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Abstract Recent progress in the deformation theory of Calabi–Yau varieties $$Y$$ with canonical singularities has highlighted the key role played by the higher Du Bois and higher rational singularities, and especially by the so-called $$k$$-liminal singularities for $$k\ge 1$$. The goal of this paper is to show that certain aspects of this study extend naturally to the $$0$$-liminal case as well, that is, to Calabi–Yau varieties $$Y$$ with Gorenstein log canonical, but not canonical, singularities. In particular, we show the existence of first order smoothings of $$Y$$ in the case of isolated $$0$$-liminal hypersurface singularities, and extend Namikawa’s unobstructedness theorem for deformations of singular Calabi–Yau three-folds $$Y$$ with canonical singularities to the case where $$Y$$ has an isolated $$0$$-liminal lci singularity under suitable hypotheses. Finally, we describe an interesting series of examples.
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Du Bois property of log centers
The analytic aspects of multiplier ideals, log canonical thresholds and log canonical centers played an important role in several papers of Demailly, including [DEL00, Dem01, DK01, Dem12, DP14, Dem16, CDM17, Dem18]. Log canonical centers are seminormal by [Amb03, Fuj17], even Du Bois by [KK10, KK20]. This has important applications to birational geometry and moduli theory; see [KK10, KK20] or [Kol23, Sec.2.5]. We recall the concept of Du Bois singularities in Definition-Theorem 4. An unusual aspect is that this notion makes sense for complex spaces that have irreducible components of different dimension. This is crucial even for the statement of our theorem. In this note we generalize the results of [KK10, KK20] by showing that if a closed subset is 'close enough' to being a union of log canonical centers, then it is Du Bois. The minimal discrepancy is a nonnegative rational number, that measures the deviation from being a union of log canonical centers. The log canonical gap gives the precise notion of 'closeness.'
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- PAR ID:
- 10581606
- Publisher / Repository:
- International Press
- Date Published:
- Journal Name:
- Pure and Applied Mathematics Quarterly
- Volume:
- 21
- Issue:
- 3
- ISSN:
- 1558-8599
- Page Range / eLocation ID:
- 1195 to 1207
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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