Remark on complements on surfaces

Abstract

We give an explicit characterization on the singularities of exceptional pairs in any dimension. In particular, we show that any exceptional Fano surface is $\frac {1}{42}$ -lc. As corollaries, we show that any $\mathbb R$ -complementary surface X has an n-complement for some integer $n\leq 192\cdot 84^{128\cdot 42^5}\approx 10^{10^{10.5}}$ , and Tian’s alpha invariant for any surface is $\leq 3\sqrt {2}\cdot 84^{64\cdot 42^5}\approx 10^{10^{10.2}}$ . Although the latter two values are expected to be far from being optimal, they are the first explicit upper bounds of these two algebraic invariants for surfaces.

Keywords

• 14E30
• 14E05
• 14J17