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Skoda's 1972 result on ideal generation is a crucial ingredient in theanalytic approach to the finite generation of the canonical ring and theabundance conjecture. Special analytic techniques developed by Skoda, otherthan applications of the usual vanishing theorems and L2 estimates for thed-bar equation, are required for its proof. This note (which is part of alecture given in the 60th birthday conference for Lawrence Ein) gives asimpler, more straightforward proof of Skoda's result, which makes it a naturalconsequence of the standard techniques in vanishing theorems and solving d-barequation with L2 estimates. The proof involves the following three ingredients:(i) one particular Cauchy-Schwarz inequality for tensors with a special factorwhich accounts for the exponent of the denominator in the formulation of theintegral condition for Skoda's ideal generation, (ii) the nonnegativity ofNakano curvature of the induced metric of a special co-rank-1 subbundle of atrivial vector bundle twisted by a special scalar weight function, and (iii)the vanishing theorem and solvability of d-bar equation with L2 estimates forvector bundles of nonnegative Nakano curvature on a strictly pseudoconvexdomain. Our proof gives readily other similar results on ideal generation.

Skoda's Ideal Generation from Vanishing Theorem for Semipositive Nakano Curvature and Cauchy-Schwarz Inequality for Tensors