{-Ind} ({\mathcal P}^{e,0}, {\mathcal R}^{e,0})\) where \({\mathcal P}^{e,0}\) are the even and odd part of the Calderon projector. Using these results, the author proves gluing formulae for the index of \(\underline{\partial}\), a cocycle condition for these indices, and finally (Th. 8) conditions that ensures the vanishing of \(\text{Ind}(\underline{\partial}^e_{X^{c}}, [\text{Id}-{\mathcal R}^0_0), {\mathcal R}^{e}_{1}])\) where \(X_c= \varphi^{-1} ((-\infty,c))\) with \(\varphi\) an exhaustion function for \(X\), \(c\in{\mathbb R}\), \(X^c=X-X_{c}\), \(c\) sufficiently large a regular value of \(\varphi,{\mathcal R}_0,{\mathcal R}_1\), the modified pseudoconvex \(\overline{\partial}\)-Neumann condition defined by \(S_0\), respectively, \(S_1\), where \(S_0\) is the classical Szegő projector defined on \(bX_c\), \(S_1\), the classical Szegő projector defined on \(bX\). The second part is devoted to the main result of the paper which is the following: Let \(X_+\) be a strictly pseudoconvex surface with boundary a CR manifold \((Y,T_b^{0,1}Y)\) and suppose that this boundary is also the boundary of a strictly pseudoconcave complex manifold \(X_-\) which contains a positive, compact holomorphic curve \(Z\). Let \(S_0\) be the classical Szegő projector defined by the CR structure on \(Y\). If \(H_c^2(X_-;\Theta)=0\) and \(\deg NZ\geq2g-1\) where \(g\) is the genus of \(Z\) and \(NZ\) the normal bundle of \(Z\) and \(\Theta\) is the tangent sheaf of \(X_-\), then there exist a constant \(M\) such that for a sufficiently small embeddable perturbations of the CR structure, with Szegő projector \(S_1\), we have \(| R\text{-Ind}(S_0,S_1)|\leq M\). As a corollary it results that the set of small embeddable deformation of \((Y,T^{0,1}Y)\) is closed in the \(C^\infty\) topology, which is a significative generalization of \textit{L. Lempert}'s result, treating domains in \(\mathbb C^2\) [Math. Ann. 300, No.~1, 1--15 (1994; Zbl 0817.32009)]. The proof of this theorem uses the previous results on the relative indices, a Banach version of Kiremidjan's theorem [\textit{G. M. Henkin} and \textit{C. L. Epstein}, Contemp. Math. 205, 51--67 (1997; Zbl 0886.32005)], the excision theorem of Gromov-Lawson, and the results of Stipsicz on the topology of Stein filligs of circle bundles over Riemann surfaces [\textit{A. I. Stipsicz}, Mich. Math. J. 51, No.~2, 327--337 (2003; Zbl 1043.53066)].