Abstract
Let $X$ denote a smooth projective variety of dimension $n$ defined over the field of complex numbers such that the anti-canonical line bundle $-K_X$ of $X$ is nef and big with $h^{0}(-K_{X})>0$, and let $L$ be a nef and big line bundle on $X$. In this paper, we consider the dimension of the global sections of $K_{X}+mL$ with $m\geq n-1$ for this case. In particular, under the assumption that $K_{X}+(n-1)L$ is nef, we prove that $h^{0}(K_{X}+(n-1)L)>0$ if $6\leq n\leq 9$ and $L$ is ample.
Funding Statement
This research was supported by JSPS KAKENHI Grant Number 16K05103.
Acknowledgment
The author would like to thank the referee for giving very useful comments and suggestions.
Citation
Yoshiaki Fukuma. "On the dimension of the global sections of adjoint bundles for quasi-polarized manifold whose anti-canonical bundle is effective, nef and big." Kodai Math. J. 45 (1) 1 - 18, March 2022. https://doi.org/10.2996/kmj/kmj45101
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