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 . 0. Introduction Let D (CPn , n ∈ N, be a pseudoconvex domain and denote by δ∂D(P) the Fubini– Study distance from P ∈ D to the boundary ∂D of D. Takeuchi [21] found that the strict subharmonicity of the function −log tan−1|z| on C \ {0} leads the strict plurisubharmonicity of the function −log δ∂D on D, and solved the Levi problem for CPn . The inequality i∂∂̄(−log δ∂D)≥ 3ωF S on D is nowadays called the ‘Takeuchi’s inequality’ (cf. [1, 7, 9, 20, 22]). Recently, many mathematicians have been interested in the following problem: ‘Is there a smooth closed Levi-flat real hypersurface in CPn if n ≥ 2?’, where a real hypersurface M ⊂ CPn is said to be Levi-flat if its complement CPn \ M is locally pseudoconvex or equivalently locally Stein. When n ≥ 3, Lins Neto [11] proved the non-existence in the real analytic case, and Siu [19] proved it in the smooth case. When n = 2, the non-existence problem is still open even in the real analytic case. Then Takeuchi’s inequality is one of the key points to approach the non-existence problem or related topics. His paper [21] is frequently cited even now although he wrote it over 50 years ago (for example, see Adachi [1], Adachi and Brinkschulte [2], Brinkschulte [5], Brunella [6], Fu and Shaw [8], Harrington and Shaw [10], Ohsawa [16, 17], and Ohsawa and Sibony [18]). It follows from Takeuchi’s theorem that if S is a complex hypersurface in CPn and if δS denotes the Fubini–Study distance to S, then the function−log δS is strictly plurisubharmonic 2010 Mathematics Subject Classification: Primary 32E40, 32C25.