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Singularities on normal varieties. (English) Zbl 1179.14003

This is a very nice foundational article about singularities of normal varieties. The authors define generalizations of the notions of multiplier ideals, adjoint ideals, log canonical, log terminal, canonical, and terminal singularities without any additional assumptions of the ambient normal variety \(X\). Prior to their work, a common additional assumption was that the canonical divisor \(K_X\) is \(\mathbb{Q}\)-Cartier, or some \(\mathbb{Q}\)-Cartier perturbation of it, \(K_X+\Delta\), was used. These assumptions are needed to define the relative canonical divisor of a resolution of singularities of \(X\). The issues are: what to do in the case when \(K_X\) is not \(\mathbb{Q}\)-Cartier, or if there is some geometrically-meaningful way of choosing \(\Delta\). These issues are clarified in this paper.
As consequence, the authors obtain generalizations of well-known results to this setting, such as: “log terminal \(\Rightarrow \) rational”, “log canonical \(\Rightarrow\) Du Bois”, subadjunction, deformation invariance of canonical singularities, plurigenera, and numerical Kodaira dimension.
The method of paper is the following. Let \(f:Y\rightarrow X\) be a resolution of singularities. For prime divisors \(E\) on \(Y\), let \(\mathrm{val}_E\) denote the valuation corresponding to \(E\) on the rational functions on \(X\). For a coherent fractional ideal sheaf \(\mathcal{I}\subset \mathcal{K}_X\), define \[ \mathrm{val}_E(\mathcal{I}):=\min\{\mathrm{val}_E\phi\;|\;\phi \in\mathcal{I}(U),\;U\cap f(E)\neq\emptyset\;\}. \] For a divisor \(D\) on \(X\), define \[ f^*D:=\sum_E \left(\lim_{k\rightarrow\infty}\frac{\mathrm{val}_E(\mathcal{O}_X(-k!D))}{k!}\right)\cdot E. \] Then the issue about \(K_X\) not being \(\mathbb{Q}\)-Cartier is resolved by using \(f^*(-K_X)\) and \(-f^*(K_X)\) to define two “relative canonical divisors” \(K_{Y/X}\) and \(K^-_{Y/X}\), respectively. About the second issue, the choice of \(\Delta\), it is shown that the newly-defined multiplier ideal of a pair \((X,Z)\) is the unique maximal element of the set of multiplier ideals defined as usual using \(\mathbb{Q}\)-Cartier divisors \(K_X+\Delta\).
There are various technical difficulties encountered (e.g. the use of limiting relative canonical divisors), but dealt with by the authors in a very readable manner. The naturality of their construction, and the generalizations of the results mentioned above, follows from the two characterization theorems of the newly-defined notions of multiplier ideals and canonicity in terms of older, more familiar, terminology.
The authors point out some open questions: the relation between their multiplier ideals and the generalized test ideals (they agree in the toric case); the discreteness of the jumping numbers; does canonical imply rational, or log canonical, in this more general setting as well?

MSC:

14B05 Singularities in algebraic geometry
14F18 Multiplier ideals
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14J17 Singularities of surfaces or higher-dimensional varieties
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References:

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