Jacobian discrepancies and rational singularities. (English) Zbl 1334.14022

de Fernex, Tommaso; Docampo, Roi

J. Eur. Math. Soc. (JEMS) 16, No. 1, 165-199 (2014).

The canonical sheaf \(\omega_X\) of a smooth complex projective variety \(X\) has two main useful properties: it satisfies Kodaira vanishing and Serre duality. On a singular (say, normal and Cohen-Macaulay) variety, for Serre duality there is the dualizing sheaf \(\omega_X\), while Kodaira vanishing is satisfied by the Grauert-Riemenschneider sheaf \(f_* \omega_Y\), where \(f: Y \to X\) is a resolution of singularities. We have an inclusion \(f_* \omega_Y \subset \omega_X\), but in general it is strict. Intuitively, this means that a holomorphic \(n\)-form (\(n = \dim X\)) on the smooth locus of \(X\) need not extend to a holomorphic \(n\)-form on \(Y\). On the other hand, every \(n\)-form extends after multiplication with a function vanishing to sufficiently high order along the singular locus and this is quantified by the colon ideal sheaf \((f_* \omega_Y : \omega_X)\).
The main result of the present paper (Theorem C) describes this colon ideal as a Jacobian multiplier ideal, for an arbitrary normal variety \(X\): \[ (f_* \omega_Y : \omega_X) := \{ g \in \mathcal O_X \;|\; g \cdot \omega_X \subset f_* \omega_Y \} = \mathcal J^{\diamond}(\mathfrak d_X^{-1}). \] To give a meaning to the term on the right-hand side, the authors consider the Jacobian discrepancy, which measures the difference between the Jacobian ideal of the resolution morphism and the Jacobian ideal of \(X\). Unlike the usual discrepancy, this definition does not need any \(\mathbb Q\)-Gorenstein assumption. For locally complete intersections, the Jacobian discrepancy coincides with the usual discrepancy, but for general \(\mathbb Q\)-Gorenstein varieties it does not. One should also remark that this notion of discrepancy is quite different from the one introduced by T. de Fernex and C. D. Hacon in [Compos. Math. 145, No. 2, 393–414 (2009; Zbl 1179.14003)], since asymptotic considerations are not taken into account. Once the definition of Jacobian discrepancy is in place, one may define the notions of J-canonical and J-log canonical pairs and Jacobian multiplier ideals \(\mathcal J^{\diamond}(\cdots)\) as usual.
The ideal \(\mathfrak d_X \subset \mathcal O_X\) is called the lci-defect ideal of \(X\). It is defined as \(\sum_V \mathfrak d_{X, V}\), where the sum runs over all reduced locally complete intersection schemes \(V \supset X\) of the same dimension. For each such \(V\), we may write \(V = X \cup X'\), where \(X'\) is the union of all the irreducible components of \(V\) different from \(X\). Then \(\mathfrak d_{X, V}\) is defined as the ideal of \(X \cap X'\) in \(X\).
Several further results are shown in the paper. Theorem B asserts that the Jacobian discrepancies can be read off from the jet schemes of the variety in question. Corollary 7.2 says that \(X\) has rational singularities if and only if \((X, \mathfrak d_X^{-1})\) is J-canonical and Cohen-Macaulay, while \(X\) is Du Bois and CM if and only if \((X, \mathfrak d_X^{-1})\) is J-log canonical and CM. For \(\mathbb Q\)-Gorenstein varieties, Theorem A gives a similar equivalence with \(\mathfrak d_X^{-1}\) replaced by \(\mathfrak d_{r, X}^{1/r} \cdot \mathfrak d_X^{-1}\), where \(\mathfrak d_{r, X}\) is the lci-defect ideal of level \(r\), depending on the choice of an integer \(r\) such that \(rK_X\) is Cartier.
These results can be seen as converses to the theorems of Elkik and Kollár-Kovács that canonical singularities are rational, resp. that log canonical singularities are Du Bois. An example is also given to show that without the Cohen-Macaulay condition, \((X, \mathfrak d_X^{-1})\) J-canonical does not imply \(X\) Du Bois.

Reviewer: Patrick Graf (Bayreuth)

MSC:

14J17	Singularities of surfaces or higher-dimensional varieties
14F18	Multiplier ideals
14E18	Arcs and motivic integration

Keywords:

discrepancy; Jacobian; adjunction; Nash blow-up; jet scheme; multiplier ideal; rational singularity; Du Bois singularity

Citations:

Zbl 1179.14003

References:

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