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On a conjecture of Tian. (arXiv:1508.04090v2 [math.AG] UPDATED)
来源于:arXiv
We study Tian's $\alpha$-invariant in comparison with the
$\alpha_1$-invariant for pairs $(S_d,H)$ consisting of a smooth surface $S_d$
of degree $d$ in the projective three-dimensional space and a hyperplane
section $H$. A conjecture of Tian asserts that $\alpha(S_d,H)=\alpha_1(S_d,H)$.
We show that this is indeed true for $d=4$ (the result is well known for
$d\leqslant 3$), and we show that $\alpha(S_d,H)<\alpha_1(S_d,H)$ for
$d\geqslant 8$ provided that $S_d$ is general enough. We also construct
examples of $S_d$, for $d=6$ and $d=7$, for which Tian's conjecture fails. We
provide a candidate counterexample for $S_5$. 查看全文>>