[Paper Review] On Clifford double mirrors of toric complete intersections

Lev Borisov, Zhan Li|arXiv (Cornell University)|Jan 5, 2016

Advanced Combinatorial Mathematics34 references3 citations

TL;DR

This paper introduces a unified toric framework for constructing noncommutative double mirrors of complete intersections in toric varieties, generalizing Kuznetsov's Clifford double mirror construction. By leveraging reflexive Gorenstein cones and derived categories of sheaves of Clifford algebras, it establishes derived equivalences that explain sporadic examples and reveal combinatorial and physical underpinnings of double mirror phenomena in Calabi-Yau geometry.

ABSTRACT

We present a construction of noncommutative double mirrors to complete intersections in toric varieties. This construction unifies existing sporadic examples and explains the underlying combinatorial and physical reasons for their existence.

Motivation & Objective

• To unify sporadic examples of double mirrors in Calabi-Yau geometry through a systematic toric construction.
• To explain the combinatorial and physical origins of derived equivalences in double mirror pairs.
• To generalize Kuznetsov’s Clifford double mirror construction to complete intersections in toric varieties.
• To extend the notion of double mirrors beyond birational geometry to include noncommutative and non-toric examples.
• To provide a derived categorical framework for noncommutative Calabi-Yau varieties using reflexive Gorenstein cones.

Proposed method

• The construction uses reflexive Gorenstein cones to encode the geometry of complete intersections in toric varieties.
• It defines a sheaf of even Clifford algebras over a projective space, arising from a decomposition of the dual degree element.
• The method relies on a regular simplicial fan in the dual cone to define the noncommutative variety via a derived category.
• It employs a coefficient function on lattice points to encode defining equations of hypersurfaces.
• The derived category of coherent sheaves on the double mirror is shown to be equivalent to that of the original complete intersection under flatness and central fan conditions.
• The framework generalizes to higher-rank Clifford algebras and includes examples without flatness or central fan assumptions.

Experimental results

Research questions

• RQ1 How can Kuznetsov’s Clifford double mirror construction be generalized to complete intersections in toric varieties?
• RQ2 What combinatorial and geometric conditions ensure derived equivalence between a complete intersection and its noncommutative double mirror?
• RQ3 Why do certain noncommutative varieties arise as double mirrors of complete intersections in a toric setting?
• RQ4 Can the double mirror phenomenon be understood through the derived category of a sheaf of Clifford algebras over a toric variety?
• RQ5 What is the role of flatness and the central fan in ensuring derived equivalence in this construction?

Key findings

• The construction produces a noncommutative double mirror as the derived category of a sheaf of even Clifford algebras over a projective space.
• For the (2,2,2,2)-complete intersection in ℂℙ⁷, the double mirror is derived equivalent to the original variety when the flatness and central fan conditions are satisfied.
• In the absence of flatness, the derived category of the double mirror fails to be equivalent to that of the original variety due to singularities in the target space.
• The method generalizes to (2,2,2)-complete intersections in ℂℙ⁵ and yields derived equivalences even when the mirror is not a smooth variety.
• The framework includes examples of double mirrors without a central fan and without flatness, suggesting a broader Calabi-Yau geometry beyond standard assumptions.
• The paper identifies limitations in covering non-toric examples like Pfaffian-Grassmannian or Reye congruence double mirrors, indicating directions for future work.