On characteristic forms on complex manifolds

Abstract

Let X be a complex manifold of dimension n ⩾ 2. In this paper, we define Weyl forms P k ( π), 0 ⩽ k ⩽ n, a kind of characteristic forms, on X by means of a C′-projective connection π and prove the formula c k= ∑ j=0 k n+1−j k−j ((n+1) −c 1(θ)) k−j P j(π) 0⩽k⩽n where the c k ( θ) are the Chern forms defined by a suitable C′-affine connection θ (Theorem 3.1). For example, we have P 0( π) = 1. p 1(π)= 0, P 2(π) = c 2(θ) − {2(n + D} 1nc 1 2(θ) . P 3(π) = c 3(θ) − (n + 1) 1(n − 1) c 1(θ)c 2(θ)+ {3(n + 1) 3} 1n(n − 1) c 1 3(θ) . P 4(π) = c 4(θ) − (n + 1) 1(n − 2)c 1(θ)c 3(θ)+ {2(n + 1) 2} 1(n − 1)(n − 2) c 1 2(θ)c 2(θ)− {8(n + 1) 3} 1(n − 1)(n − 2) c 1 4(θ) . If X is compact, Kachler, and admits a holomorphic projective connection, then the Weyl forms are d-exact. Hence, in this case, our formula reduces to the formula concerning the Chern classes: c k=(n+1) −k n+1 k c k 1, 1⩽k⩽n The latter was claimed by Gunning without the keahlerity assumption, but unfortunately his proof was not correct (cf. Remark 3.39). A direct rigorous proof of formula (∗) under the kaehlerity assumption can be found in the work of Kobayashi and Ochiai.