[Paper Review] Mirror Symmetry and Toric Degenerations of Partial Flag Manifolds

Victor V. Batyrev, Ionuţ Ciocan-Fontanine+2 more|arXiv (Cornell University)|Mar 24, 1998

Algebraic Geometry and Number Theory11 references4 citations

TL;DR

This paper proposes a mirror symmetry construction for complete intersections in partial flag manifolds by degenerating the flag variety to a Gorenstein toric Fano variety, building on prior work for Grassmannians and flag manifolds. The key contribution is a generalized proof of the Gonciulea-Lakshmibai conjecture on the singular locus of the toric degeneration via a small crepant desingularization.

ABSTRACT

In this paper we propose and discuss a mirror construction for complete intersections in partial flag manifolds $F(n_1, ..., n_l, n)$. This construction includes our previous mirror construction for complete intersection in Grassmannians and the mirror construction of Givental for complete flag manifolds. The key idea of our construction is a degeneration of $F(n_1, ..., n_l, n)$ to a certain Gorenstein toric Fano variety $P(n_1, ..., n_l, n)$ which has been investigated by Gonciulea and Lakshmibai. We describe a natural small crepant desingularization of $P(n_1, ..., n_l, n)$ and prove a generalized version of a conjecture of Gonciulea and Lakshmibai on the singular locus of $P(n_1, ..., n_l, n)$.

Motivation & Objective

• To extend mirror symmetry constructions from Grassmannians and complete flag manifolds to general partial flag manifolds.
• To provide a unified framework for mirror symmetry in complete intersections within partial flag varieties.
• To investigate the singularities of the toric degeneration $ P(n_1, \dots, n_l, n) $ proposed by Gonciulea and Lakshmibai.
• To prove a generalized version of the Gonciulea-Lakshmibai conjecture on the singular locus of the toric variety $ P(n_1, \dots, n_l, n) $.

Proposed method

• Degenerate the partial flag manifold $ F(n_1, \dots, n_l, n) $ to a Gorenstein toric Fano variety $ P(n_1, \dots, n_l, n) $ using a flat degeneration in the Hilbert scheme.
• Utilize the toric structure of $ P(n_1, \dots, n_l, n) $ to define a mirror Landau-Ginzburg model via Laurent polynomial potentials.
• Construct a small crepant desingularization of $ P(n_1, \dots, n_l, n) $ to resolve its singularities while preserving the Calabi-Yau condition.
• Apply techniques from toric geometry and birational geometry to analyze the singular locus of $ P(n_1, \dots, n_l, n) $, particularly focusing on the codimension of singular loci.
• Generalize the conjecture of Gonciulea and Lakshmibai on the singular locus of $ P(n_1, \dots, n_l, n) $ and prove it using the desingularization.
• Verify compatibility with existing mirror constructions for Grassmannians and complete flag manifolds by showing consistency in the degeneration limit.

Experimental results

Research questions

• RQ1 How can mirror symmetry be systematically constructed for complete intersections in partial flag manifolds?
• RQ2 What is the structure of the singular locus in the toric degeneration $ P(n_1, \dots, n_l, n) $ of a partial flag manifold?
• RQ3 Can the generalized Gonciulea-Lakshmibai conjecture on the singular locus of $ P(n_1, \dots, n_l, n) $ be proven using a small crepant desingularization?
• RQ4 How does the proposed mirror construction unify previous results for Grassmannians and complete flag manifolds?
• RQ5 What role does the toric degeneration play in realizing the mirror symmetry correspondence for these flag varieties?

Key findings

• The partial flag manifold $ F(n_1, \dots, n_l, n) $ admits a flat degeneration to a Gorenstein toric Fano variety $ P(n_1, \dots, n_l, n) $, which serves as the central fiber of the degeneration.
• A small crepant desingularization of $ P(n_1, \dots, n_l, n) $ is explicitly constructed, resolving its singularities while preserving the Calabi-Yau condition.
• The generalized Gonciulea-Lakshmibai conjecture is proven: the singular locus of $ P(n_1, \dots, n_l, n) $ has codimension at least 3.
• The mirror construction for complete intersections in $ F(n_1, \dots, n_l, n) $ is shown to be compatible with Givental's mirror for complete flag manifolds and the authors' prior construction for Grassmannians.
• The toric degeneration provides a geometric framework that unifies mirror symmetry for complete intersections in various flag varieties, including partial and complete ones.