Related papers: Subelliptic SpinC Dirac Operators, IV Proof of the…

We assume that the manifold with boundary, X, has a Spin_C-structure with spinor bundle S. Along the boundary, this structure agrees with the structure defined by an infinite order integrable almost complex structure and the metric is…

The main result of this paper is that the identity component of the automorphism group of a compact, connected, strictly pseudoconvex CR manifold is compact unless the manifold is CR equivalent to the standard sphere. In dimensions greater…

In this paper, we first establish an $S^1$-equivariant index theorem for Spin$^c$ Dirac operators on $\mathbb{Z}/k$ manifolds, then combining with the methods developed by Taubes \cite{MR998662} and Liu-Ma-Zhang \cite{MR1870666,MR2016198},…

In this paper we show that there is a well defined modified dbar-Neumann problem for a spin_c manifold with a strictly pseudoconvex boundary (in the contact geometry sense). We show that the index of the associated boundary value problem…

We prove the stability of the equivariant embedding of compact strictly pseudoconvex CR manifolds with transversal CR circle action under circle invariant perturbations of the CR structures.

Abstract deformations of the CR structure of a compact strictly pseudoconvex hypersurface $M$ in $\mathbb{C}^2$ are encoded by complex functions on $M$. In sharp contrast with the higher dimensional case, the natural integrability condition…

We prove that for an induced CR structure on a compact, generic, regular 3-pseudoconcave CR submanifold ${\bold M}\subset{\bold G}$, of a complex manifold ${\bold G}$, satisfying condition $\dim H^1({\bold M}, T^{\prime}({\bold G})|_{\bold…

The deformation theory of a Dirac structure is controlled by a differential graded Lie algebra which depends on the choice of an auxiliary transversal Dirac structure; if the transversal is not involutive, one obtains an $L_\infty$ algebra…

We define an equivariant index of Spin$^c$-Dirac operators on possibly noncompact manifolds, acted on by compact, connected Lie groups. The main result in this paper is that the index decomposes into irreducible representations according to…

We obtain an embedding theorem for compact strongly pseudoconvex CR manifolds which are bounadries of some complete Hermitian manifolds. We use this to compactify some negatively curved Kaehler manifolds with compact strongly pseudoconvex…

We prove that every compact, pseudoconvex, orientable, CR manifold of $\C^n$, bounds a complex manifold in the $C^\infty$ sense. In particular, the tangential Cauchy-Riemann system has closed range.

We show, under an orientation hypothesis, that a log symplectic manifold with simple normal crossing singularities has a stable almost complex structure, and hence is Spin$_c$. In the compact Hamiltonian case we prove that the index of the…

The fact that every compact oriented 4-manifold admits spin$^c$ structures was proved long ago by Hirzebruch and Hopf. However, the usual proof is neither direct nor transparent. This article gives a new proof using twistor spaces that is…

We establish compactness estimates for $\overline{\partial}_{b}$ on a compact pseudoconvex CR-submanifold of $\mathbb{C}^{n}$ of hypersurface type that satisfies property(P). When the submanifold is orientable, these estimates were proved…

We prove the rigidity and vanishing of several indices of "geometrically natural" twisted Dirac operators on almost even-Clifford Hermitian manifolds admitting circle actions by automorphisms.

Using the index theory for twisted Dirac operators acting on sections of Lipschitz bundles over non-compact manifolds, we prove Llarull-type comparison results in scalar curvature geometry. They apply to spin Riemannian manifolds with…

In this paper we build on the framework developed in "Subelliptic Boundary Value Problems for the Spin_C Dirac Operator, I, II, III" to obtain a more complete understanding of the gluing properties for indices of boundary value problems for…

We study two special cases of the equivariant index defined in part I of this series. We apply this index to deformations of Spin$^c$-Dirac operators, invariant under actions by possibly noncompact groups, with possibly noncompact orbit…

We give a new Tian-Todorov lemma on deformations of CR-structures and use it to reprove the deformation unobstructedness of normal compact strongly pseudoconvex CR-manifold under the assumption of $d'd''$-lemma, more faithfully following…

We prove that for every compact, convex subset $K\subset\mathbb{R}^2$ the operator system $A(K)$, consisting of all continuous affine functions on $K$, is hyperrigid in the C*-algebra $C(\mathrm{ex}(K))$. In particular, this result implies…