Related papers: The Dolbeault operator on Hermitian spin surfaces

S^4 is not a complex manifold, but it is sufficient to remove one point to make it complex. Using supersymmetry methods, we show that the Dolbeault complex (involving the holomorphic exterior derivative and its Hermitian conjugate) can be…

In the first part we use Gromov's K--area to define the K--area homology which stabilizes into singular homology on the category of pairs of compact smooth manifolds. The second part treats the questions of certain curvature gaps. For…

The smooth hermitian representations of a split reductive p-adic group whose restriction to a maximal hyperspecial compact subgroup contain a single K-type with Iwahori fixed vectors have been studied in [D. Barbasch, A. Moy, Classification…

The $k$-Cauchy-Fueter operator $ D_0^{(k) } $ on one dimensional quaternionic space $\mathbb{H}$ is the Euclidean version of helicity $\frac k 2$ massless field operator on the Minkowski space in physics. The $k$-Cauchy-Fueter equation for…

We determine the spectrum of Kostant's cubic Dirac operator $D^{1/3}$ on locally symmetric Lorentzian manifolds of the form $\Gamma\backslash {\rm Osc}_1$, where ${\rm Osc}_1$ is the four-dimensional oscillator group and $\Gamma\subset {\rm…

In this work we prove that any unitary Sobolev $W^{1,2}$ connection of an Hermitian bundle over a 2-dimensional K\"ahler manifold whose curvature is $(1,1)$ defines a smooth holomorphic structure. We prove moreover that such a connection…

In this paper, we suggest a construction of determinant lines of finitely generated Hilbertian modules over finite von Neumann algebras. Nonzero elements of the determinant lines can be viewed as volume forms on the Hilbertian modules.…

We consider a complete nonsingular variety $X$ over $\bC$, having a normal crossing divisor $D$ such that the associated logarithmic tangent bundle is generated by its global sections. We show that $H^i\big(X, L^{-1} \otimes \Omega_X^j(\log…

The issue of general covariance of spinors and related objects is reconsidered. Given an oriented manifold $M$, to each spin structure $\sigma$ and Riemannian metric $g$ there is associated a space $S_{\sigma, g}$ of spinor fields on $M$…

Let $(X,g)$ be a compact $n$-dimensional smooth Riemannian manifold with a lower bound on the average of the lowest $n-p$ eigenvalues of the curvature operator and the diameter of $X$ is bounded above by $D>0$. In this article, we…

Let $X$ be a smooth projective variety acted on by a reductive group $G$. Let $L$ be a positive $G$-equivariant line bundle over $X$. We use the Witten deformation of the Dolbeault complex of $L$ to show, that the cohomology of the sheaf of…

We study semi-Riemannian submanifolds of arbitrary codimension in a Lie group $G$ equipped with a bi-invariant metric. In particular, we show that, if the normal bundle of $M \subset G$ is closed under the Lie bracket, then any normal…

Metrics of constant negative curvature on a compact Riemann surface are critical points of the Liouville action functional, which in recent constructions is rigorously defined as a class in a Cech-de Rham complex with respect to a suitable…

We analyze the spectral properties of a self-adjoint second-order differential operator $\hat{C}$, defined on the Hilbert space $L^2([-v_c, v_c])$ with Dirichlet boundary conditions. We derive the discrete spectrum $\{C_n\}$, prove the…

In this paper, we consider general $k$th-mixed curvature $\mathcal{C}^{(k)}_{\alpha,\beta}$ ($\beta\neq0$) for Hermitian manifolds, which is a convex combination of the $k$th Chern Ricci curvature and holomorphic sectional curvature. We…

Given a closed connected spin manifold M with non-negative and somewhere positive scalar curvature, we show that the Dirac operator twisted with any flat Hilbert module bundle is invertible.

In the first part of this paper we consider compact algebraic manifolds M^2n with an algebraic (n-1)-Torus action. We show that there is a T-invariant meromorphic section $\sigma$ of the canonical bundle of M. Any such $\sigma$ defines a…

We compute the cohomology spaces for the tautological bundle tensor the determinant bundle on the punctual Hilbert scheme H of subschemes of length n of a smooth projective surface X. We show that for L and A invertible vector bundles on X,…

We compute the spectrum of the Dirac operator on 3-dimensional Heisenberg manifolds. The behavior under collapse to the 2-torus is studied. Depending on the spin structure either all eigenvalues tend to $\pm\infty$ or there are eigenvalues…

We establish a lower bound for the eigenvalues of the Dirac operator defined on a compact K\"ahler-Einstein manifold of positive scalar curvature and endowed with particular ${\rm spin}^c$ structures. The limiting case is characterized by…