Abstract
In this paper we develop a systematical way to compute the integral \(\int _{\mathbb {P}^{r-1}} \frac{|\sum _{i=1}^{r}f_iX_i|^2(\sum h_{i\bar{j}}X_i\bar{X}_j)^{k}}{(\sum _{j=1}^p|\sum _{i=1}^{r} g_{ji} X_i|^2)^{r+k+1}}h_0\,\mathrm{d}V_X \) on the projective space \(\mathbb {P}^{r-1}\). Then by this average technique we can link Skoda’s two division theorems together and by the algebraic verification of the fundamental inequality applied to the division theorems we can derive some vanishing theorems.
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References
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I want to thank my adviser professor Siu who has incredible patient and superb knowledge to help me finish this paper, and professor Schmid and professor Chi Chen-Yu for useful discussion on the representation theory in integration.
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Sung, Y. Average technique and its algebraic geometric applications. Math. Ann. 362, 335–387 (2015). https://doi.org/10.1007/s00208-014-1115-6
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DOI: https://doi.org/10.1007/s00208-014-1115-6