Weighted multiplier ideals of reduced divisors

Olano, Sebastian

doi:10.1007/s00208-021-02306-3

Cited by 2 publications

(11 citation statements)

References 22 publications

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“…The description in Proposition 4.3 for coincides with the description in [Ola22, Proposition 3] since is an inclusion. The complex also has a simple description.…”

Section: Characterizationsmentioning

confidence: 56%

“…Recall that 0-th weighted Hodge ideals, or weighted multiplier ideals, form a sequence of ideals interpolating between the adjoint ideal and a multiplier ideal. This is the case, as (see, for instance, [Ola22, Theorem A]) and for [BS05]. These two ideals identify if a singularity is respectively rational or log-canonical.…”

Section: Introductionmentioning

confidence: 84%

“…For the rest of the proof, we use right -modules. By [Ola22, Proposition 4.1], Around z , is generated by where is the standard generator of . It is clear that is generated by The result follows from the equation .…”

Section: Characterizationsmentioning

confidence: 99%

“…For this reason, , for all . An argument without the use of the V -filtration in the case of is described in [Ola22, §10].…”

Section: Characterizationsmentioning

confidence: 99%

“…In this paper, we continue the study of weighted Hodge ideals that started in [Ola22], where the focus was the -th weighted Hodge ideals, also called weighted multiplier ideals. We show that several results satisfied by the weighted multiplier ideals can be generalized under suitable conditions.…”

Section: Introductionmentioning

confidence: 99%

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Weighted Hodge ideals of reduced divisors

Olano

2023

Forum of Mathematics, Sigma

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We study the Hodge and weight filtrations on the localization along a hypersurface, using methods from birational geometry and the V-filtration induced by a local defining equation. These filtrations give rise to ideal sheaves called weighted Hodge ideals, which include the adjoint ideal and a multiplier ideal. We analyze their local and global properties, from which we deduce applications related to singularities of hypersurfaces of smooth varieties.

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“…The description in Proposition 4.3 for coincides with the description in [Ola22, Proposition 3] since is an inclusion. The complex also has a simple description.…”

Section: Characterizationsmentioning

confidence: 56%

Section: Introductionmentioning

confidence: 84%

Section: Characterizationsmentioning

confidence: 99%

“…For this reason, , for all . An argument without the use of the V -filtration in the case of is described in [Ola22, §10].…”

Section: Characterizationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Weighted Hodge ideals of reduced divisors

Olano

2023

Forum of Mathematics, Sigma

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On a Conjecture of Bitoun and Schedler

Mustaţă

Olano

2023

International Mathematics Research Notices

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Suppose that $X$ is a smooth complex algebraic variety of dimension $\geq 3$ and $f$ defines a hypersurface $Z$ in $X$, with a unique singular point $P$. Bitoun and Schedler conjectured that the ${\mathcal {D}}$-module generated by $\tfrac {1}{f}$ has length equal to $g_{P}(Z)+2$, where $g_{P}(Z)$ is the reduced genus of $Z$ at $P$. We prove that this length is always $\geq g_{P}(Z)+2$ and equality holds if and only if $\tfrac {1}{f}$ lies in the ${\mathcal {D}}$-module generated by $I_{0}(f)\tfrac {1}{f}$, where $I_{0}(f)$ is the multiplier ideal ${\mathcal {J}}(f^{1-\epsilon })$, with $0<\epsilon \ll 1$. In particular, we see that the conjecture holds if the pair $(X,Z)$ is log canonical. We can also recover, with an easy proof, the result of Bitoun and Schedler saying that the conjecture holds for weighted homogeneous isolated singularities. On the other hand, we give an example (a polynomial in $3$ variables with an ordinary singular point of multiplicity $4$) for which the conjecture does not hold.

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Weighted multiplier ideals of reduced divisors

Cited by 2 publications

References 22 publications

Weighted Hodge ideals of reduced divisors

Weighted Hodge ideals of reduced divisors

On a Conjecture of Bitoun and Schedler

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