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On topological invariants of algebraic threefolds with ($\mathbb Q$-factorial) singularities

Antonella Grassi, Timo Weigand, with an Appendix by V. Srinivas

TL;DR

This work develops a coherent framework linking the topology of threefolds with terminal, canonical, and $\mathbb{Q}$-factorial singularities to Lie algebras and their representations arising from elliptic/genus-one fibrations. It establishes local-to-global principles for rational homology, rational Poincaré duality, and deformations, expresses deformations in terms of Milnor and Tyurina data, and derives Euler-characteristic and Betti-number relations that incorporate singularities. A central theme is the Grothendieck–Brieskorn program: associating gauge algebras to discriminant strata, using Tate’s algorithm and intersection data to read off non-abelian and abelian factors, and describing unlocalized and localized representations tied to codimension-one and codimension-two strata. These results are framed in a physics-motivated F-theory context, yielding a birationally invariant invariant $\mathcal{R}$ and a conjectural extension of Kodaira’s fiber classification to birationally equivalent genus-one fibrations, with implications for higher-dimensional elliptic fibrations. Overall, the paper unifies singularity theory, algebraic topology, and gauge-theoretic data to illuminate how topology, deformation theory, and representation theory intertwine in Calabi–Yau threefolds with singularities.

Abstract

We study local, global and local-to-global properties of threefolds with certain singularities. We prove criteria for these threefolds to be rational homology manifolds and conditions for threefolds to satisfy rational Poincaré duality. We relate the topological Euler characteristic of elliptic Calabi-Yau threefolds with $\mathbb Q$-factorial terminal singularities to dimensions of Lie algebras and certain representations, Milnor and Tyurina numbers and other birational invariants of an elliptic fibration. We give an interpretation in terms of complex deformations. We state a conjecture on the extension of Kodaira's classification of singular fibers on relatively minimal elliptic surfaces to the class of birationally equivalent relatively minimal genus one fibered varieties and we give results in this direction.

On topological invariants of algebraic threefolds with ($\mathbb Q$-factorial) singularities

TL;DR

This work develops a coherent framework linking the topology of threefolds with terminal, canonical, and -factorial singularities to Lie algebras and their representations arising from elliptic/genus-one fibrations. It establishes local-to-global principles for rational homology, rational Poincaré duality, and deformations, expresses deformations in terms of Milnor and Tyurina data, and derives Euler-characteristic and Betti-number relations that incorporate singularities. A central theme is the Grothendieck–Brieskorn program: associating gauge algebras to discriminant strata, using Tate’s algorithm and intersection data to read off non-abelian and abelian factors, and describing unlocalized and localized representations tied to codimension-one and codimension-two strata. These results are framed in a physics-motivated F-theory context, yielding a birationally invariant invariant and a conjectural extension of Kodaira’s fiber classification to birationally equivalent genus-one fibrations, with implications for higher-dimensional elliptic fibrations. Overall, the paper unifies singularity theory, algebraic topology, and gauge-theoretic data to illuminate how topology, deformation theory, and representation theory intertwine in Calabi–Yau threefolds with singularities.

Abstract

We study local, global and local-to-global properties of threefolds with certain singularities. We prove criteria for these threefolds to be rational homology manifolds and conditions for threefolds to satisfy rational Poincaré duality. We relate the topological Euler characteristic of elliptic Calabi-Yau threefolds with -factorial terminal singularities to dimensions of Lie algebras and certain representations, Milnor and Tyurina numbers and other birational invariants of an elliptic fibration. We give an interpretation in terms of complex deformations. We state a conjecture on the extension of Kodaira's classification of singular fibers on relatively minimal elliptic surfaces to the class of birationally equivalent relatively minimal genus one fibered varieties and we give results in this direction.

Paper Structure

This paper contains 31 sections, 42 theorems, 27 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Motivation: smooth Calabi-Yau varieties
  3. Terminal, canonical, klt, ($\mathbb{Q}$-)factoriality
  4. Terminal, canonical, klt
  5. Factorial, $\mathbb{Q}$-factorial; the geometric definitions
  6. Almost factoriality and other properties; $\mathbb{Q}$-factorializations
  7. The algebraic definition
  8. Geometry
  9. $\mathbb{Q}$-factorializations
  10. $\mathbb{Q}$-factoriality and the "Calabi-Yau condition"
  11. Cohomologies, local and global (analytic) $\mathbb{Q}$-factoriality, Rational Poincaré
  12. Rational homology manifolds
  13. Rational homology manifolds and $\mathbb{Q}$-factoriality
  14. Rational Poincaré Duality
  15. The third Betti number (and complex deformations of Calabi-Yau threefolds)
  16. ...and 16 more sections

Key Result

Theorem 3.3

Klt singularities, hence canonical and terminal singularities, are rational and in particular Cohen-Macaulay.

Theorems & Definitions (109)

  • Definition 2.1
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3: See for example 6.2.12, IshiiLibro
  • Remark 3.4
  • Definition 3.5
  • Definition 3.6
  • Proposition 3.7: Reid-Ue; Corollary 5.1 Kawamata1988; LinUFD
  • Definition 3.8
  • Definition 3.9
  • ...and 99 more