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Abstract We prove that the nonvarying strata of abelian and quadratic differentials in low genus have trivial tautological rings and are affine varieties. We also prove that strata ofk-differentials of infinite area are affine varieties for allk. Vanishing of homology in degree higher than the complex dimension follows as a consequence for these affine strata. Moreover we prove that the stratification of the Hodge bundle for abelian and quadratic differentials of finite area is extremal in the sense that merging two zeros in each stratum leads to an extremal effective divisor in the boundary. A common feature throughout these results is a relation of divisor classes in strata of differentials as well as its incarnation in TeichmĂĽller dynamics.
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Higher multiplier ideals
We associate a family of ideal sheaves to any ℚ-effective divisor on a complex manifold, called higher multiplier ideals, using the theory of mixed Hodge modules and 𝑉-filtrations.This family is indexed by two parameters, an integer indicating the Hodge level and a rational number, and these ideals admit a weight filtration.When the Hodge level is zero, they recover the usual multiplier ideals.We study the local and global properties of higher multiplier ideals systematically.In particular, we prove vanishing theorems and restriction theorems, provide criteria for the nontriviality, and introduce the center of minimal exponent (generalizing the notion of minimal log canonical center).The main idea is to exploit the global structure of the 𝑉-filtration along an effective divisor using the notion of twisted Hodge modules.As applications, we prove new cases of conjectures by Debarre, Casalaina–Martin, and Grushevsky on singularities of theta divisors on principally polarized abelian varieties.
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- Award ID(s):
- 2301526
- PAR ID:
- 10674695
- Publisher / Repository:
- DeGruyter
- Date Published:
- Journal Name:
- Journal fĂĽr die reine und angewandte Mathematik (Crelles Journal)
- ISSN:
- 0075-4102
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript
- Journal Article:
- https://doi.org/10.1515/crelle-2025-0097
