THE STRUCTURE OF THE SPACE OF AFFINE K¨AHLER CURVATURE TENSORS AS A COMPLEX MODULE

摘要

In dimension m ≥ 4, results of Strichartz decompose the space A of affine curvaturentensors as a direct sum of 3 modules in the real setting and results of Bokan give ancorresponding finer decomposition of A in the Riemannian setting as the direct sum of 8nirreducible modules. In dimension m ≥ 8, results of Matzeu and Nikˇcevi´c decompose thenspace K of affine K¨ahler curvature tensors as the direct sum of 12 irreducible modulesnin the Hermitian setting (i.e. given an auxiliary inner product which is invariant undernthe given almost complex structure). In this paper, we decompose K as a direct sumnof six irreducible modules in the complex setting in dimension m ≥ 8. Correspondingndecompositions into fewer modules are given in dimension m = 4 and m = 6.