Multiplicity and invariants in birational geometry (Q511852)

The author studies the following conjecture of Watanabe: Let \(X\) be an \(n\)-dimensional variety of locally a complete intersection with canonical singularities. Then \(\mathrm{mult}_xX\leq 2^{n-1}\) for a closed point \(x\in X\). This conjecture is known (and motivated) by the work of Watanabe when \(X\) is in addition a quotient singularity. The main result of this paper shows the conjecture holds when \(n\leq 32\). The author actually obtained some more detailed estimates on multiplicities using the log canonical threshold and the minimal log discrepancy.