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Let $X$ be a smooth variety over a field $k=\bar k$, $\rm char$ $k=0$. Assume that $\mathcal{L}$ is a nef and big line bundle. Is there some bound saying that $$ h^1(\mathcal{L}^{\otimes n}) \le f(n) $$ where $f(n)$ is a function only depending on $n$? For example, a polynomial. And how about the case when $\rm char$ $k>0$? |
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In general, for any line bundle $L$ over $X$, you have $h^i(X, L^{\otimes n})\leqslant C m^n$ where $m=\dim X$. (write $L$ as the difference of two very ample line bundles, and use the usual restriction exact sequence) In the case where $L$ is nef, then one can say more: $h^i(X,L^{\otimes n})\leqslant C m^{n-i}$ using Fujita's theorem (cf Lazarsfeld, Positivity in Algebraic Geometry, 1.2.29 "growth of cohomology"). But in the case $i=1$, you don't need Fujita, and this can be done more basically. (see e.g Debarre's "Higher dimensional Algebraic Geometry", Proposition 1.31 p21) |
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