𝒟-Modules in Birational Geometry

Popa, Mihnea

doi:10.1142/9789813272880_0077

Cited by 5 publications

(12 citation statements)

References 39 publications

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“…In the special case when D has an isolated quasihomogeneous singularity, by analogy with the reduced case in [Sai09], this result was conjectured in [Pop18] and proved in [Zha18]. Note also that Theorem E recovers the second statement of [MP19, Theorem 10.1], namely that the filtration on M(f −α ) is always generated at level n − 1.…”

Section: A Introductionsupporting

confidence: 55%

Hodge filtration, minimal exponent, and local vanishing

Mustaţă

Popa

2019

Invent. math.

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We bound the generation level of the Hodge filtration on the localization along a hypersurface in terms of its minimal exponent. As a consequence, we obtain a local vanishing theorem for sheaves of forms with log poles. These results are extended to Q-divisors, and are derived from a result of independent interest on the generation level of the Hodge filtration on nearby and vanishing cycles.1 As mentioned above, for n = 2 the filtration is always generated at level 0.

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Section: A Introductionsupporting

confidence: 55%

Hodge filtration, minimal exponent, and local vanishing

Mustaţă

Popa

2019

Invent. math.

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“…In §4, we return to the other extreme: log-canonical singularities, and the even milder klog-canonical singularities with k ≥ 1 (k = 0 being the log-canonical case). This is inspired by the recent work of Mustata-Popa on Hodge ideals [MP19, MP18, MP20b, MP20a, MOP20,Pop18], and some recent work of M. Saito and his collaborators [Sai16,JKYS19a]. After delineating the relationships between the various birational and Hodge-theoretic invariants of singularities (log-canonical threshold, jumping numbers, generation level, period exponent, etc.)…”

Section: Introductionmentioning

confidence: 99%

Hodge theory of degenerations, (II): vanishing cohomology and geometric applications

Kerr¹,

Laza²

2020

Preprint

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We study the mixed spectrum and vanishing cohomology for several classes of (isolated and nonisolated) hypersurface singularities, and how they contribute to the limiting mixed Hodge structure of a smoothing. Applications are given to several types of singularities arising in KSBA and GIT compactifications and mirror symmetry, including nodes, k-log-canonical singularities, singularities with Calabi-Yau tail, normal-crossing degenerations, slc surface singularities, and the J k,∞ series. ContentsIntroduction Set-up and overview of results Structure of the paper Synopsis of Part I ([KL19]) Acknowledgement 1. Mixed spectra of isolated singularities 2. The quasi-homogeneous case 2.1. Weighted Fermat singularities and deformations 2.2. Nodes on odd-dimensional hypersurfaces 3. Singularities with Calabi-Yau tail 4. Isolated hypersurface singularities: birational invariants 5. Isolated hypersurface singularities: spectral combinatorics 5.1. Toric geometry approach 5.2. Brieskorn lattice approach 6. Sebastiani-Thom formula 6.1. Application to k-log-canonical singularities 7. Non-isolated singularities 7.1. Remarks on rational, du Bois, normal-crossing, and k-lc singularities 7.2. The SSS formula 7.3. Some non-isolated slc surface singularities 7.4. The J k,∞ series 7.5. Clemens-Schmid discrepancies for nodal total spaces References

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“…For comparison, the list of properties of Hodge ideals in the case when D is reduced is summarized in [Pop18,§4]. While much of the story carries over to the setting of Q-divisors -besides of course the connection with the classical Hodge theory of the complement U = X D, which only makes sense in the reduced case -there are a few significant points where the picture becomes more intricate.…”

Section: A Introductionmentioning

confidence: 99%

Hodge ideals for Q-divisors: birational approach

Mustaţă¹,

Popa²

2019

Journal De L’École Polytechnique — Mathématiques

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We develop the theory of Hodge ideals for Q-divisors by means of log resolutions, extending our previous work on reduced hypersurfaces. We prove local (non-)triviality criteria and a global vanishing theorem, as well as other analogues of standard results from the theory of multiplier ideals, and we derive a new local vanishing theorem. The connection with the V -filtration is analyzed in a sequel.

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𝒟-Modules in Birational Geometry

Cited by 5 publications

References 39 publications

Hodge filtration, minimal exponent, and local vanishing

Hodge filtration, minimal exponent, and local vanishing

Hodge theory of degenerations, (II): vanishing cohomology and geometric applications

Hodge ideals for Q-divisors: birational approach

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