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Jumping coefficients of multiplier ideals. (English) Zbl 1061.14003
Let us consider a non singular complex variety $$X$$ and some sheaf of ideals $$\mathbf a$$. A log resolution of $$\mathbf a$$ is a proper birational map $$\pi :X'\longrightarrow X$$, such that $$X'$$ is non singular, $${\mathbf a}{\mathcal O}_{X'}={\mathcal O}_{X'}(-F)$$ is a normal crossing divisor and each component is smooth. For each real number $$c$$ the multiplier ideal is defined by $${\mathcal I}(X,\mathbf a^c)=\pi_*({\mathcal O}_{X'}(K_{X'/X}-[cF]))$$ where $$K_{X'/X}$$ is the relative canonical divisor and [] means integer part. For $$x\in X$$, it is clear that there exist some rational numbers $$\xi_i$$ such that $${\mathcal I}(X,\mathbf a^c)_x={\mathcal I}(X,\mathbf a^{\xi_i})_x$$ for all $$c\in [\xi_i,\xi_{i+1}[$$. The numbers $$\xi_i$$ are called jumping coefficients of multiplier ideals. In this interesting paper the authors prove that jumping coefficients appears naturally as deep invariants of singularities, for example:
Let $$f$$ be a non zero polynomial and $$\xi$$ a jumping coefficient of the ideal generated by $$f$$, in the interval $$(0,1]$$, then $$-\xi$$ is a zero of the Bernstein polynomial of $$f$$.
The authors give a uniform Artin-Rees bound for multiplier ideals, specially the case of a principal ideal.
Reviewer’s remark: I should mention that the reviewer has constructed with many details a log resolution for integrally closed ideals and hypersurfaces non degenerated with respect to their Newton polyhedra, even for a complete intersection non degenerated with respect to their Newton polyhedra. Howald’s results follow from this easily; see M. Morales [Bull. Soc. Math. Fr. 112, 325–341 (1984; Zbl 0564.32006); Compos. Math. 64, 311–327 (1987; Zbl 0648.14005)].

##### MSC:
 14B05 Singularities in algebraic geometry 32S05 Local complex singularities 13H05 Regular local rings
Siu, Y.-T.
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