Kählerian twistor spinors

Abstract

Abstract. On a Kähler spin manifold, Kählerian twistor spinors are a natural analogue of twistor spinors on Riemannian spin manifolds. They are defined as sections in the kernel of a first order differential operator adapted to the Kähler structure, called Kählerian twistor (Penrose) operator. We study Kählerian twistor spinors and give a complete description of compact Kähler manifolds of constant scalar curvature admitting such spinors. As in the Riemannian case, the existence of Kählerian twistor spinors is related to the lower bound of the spectrum of the Dirac operator. IntroductionThe purpose of this paper is to study the analogue of twistor spinors on Kähler spin manifolds and to describe the manifolds that admit such spinors.On a Riemannian spin manifold (M, g), a special class of spinors exists, the so-called twistor spinors. They are defined as sections in the kernel of a natural first order operator, the twistor operator, which is given by the projection of the covariant derivative onto the Cartan summand of the tensor product T * M ⊗ ΣM (where T * M is the cotangent bundle and ΣM is the spinor bundle). More precisely, a twistor spinor ϕ ∈ Γ(ΣM) is a solution of the equationThe author thanks Graduiertenkolleg 1269 "Global Structures in Geometry and Analysis" for financial support and the Centre de Mathématiques "Laurent Schwartz" de l'École Polytechnique for hospitality during part of the preparation of this work, within the French-German cooperation project Procope no. 17825PG. The problem of finding optimal lower bounds for the eigenvalues of the Dirac operator on compact manifolds was first considered in 1980 by Th. Friedrich, [7]. He proved that on a compact spin manifold (M n , g) of positive scalar curvature S, the first eigenvalue λ of D satisfiesThe limiting case of this equality is characterized by the existence of real Killing spinors or equivalently by constant scalar curvature and the existence of twistor spinors. The general geometric description of simply connected manifolds carrying Killing spinors was obtained in 1993 by Ch. Bär, [4].As shown by O. Hijazi in 1984, [11], Kähler spin manifolds cannot bear any nontrivial Killing spinors. Moreover, in 1992 K.-D. Kirchberg proved, [16], that if the scalar curvature is nonzero, then nontrivial twistor spinors cannot exist. It is thus natural to ask for an analogue class of spinors on Kähler manifolds, defined by a twistorial equation adapted to the Kähler structure. These spinors are called Kählerian twistor spinors and are defined in the following way. On a Kähler spin manifold (M 2m , g, J), the spinor bundle ΣM splits into U(m)-irreducible subbundles: ΣM = ⊕ m r=0 Σ r M, where Σ r M is the eigenbundle of the Clifford multiplication with the Kähler form for the eigenvalue i(2r − m). For each 0 ≤ r ≤ m, a Kählerian twistor operator is defined by the projection of the covariant derivative onto the Cartan summand of the tensor product T * M ⊗ Σ r M. The sections in the kernel of this first order differential operator are the Kählerian twisto...

Eigenvalue estimates for the Dirac operator on Kähler–Einstein manifolds of even complex dimension

“…the image of D is contained in the kernel of the Clifford multiplication. Using the orthogonal decomposition (33), one proves the following proposition by a straightforward calculation [11].…”

“…, m}. By (11) and (14), then the eigenvalue equation Dψ = λψ is equivalent to the system of equations…”

Invariant Dirac Operators, Harmonic Spinors, and Vanishing Theorems in CR Geometry

“…In the non-extremal cases, i.e., for = ±m, the twistor equation is overdetermined. In fact, similar as in [19], we suppose the existence of a twistor connection such that CR twistor spinors correspond to parallel sections in certain twistor bundles. This would imply that for = ±m the CR invariants p are numbers.…”

A Note on the Conformal Invariance of G-Generalized Gradients

“…. , m, which act on sections of the irreducible subbundles Σ r M of the spinor bundle (see [19] for details).…”

“…If we consider the spinor bundle, as in Example 2.4, we have as sections in the kernel of the Dirac operator the so-called harmonic spinors and in the kernel of the twistor operator the so-called twistor spinors. On a Kähler manifold such examples are provided by the so-called Kählerian twistor spinors, which are defined as sections in the kernel of the U( n 2 )-generalized gradient called Kählerian twistor operator (see [19]). An important application of Corollary 3.16 is the construction of non-trivial solutions in the kernel of G-generalized gradients starting from trivial ones, given for instance by parallel sections, which are well understood in terms of the holonomy representation.…”