On boundedness of divisors computing minimal log discrepancies for surfaces

Abstract

Let $\hat{I}“$ be a finite set, and $X\ni x$ a fixed klt germ. For any lc germ $(X\ni x,B:={\sum }_i{b}_i{B}_i)$ such that $b_i\in \hat{I}“$, Nakamura's conjecture, which is equivalent to the ACC conjecture for minimal log discrepancies for fixed germs, predicts that there always exists a prime divisor $E$ over $X\ni x$, such that $a(E,X,B)=\mathrm{mld}(X\ni x,B)$, and $a(E,X,0)$ is bounded from above. We extend Nakamura's conjecture to the setting that $X\ni x$ is not necessarily fixed and $\hat{I}“$ satisfies the DCC, and show it holds for surfaces. We also find some sufficient conditions for the boundedness of $a(E,X,0)$ for any such $E$.

38 pages, the main part of this paper will appear in J. Inst. Math. Jussieu., and Appendix A (A simple proof of ACC for minimal log discrepancies for surfaces) will appear in Acta Math. Sin. (Engl. Ser.)