Its Square

Abstract

The main goal of this chapter is Theorem 6.1, which says that the square of the spin-c Dirac operator D is equal to the Laplace operator plus a zero order term, given by curvature expressions. The contribution from the curvature of L will be responsible for the Chern characters ch (L _j ) in Proposition 13.2. On the other hand, the term with one half of the curvature of K^* leads, by combining the corresponding factors in (11.17) and (12.12) with the real determinants, to the complex determinants in Proposition 13.1 and Proposition 13.2, respectively. Theorem 6.1 is followed by a comparison of the spin-c Dirac operator with the spinor Dirac operator, which exists if Mis provided with a spin structure. We conclude this chapter with the description, in Proposition 6.1, of what happens with the formula for D² in the Kähler case when D is equal to the Dolbeault-Dirac operator.