[Paper Review] Categorical crepant resolutions of singularities and the Tits-Freudenthal magic square

Roland Abuaf|arXiv (Cornell University)|Jul 5, 2013

Algebraic Geometry and Number Theory23 references3 citations

TL;DR

This paper establishes that the tangent developables of the four symmetric spaces in the third row of the Tits-Freudenthal magic square—corresponding to complexifications of real, complex, quaternionic, and octonionic Grassmannians—admit categorical crepant resolutions of singularities. Using derived categories and properties of Gorenstein, rational singularities, the author proves these resolutions exist via a weakly crepant categorical framework, extending classical crepant resolution theory to cases where geometric resolutions fail.

ABSTRACT

We prove that the tangent developables of the varieties appearing in the third row of the Tits-Freudenthal magic square admit categorical crepant resolutions of singularities.

Motivation & Objective

• To extend the notion of crepant resolution beyond geometric resolutions to categorical resolutions for singular varieties.
• To address the scarcity of geometric crepant resolutions in cases like odd-dimensional Veronese cones by introducing categorical alternatives.
• To prove the existence of categorical crepant resolutions for tangent developables of varieties in the third row of the Tits-Freudenthal magic square.
• To establish that these resolutions are weakly crepant, satisfying key derived category adjunction and Serre functor conditions.
• To demonstrate that the derived categories of these tangent developables admit fully faithful embeddings into derived categories of resolutions, supporting a categorical minimality conjecture.

Proposed method

• The paper uses the formalism of categorical crepant resolutions as defined by Kuznetsov, embedding a triangulated category T into the derived category of a smooth resolution via a fully faithful functor.
• It applies the notion of weakly crepant resolution, requiring that the left adjoint to the pushforward is isomorphic to the right adjoint, ensuring compatibility with dualizing complexes.
• The construction relies on the derived category D^b(X) of bounded complexes of coherent sheaves on a Gorenstein variety X with rational singularities.
• The author verifies that the derived category of the tangent developable satisfies the conditions for a weakly crepant categorical resolution using the adjunction formula and properties of exceptional divisors.
• It leverages the fact that the varieties in question arise from complex composition algebras (R, C, H, O), allowing a uniform treatment via the Tits-Freudenthal magic square.
• The proof uses the fact that the singularities of the tangent developables are terminal and Q-factorial, and applies a criterion from Debarre and Kollar to conclude Q-factoriality and hence the existence of categorical resolutions.

Experimental results

Research questions

• RQ1 Do the tangent developables of the varieties in the third row of the Tits-Freudenthal magic square admit categorical crepant resolutions despite lacking geometric ones?
• RQ2 Can the notion of crepant resolution be extended to non-geometric settings using derived categories and triangulated categories?
• RQ3 Is the derived category of the tangent developable of such varieties a weakly crepant categorical resolution satisfying the adjunction and Serre functor conditions?
• RQ4 Are these categorical resolutions minimal in the sense of the Bondal-Orlov conjecture, with fully faithful embeddings into other resolutions?
• RQ5 What is the role of Q-factoriality and terminal singularities in ensuring the existence of such categorical resolutions?

Key findings

• The tangent developable of the symplectic Grassmannian G_ω(3,6) ⊂ P^13 admits a categorical crepant resolution.
• The Grassmannian G(3,6) ⊂ P^19 also admits a categorical crepant resolution via its derived category.
• The spinor variety S_12 ⊂ P^31 and the octonionic Grassmannian G_ω(O^3,O^6) ⊂ P^55 both admit categorical crepant resolutions.
• The resolutions are weakly crepant, satisfying Lπ_*^! ≅ Lπ_*^*, ensuring compatibility with dualizing complexes.
• The varieties are Q-factorial with terminal singularities, which supports the existence of such categorical resolutions.
• The construction is uniform across the four varieties, all arising from complex composition algebras over R, C, H, and O, and lying in the third row of the Tits-Freudenthal magic square.