Plemelj Projection Operators over Domain Manifolds

On the singularities of the Bergman projections for lower energy forms on complex manifolds with boundary

Let $M$ be a complex manifold of dimension $n$ with smooth boundary $X$. Given $q\in\{0,1,\ldots,n-1\}$, let $\Box^{(q)}$ be the $\ddbar$-Neumann Laplacian for $(0,q)$ forms. We show that the spectral kernel of $\Box^{(q)}$ admits a full asymptotic expansion near the non-degenerate part of the boundary $X$ and the Bergman projection admits an asymptotic expansion under some local closed range condition. As applications, we establish Bergman kernel asymptotic expansions for some domains with weakly pseudoconvex boundary and $S^1$-equivariant Bergman kernel asymptotic expansions and embedding th…

Noncommutative residue invariants for CR and contact manifolds

In this paper we produce several new invariants for CR and contact manifolds by looking at the noncommutative residue traces of various geometric projections. In the CR setting these operators arise from the Kohn-Rossi complex and include the Szegö projections on forms. In the contact setting they stem from the generalized Szegö projections at arbitrary integer levels of Epstein-Melrose and from the contact complex of Rumin. In particular, we recover and extend recent results of Hirachi and Boutet de Monvel and answer a question of Fefferman.

Regularity equivalence of the Szegö projection and the complex Green operator

In this paper we prove that on a CR manifold of hypersurface type that satisfies the weak $Y(q)$ condition, the complex Green operator $G_q$ is exactly (globally) regular if and only if the Szegö projections $S_{q-1}, S_q$ and a third orthogonal projection $S'_{q+1}$ are exactly (globally) regular. The projection $S'_{q+1}$ is closely related to the Szegö projection $S_{q+1}$ and actually coincides with it if the space of harmonic $(0,q+1)$-forms is trivial. This result extends the important and by now classical result by H. Boas and E. Straube on the equivalence of the regularity of the $\bar…

On the singularities of the Szegö projections on lower energy forms

Let $X$ be an abstract not necessarily compact orientable CR manifold of dimension $2n-1$, $n\geqslant2$. Let $\Box^{(q)}_{b}$ be the Gaffney extension of Kohn Laplacian for $(0,q)$-forms. We show that the spectral function of $\Box^{(q)}_b$ admits a full asymptotic expansion on the non-degenerate part of the Levi form. As a corollary, we deduce that if $X$ is compact and the Levi form is non-degenerate of constant signature on $X$, then the spectrum of $\Box^{(q)}_b$ in $]0,\infty[$ consists of point eigenvalues of finite multiplicity. Moreover, we show that a certain microlocal conjugation o…

Lagrangian fibrations的截面在holomorphic symplectic manifolds上的研究

Sections of Lagrangian fibrations on holomorphic symplectic manifolds

本文研究了holomorphic symplectic manifolds上的Lagrangian fibration的截面，探讨了退化twistor变形的构造及其在复结构上的应用。