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

Antonella Grassi, Timo Weigand+1 more|arXiv (Cornell University)|Apr 6, 2018

Advanced Topics in Algebra6 citations

TL;DR

This paper establishes a correspondence between topological invariants of Calabi-Yau threefolds with $\mathbb{Q}$-factorial terminal singularities and Lie algebras with their representations, using stratified singular loci of elliptic fibrations. It proves a formula linking the topological Euler characteristic to Lie algebra dimensions, Milnor and Tyurina numbers, and birational invariants, while conjecturing an extension of Kodaira’s singular fiber classification to birationally equivalent genus one fibered threefolds.

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.

Motivation & Objective

• To establish a mathematical framework connecting topological invariants of singular Calabi-Yau threefolds to Lie algebras and their representations.
• To prove that rational Poincaré duality and rational homology manifold conditions hold under specific $\mathbb{Q}$-factorial and klt singularity criteria.
• To relate the topological Euler characteristic of elliptic Calabi-Yau threefolds to Lie algebra dimensions, Milnor and Tyurina numbers, and birational invariants.
• To extend Kodaira’s classification of singular fibers on elliptic surfaces to birationally equivalent relatively minimal genus one fibered threefolds.
• To formalize a 'Grothendieck-Brieskorn' program linking singularities and Lie algebras through local-to-global principles in fibrations.

Proposed method

• Uses stratified singular loci of elliptic fibrations to assign non-abelian gauge algebras and representations to codimension one and two strata.
• Applies Mayer-Vietoris theorem to compute the topological Euler characteristic via localized contributions from singular fibers.
• Employs local analytic $\mathbb{Q}$-factorialization for isolated klt singularities to derive global cohomological conditions.
• Relies on rational Poincaré duality and rational homology manifold criteria via $\mathbb{Q}$-factoriality and local-to-global principles.
• Introduces a formula (Theorem 9.4) expressing the Euler characteristic as a sum over Lie algebra dimensions and representation contributions.
• Uses deformation theory and the gravitational anomaly cancellation condition to validate the correspondence in physical models.

Experimental results

Research questions

• RQ1 How can topological invariants of Calabi-Yau threefolds with $\mathbb{Q}$-factorial terminal singularities be related to Lie algebras and their representations?
• RQ2 What conditions ensure that such threefolds are rational homology manifolds or satisfy rational Poincaré duality?
• RQ3 Can the correspondence between elliptic fibrations and gauge algebras be extended beyond smooth Calabi-Yau varieties to singular, $\mathbb{Q}$-factorial cases?
• RQ4 To what extent does the Kodaira classification of singular fibers generalize to birationally equivalent genus one fibered threefolds?
• RQ5 How do Mordell-Weil group rank and multiple fibers affect the gauge algebra and hypermultiplet spectrum in the presence of singularities?

Key findings

• The topological Euler characteristic of an elliptic Calabi-Yau threefold with $\mathbb{Q}$-factorial terminal singularities is given by a formula involving Lie algebra dimensions, Milnor and Tyurina numbers, and birational invariants (Theorem 9.4).
• The formula matches the gravitational anomaly cancellation condition in F-theory when the genericity assumption holds and $h^{1,1}(X) = 1 + h^{1,1}(B) + ext{rk}(rak{g})$.
• The contribution of uncharged localized hypermultiplets is given by $H_{unch}^{loc} = \sum_P \tau(P)$, where $\tau(P)$ is the Tyurina number at each singular point $P$.
• The gauge algebra and unlocalized representations are birational invariants of the $\mathbb{Q}$-factorial terminal minimal model, as proven in Corollary 7.5 and Theorem 9.7.
• Codimension two strata $Q$ in the discriminant locus correspond to tensor product representations of $\mathfrak{g}_i \oplus \mathfrak{g}_j$, with dimensions determined by the fiber type over $Q$.
• Conjecture 9.8 proposes an extension of Kodaira’s singular fiber classification to birationally equivalent genus one fibered threefolds, assigning gauge algebras and representations to stratified discriminant loci.