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Global smoothing of singular Fano and Calabi-Yau varieties

Anda Tenie

TL;DR

The paper develops global smoothing criteria for Fano and Calabi–Yau varieties with isolated rational lci singularities by linking local deformation directions to global deformations through the framework of 1-rational and 1-Du Bois singularities. It establishes that Fano and CY varieties can be deformed to stricter singularity types (1-rational or 1-Du Bois) or become smoothable under natural global conditions, including a Hodge–Du Bois numerical criterion in the presence of 1-liminal singularities. The results generalize classical threefold smoothing theorems (Namikawa–Steenbrink, Friedman, Gross, Friedman–Laza) to higher dimensions and to lci singularities beyond hypersurfaces, with unobstructedness statements $\mathbb{T}^i_Y=0$ for $i\ge 2$ in the Fano case and analogous CY unobstructedness. The work provides a unified deformation-theoretic approach to smoothing that leverages the Du Bois framework, Hodge theory, and local-to-global exact sequences, yielding new smoothing criteria and broadening the scope of known smoothability results.

Abstract

We study the problem of smoothing Fano and Calabi-Yau varieties with isolated rational lci singularities. For Fano varieties we show that any such $Y$ admits a deformation to a Fano variety with only $1$-rational singularities, and if all the singularities of $Y$ are not $1$-rational, then $Y$ is smoothable. For Calabi-Yau varieties, we show first that any such $Y$ deforms to a Calabi-Yau variety with only $1$-Du Bois singularities. Moreover, if all the singularities of $Y$ are not $1$-Du Bois then $Y$ is smoothable. When allowing $1$-liminal singularities, we give a global criterion in terms of the Hodge-Du Bois numbers of $Y$ which ensures $Y$ is smoothable. These theorems recover and generalize results for threefolds of Friedman, Namikawa, Namikawa-Steenbrink, Gross, and Friedman-Laza. In higher dimensions, our results provide alternative smoothing conditions and also extend the work of Friedman-Laza from the case of hypersurface singularities to lci singularities.

Global smoothing of singular Fano and Calabi-Yau varieties

TL;DR

The paper develops global smoothing criteria for Fano and Calabi–Yau varieties with isolated rational lci singularities by linking local deformation directions to global deformations through the framework of 1-rational and 1-Du Bois singularities. It establishes that Fano and CY varieties can be deformed to stricter singularity types (1-rational or 1-Du Bois) or become smoothable under natural global conditions, including a Hodge–Du Bois numerical criterion in the presence of 1-liminal singularities. The results generalize classical threefold smoothing theorems (Namikawa–Steenbrink, Friedman, Gross, Friedman–Laza) to higher dimensions and to lci singularities beyond hypersurfaces, with unobstructedness statements for in the Fano case and analogous CY unobstructedness. The work provides a unified deformation-theoretic approach to smoothing that leverages the Du Bois framework, Hodge theory, and local-to-global exact sequences, yielding new smoothing criteria and broadening the scope of known smoothability results.

Abstract

We study the problem of smoothing Fano and Calabi-Yau varieties with isolated rational lci singularities. For Fano varieties we show that any such admits a deformation to a Fano variety with only -rational singularities, and if all the singularities of are not -rational, then is smoothable. For Calabi-Yau varieties, we show first that any such deforms to a Calabi-Yau variety with only -Du Bois singularities. Moreover, if all the singularities of are not -Du Bois then is smoothable. When allowing -liminal singularities, we give a global criterion in terms of the Hodge-Du Bois numbers of which ensures is smoothable. These theorems recover and generalize results for threefolds of Friedman, Namikawa, Namikawa-Steenbrink, Gross, and Friedman-Laza. In higher dimensions, our results provide alternative smoothing conditions and also extend the work of Friedman-Laza from the case of hypersurface singularities to lci singularities.
Paper Structure (15 sections, 24 theorems, 67 equations)

This paper contains 15 sections, 24 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Statement of main results
  3. Related work
  4. Consequences of our results and relations to prior work
  5. Strategy of proof
  6. Preliminaries
  7. Hodge Theory of singularities
  8. Introduction to higher singularities
  9. Local deformations and smoothings
  10. Smoothing hypersurface singularities.
  11. Smoothing lci singularities.
  12. Exact sequences in local deformations.
  13. Smoothing Fano varieties
  14. Smoothing Calabi-Yau varieties
  15. Notation

Key Result

Theorem 1.4

(Theorem thm: lcinot1DB) Suppose $Y$ is a singular Calabi-Yau variety with rational, isolated lci singularities. Then $Y$ can be deformed to a Calabi-Yau variety whose singularities are 1-Du Bois. Moreover, if one assumes in addition that all the singularities of $Y$ are not 1-Du Bois, then $Y$ is s

Theorems & Definitions (56)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Theorem 1.9: namikawa1995global
  • Theorem 1.10: gross1997deforming
  • ...and 46 more