[Paper Review] Homological mirror symmetry for Brieskorn-Pham singularities

Masahiro Futaki, Kazushi Ueda|arXiv (Cornell University)|Dec 2, 2009

Algebraic structures and combinatorial models16 references5 citations

TL;DR

This paper establishes homological mirror symmetry for Brieskorn-Pham singularities by proving a quasi-equivalence between the derived Fukaya category of the Lefschetz fibration defined by a Brieskorn-Pham polynomial and the triangulated category of singularities of the same polynomial, graded by a rank-one abelian group. The proof relies on symplectic Picard-Lefschetz theory and a categorification of the Milnor lattice via tensor products of A∞-categories associated with A-type singularities.

ABSTRACT

We prove that the derived Fukaya category of the Lefschetz fibration defined by a Brieskorn-Pham polynomial is equivalent to the triangulated category of singularities associated with the same polynomial together with a grading by an abelian group of rank one. Symplectic Picard-Lefschetz theory developed by Seidel is an essential ingredient of the proof.

Motivation & Objective

• To establish homological mirror symmetry for Brieskorn-Pham singularities, a class of hypersurface singularities including ADE types and exceptional unimodal singularities.
• To categorify the Milnor lattice of Brieskorn-Pham singularities using A∞-categories derived from A-type singularities.
• To prove an equivalence between the derived Fukaya category of the associated Lefschetz fibration and the graded triangulated category of singularities.
• To extend known results in the n=2 case to arbitrary n, using symplectic and homological algebra techniques.
• To relate the category of singularities to coherent sheaves on a quotient stack under the Calabi-Yau/Landau-Ginzburg correspondence when the weights satisfy the condition ∑1/pi = 1.

Proposed method

• Use symplectic Picard-Lefschetz theory (Seidel) to inductively compute the derived Fukaya category of the Lefschetz fibration Wp : C^n → C defined by a Morsified Brieskorn-Pham polynomial.
• Define the A∞-category Ap−1 as a differential graded category with objects C1,…,Cp and morphisms given by C·idCi in degree 0, C[−1] in degree 1, and zero otherwise.
• Construct the Fukaya category Fuk Wp as a quasi-equivalence to the tensor product Ap1−1 ⊗⋯⊗Apn−1 of such A∞-categories.
• Equip the coordinate ring Ap = C[x1,…,xn]/(fp) with a grading by a rank-one abelian group L(p) generated by ⃗x1,…,⃗xn,⃗c with relations pi⃗xi = ⃗c.
• Define the graded triangulated category of singularities Dgr_Sg(Ap) as the quotient Db(gr Ap)/Dperf(gr Ap), and show it is equivalent to Db(Ap1−1 ⊗⋯⊗Apn−1).
• Use a free resolution of the skyscraper sheaf at the origin (Lemma 4.1) to compute RHom between graded modules and identify the Yoneda product structure with the A∞-category.

Experimental results

Research questions

• RQ1 Can homological mirror symmetry be established for Brieskorn-Pham singularities beyond the n=2 case?
• RQ2 Is the derived Fukaya category of the Lefschetz fibration associated with a Brieskorn-Pham polynomial quasi-equivalent to a tensor product of A∞-categories of A-type singularities?
• RQ3 Does the graded triangulated category of singularities of a Brieskorn-Pham singularity admit a full exceptional collection isomorphic to the tensor product of Api−1 categories?
• RQ4 Under what conditions does the category of singularities become equivalent to the derived category of coherent sheaves on a quotient stack?
• RQ5 Can the Milnor lattice of a Brieskorn-Pham singularity be categorified via symplectic Picard-Lefschetz theory?

Key findings

• The derived Fukaya category of the Lefschetz fibration defined by a Brieskorn-Pham polynomial is quasi-equivalent to the tensor product of A∞-categories Ap1−1 ⊗⋯⊗Apn−1.
• The triangulated category of singularities Dgr_Sg(Ap) for the Brieskorn-Pham singularity is equivalent to the bounded derived category of the tensor product category Db(Ap1−1 ⊗⋯⊗Apn−1).
• The full exceptional collection (k(⃗n))⃗n∈I in Dgr_Sg(Ap) generates the category and is isomorphic as a graded category to Ap1−1 ⊗⋯⊗Apn−1.
• The Yoneda product structure on the exceptional collection matches the A∞-structure of the tensor product, and higher A∞-operations vanish due to degree constraints.
• When ∑1/pi = 1, the category Dgr_Sg(Ap) is equivalent to the derived category of coherent sheaves on the quotient stack Yp = [Xp/Gp], where Xp is a hypersurface in weighted projective space.
• The proof relies on a free resolution of the structure sheaf of the origin (Lemma 4.1), which allows explicit computation of RHom between graded modules and establishes the equivalence of categories.