Discreteness of log discrepancies over log canonical triples on a fixed pair (Q2922505)

Let \((X, \Delta)\) be a pair of data with a normal variety \(X\) and a boundary \(\Delta\) which is an effective \(\mathbb{R}\)-divisor such that \(K_X+ \Delta\) is an \(\mathbb{R}\)-Cartier \(\mathbb{R}\)-divisor. In the triple \((X, \Delta, \mathfrak{a})\), \(\mathfrak{a}\) is a formal product of finite many coherent ideal sheaves \({\mathfrak{a}}_j\) with real exponents \(r_j\in {\mathbb{R}}_{\geq 0}\): NEWLINE\[NEWLINE\mathfrak{a}=\prod_{j=1}^e {\mathfrak{a}}_j^{r_j}. NEWLINE\]NEWLINENEWLINENEWLINELet \(E\) be a prime divisor on a normal variety \(X'\) and \(\phi: X'\rightarrow X\) a proper birational morphism. The log discrepancy \(a_E(X, \Delta, \mathfrak{a})\) of \(E\) with respect to the triple \((X, \Delta, \mathfrak{a})\) is defined by NEWLINE\[NEWLINE a_E(X, \Delta, \mathfrak{a})=1+{\text{ord}_E}(K_{X'}-\phi^*(K_{X}+\Delta))- {\text{ord}_E}\mathfrak{a}, NEWLINE\]NEWLINE where \( {\text{ord}_E}\mathfrak{a}= \sum_{j=1}^e r_j {\text{ord}_E}\mathfrak{a}_j\).NEWLINENEWLINEThe main theorem of this paper isNEWLINENEWLINETheorem 1.1. In the triple \((X, \Delta, \mathfrak{a})\), the set NEWLINE\[NEWLINE \{ a_E(X, \Delta, \mathfrak{a}) | \mathfrak{a}_j \subset {\mathcal{O}}_X, (X, \Delta, \mathfrak{a})\text{ lc at }\eta_{\phi (E)} \} NEWLINE\]NEWLINE is discrete in \(\mathbb{R} \), where \(r_1,\dots, r_e\in {\mathbb{R}}_{\geq 0} \) and \(\eta_{\phi (E)}\) is the generic point of \(\phi (E)\).NEWLINENEWLINETheorem 1.1 is proved by developing the approach to the ACC for log canonical thresholds due to de Fernex, Ein, Mustat\(\breve{\text{a}}\) and Koll\(\acute{\text{a}}\)r.NEWLINENEWLINEThe author also discusses some applications of Theorem 1.1. For example, an immediate application of Theorem 1.1 is that the set NEWLINE\[NEWLINE\{ {\text{mld}_{\eta_Z}} (X, \Delta, \mathfrak{a}) | \mathfrak{a}_j \subset {\mathcal{O}}_X, Z\subset X\} NEWLINE\]NEWLINE is finite, where the minimal log discrepancy \({\text{mld}_{\eta_Z}}(X, \Delta, \mathfrak{a}) \) at \(\eta_Z\) is the infimum of \( a_E(X, \Delta, \mathfrak{a})\) for all prime divisors \(E\) on \(X'\) such that \(\phi(E)=Z\).