Hermitian metrics on the anti-canonical bundle of the blow-up of the projective plane at nine points

Abstract

We investigate Hermitian metrics on the anti-canonical bundle of a rational surface obtained by blowing up the projective plane at nine points. For that purpose, we pose a modified variant of an argument made by Ueda on the complex analytic structure of a neighborhood of a subvariety by considering the deformation of the complex structure.Problem 1.1. Let X be a complex manifold and Y ⊂ X be a (reduced) hypersurface. Assume that Y is compact andWe will see previous results on this problem generally in §2.1. Here, let us focus on one of the most interesting cases, which is the case where X = X Z is the one obtained

Linearization of transition functions of a semi-positive line bundle along a certain submanifold

“…See also [K4,Conjecture 2.1] and [K2,Theorem 1.1]. Towards solving this conjecture, we investigate the difference between L and L in each finite order jet along Y in the present paper.…”

Holomorphic foliation associated with a semi-positive class of numerical dimension one

“…non-emptiness of SP(c 1 ([Y ]))) and the complex analytic structure of a neighborhood of Y . In [K2,Conjecture 2.1], we conjectured that [Y ] is semi-positive if and only if the pair (Y, X) is of class (β ′ ) or (β ′′ ) in the classification of Ueda [U]. The following corollary, which follows from [K3,Theorem 1.4] and the argument in the proof of Theorem 1.4, gives an affirmative answer to [K2,Conjecture 2.1] when Y is non-singular.…”