Title: Skoda's Ideal Generation from Vanishing Theorem for Semipositive Nakano Curvature and Cauchy-Schwarz Inequality for Tensors,
Abstract: Skoda's 1972 result on ideal generation is a crucial ingredient in the
analytic approach to the finite generation of the canonical ring and the
abundance conjecture. Special analytic techniques developed by Skoda, other
than applications of the usual vanishing theorems and L2 estimates for the
d-bar equation, are required for its proof. This note (which is part of a
lecture given in the 60th birthday conference for Lawrence Ein) gives a
simpler, more straightforward proof of Skoda's result, which makes it a natural
consequence of the standard techniques in vanishing theorems and solving d-bar
equation with L2 estimates. The proof involves the following three ingredients:
(i) one particular Cauchy-Schwarz inequality for tensors with a special factor
which accounts for the exponent of the denominator in the formulation of the
integral condition for Skoda's ideal generation, (ii) the nonnegativity of
Nakano curvature of the induced metric of a special co-rank-1 subbundle of a
trivial vector bundle twisted by a special scalar weight function, and (iii)
the vanishing theorem and solvability of d-bar equation with L2 estimates for
vector bundles of nonnegative Nakano curvature on a strictly pseudoconvex
domain. Our proof gives readily other similar results on ideal generation.