Some characterizations of complex quadrics in terms of the spectrum are given.All manifolds are assumed to be compact, connected and of complex dimension /> 1. Two K~ihler manifolds are regarded as the same if one is holomorphically isometric to the other. The spectrum of p-forms for a K/ihler manifold M is denoted by SpecP(M). The complex projective n-space with the Fubini-Study metric of constant holomorphic curvature 1 is denoted by CP". The nonsingular complex quadric in CP" with the induced K~hler metric is denoted by Q,_ ~. The index -c, the Euler characteristic )6 and the arithmetic genus a of Q2 are 0,4, and 1, respectively.In this short note, we obtain the following: THEOREM 1. Let M be a Kiihler manifold with Spec z (M) = Spec 2 (Q,) (n ~> 5). If M can be holomorphically isometrically immersed in C Pm for some m, then M is Q,. THEOREM 2. Let M be a Kiihler manifold with Spec v (M) = SpecP(Q,)for a fixed p (p = 0, 1, ..., 2n). If M can be holomorphically isometrically imbedded in CP" for some m, then M is Q',. THEOREM 3. Let M be a Kfihler surface with Spec ° (M) = Spec ° (Q2)" If m satisfies any one of the followin 9 conditions: (a) Spec 1 (M)= Spec 1 (Q2); (b) a(m)/> 1 ; (c) z(m) ~< 0, or (d) x(M) ~> 4, then m is Q2" From Theorem 3 we immediately obtain the following. COROLLARY. Let M be either a product space or a rational surface. If Spec ° (M) = Spec ° (Q2), then M is Q2" 2. PROOF OF THEOREM 1By p,s, and R we denote the scalar curvature, the Ricci tensor, and the curvature tensor of the n-dimensional K~hler manifold M, respectively. The Minakshisundaram-Pleijel-Gaffney's formula for Spec z (M, 9) is given by (cf., l-l] ) exp (2k,2 t),To(4~zt)-" ~, ai,2t'. k=O i=0
