由半正Nakano曲率的消灭定理和张量的柯西-施瓦茨不等式生成Skoda理想
Skoda's Ideal Generation from Vanishing Theorem for Semipositive Nakano Curvature and Cauchy-Schwarz Inequality for Tensors
摘要：
斯柯达的1972年结果理想的一代是一个至关重要的成分分析方法的有限代的典型环和丰度猜想。特殊的分析技术开发的斯柯达，而不是通常的应用程序的消灭定理和L2估计的d-杆方程，需要证明。这说明（这是一个演讲的一部分，在60岁生日会议的劳伦斯艾因）给出了一个更简单，更直接的证明斯柯达的结果，这使得它的自然后果的标准技术在消灭定理和解决d-杆方程与L2估计。证明涉及以下三个要素：（i）一个特殊的张量的柯西-施瓦茨不等式，它在Skoda理想生成的积分条件的公式化中具有一个特殊的因子，该因子表示分母的指数，（ii）平凡向量丛的一个特殊的余秩1子丛的诱导度量的Nakano曲率的非负性，该平凡向量丛被一个特殊的标量权函数扭曲，（iii）严格拟凸区域上具有非负Nakano曲率的向量丛的具有L2估计的d-杆方程的消灭定理和可解性.我们的证明很容易给出其他类似的理想生成的结果。
英文摘要：
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.
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