[Paper Review] Non-commutative width and Gopakumar-Vafa invariants

Yukinobu Toda|arXiv (Cornell University)|Nov 6, 2014

Algebraic Geometry and Number Theory7 references4 citations

TL;DR

This paper establishes a precise link between Donovan-Wemyss's non-commutative widths for flopping curves in 3-folds and Katz's genus-zero Gopakumar-Vafa invariants. It proves that the non-commutative width is the sum of $ j^2 \cdot n_j $ over curve classes $ j[C] $, while the commutative width equals $ n_1 $, where $ n_j $ are the virtual counts of one-dimensional stable sheaves. This identifies non-commutative invariants as commutative enumerative invariants via deformation theory and autoequivalence deformation.

ABSTRACT

We show that the non-commutative widths for flopping curves on smooth 3-folds introduced by Donovan-Wemyss are described by Katz's genus zero Gopakumar-Vafa invariants.

Motivation & Objective

• To interpret Donovan-Wemyss's non-commutative and commutative widths in terms of enumerative invariants.
• To show that these widths are determined by genus-zero Gopakumar-Vafa invariants $ n_j $, which count one-dimensional stable sheaves on the 3-fold.
• To establish a bridge between non-commutative algebraic geometry and enumerative geometry via deformation of flop-flop autoequivalences.
• To correct a sign error in the literature regarding the flop-flop autoequivalence and its relation to generalized twist functors.

Proposed method

• Use the main result of Donovan-Wemyss on the non-commutative twist functor realizing Bridgeland-Chen's flop-flop autoequivalence.
• Deform the flopping contraction $ f: X \to Y $ to a family of disjoint $(-1,-1)$-curves, where the number of such curves in class $ j[C] $ is $ n_j $.
• Analyze the deformation of the non-commutative twist functor to a composition of Seidel-Thomas spherical twists along $(-1,-1)$-curves.
• Relate the Hilbert polynomial of the kernel of the non-commutative twist to that of the deformed spherical twist composition.
• Apply Grothendieck duality and kernel equivalence uniqueness to identify the kernel of the deformed functor.
• Correct a misstatement in prior work on the flop-flop autoequivalence, showing it is the inverse of the generalized twist functor.

Experimental results

Research questions

• RQ1 How are Donovan-Wemyss's non-commutative widths for flopping curves related to Gopakumar-Vafa invariants?
• RQ2 Can the non-commutative width be computed purely via commutative enumerative invariants?
• RQ3 What is the precise relationship between the contraction algebra and the genus-zero GV invariants?
• RQ4 Why does the flop-flop autoequivalence fail to be isomorphic to the generalized twist functor as previously claimed?
• RQ5 How does the deformation of the contraction $ f $ to multiple $(-1,-1)$-curves reflect the invariants $ n_j $?

Key findings

• The non-commutative width $ \mathrm{wid}(C) $ is given by $ \sum_{j=1}^{l} j^2 \cdot n_j $, where $ l $ is the scheme-theoretic length of the fiber over the singular point.
• The commutative width $ \mathrm{cwid}(C) $ equals $ n_1 $, the genus-zero Gopakumar-Vafa invariant for the class $ [C] $.
• For the $ (1,-3) $-curve example with $ R_k = \mathbb{C}[u,v,x,y]/(u^2 + v^2 y = x(x^2 + y^{2k+1})) $, $ \mathrm{wid}(C) = 3(2k+1) $ and $ \mathrm{cwid}(C) = 2k+3 $, implying $ n_1 = 2k+3 $, $ n_2 = k $.
• The result implies $ \mathrm{wid}(C) \geq \sum_{j=1}^{l} j^2 $, a stronger lower bound than previously known.
• The flop-flop autoequivalence is not isomorphic to the generalized twist functor $ T_{\mathcal{E}} $, but to its inverse $ T_{\mathcal{E}}^{-1} $, correcting a prior error.
• The kernel of the non-commutative twist functor deforms to a composition of spherical twists, and their Hilbert polynomials match, confirming the formula for $ \mathrm{wid}(C) $.