Holomorphically parametrized L2 Cramer's rule and its algebraic geometric applications

Abstract

Suppose $f,g_1,\cdots,g_p$ are holomorphic functions over $\Omega\subset\cxC^n$. Then there raises a natural question: when can we find holomorphic functions $h_1,\cdots,h_p$ such that $f=\sum g_jh_j$? The celebrated Skoda theorem solves this question and gives a $L^2$ sufficient condition. In general, we can consider the vector bundle case, i.e. to determine the sufficient condition of solving $f_i(x)=\sum g_{ij}(x)h_j(x)$ with parameter $x\in\Omega$. Since the problem is related to solving linear equations, the answer naturally connects to the Cramer's rule. In the first part we will give a proof of division theorem by projectivization technique and study the generalized fundamental inequalities. In the second part we will apply the skills and the results of the division theorems to show some applications.