Spectral geometry and the Kaehler condition for complex manifolds

Abstract

Recently, the relationship between the geometry of a smooth manifold and the spectrum of its Laplacian has been explored extensively. It is known that the Euler number and signature of a manifold are determined by the spectrum. Furthermore, the arithmetic genus of a complex manifold is determined by the spectrum of its complex Laplacian. Patodi [7] proved that it is possible to determine whether a real manifold is flat, constant curvature, or Einstein from the spectrum of the Laplace operator. He showed that it is possible to determine ira space is isometric to the standard sphere from the spectral geometry. In this paper, we answer a problem proposed by M. Berger [2]: we will show that it is possible to determine whether a complex Hermitian manifold M is Kaehler from the spectrum of the complex Laplacian ([~w acting on forms of type (0,0), (1,0), and (0, 1). John Sacks and the author [9] have used this result to show that it is possible to determine whether a complex Hermitian manifold is isometric to complex projective space with the standard metric from the spectrum of the complex Laplacian. The proofs of these results depend on certain asymptotic invariants of the eigenvalues of these operators. These invariants are computed by integrating corresponding local invariants of the metric tensor over the manifold. In this paper, we compute the second term in the asymptotic expansion for the operators Dp, q= (~* (~+(~*) on forms of type (p, q). In Section 1 we derive a combinatorial formula for the second term in the asymptotic expansion of an arbitrary operator using invariance theory and the results of McKean-Singer [6]. In Section 2, we compute all the invariant polynomials which are of order 2 in the derivatives of a Hermitian metric. This computation also uses invariance theory. In Section 3, we combine these results to compute the second term in the asymptotic expansion of the operators Dp, q. In the final section we derive the result cited above.