TAKEUCHI'S EQUALITY FOR THE LEVI FORM OF THE FUBINI-STUDY DISTANCE TO COMPLEX SUBMANIFOLDS IN COMPLEX PROJECTIVE SPACES

Abstract

A. Takeuchi showed that the negative logarithm of the Fubini-Study boundary distance function of pseudoconvex domains in the complex projective space CPⁿ, n ∈ N, is strictly plurisubharmonic and solved the Levi problem for CPⁿ. His estimate from below of the Levi form is nowadays called the ‘Takeuchi's inequality.' In this paper, we give the ‘Takeuchi's equality,' i.e. an explicit representation of the Levi form of the negative logarithm of the Fubini-Study distance to complex submanifolds in CPⁿ.