Non-commutative width and Gopakumar-Vafa invariants
In the paper under review, the author describes a relation between two types of invariants involving birational flopping contractions on smooth complex three-folds: the non-commutative width introduced by \textit{W. Donovan} and \textit{M. Wemyss} [Duke Math. J. 165, No. 8, 1397--1474 (2016; Zbl 1346.14031)] and the genus zero Gopakumar-Vafa invariants defined by \textit{S. Katz} [J. Differ. Geom. 79, No. 2, 185--195 (2008; Zbl 1142.32011)]. Let \(f:X\to Y\) be a birational flopping contraction from a smooth quasi-projective complex three-fold \(X\), contracting a single rational curve \(\mathbb P^1\cong C\subset X\) to a single point \(p\in Y\). The contraction algebra associated to \(f\), denoted by \(A_{\mathrm{con}}\), was introduced by Donovan-Wemyss in the above mentioned paper and it is a finite-dimensional algebra given by the universal non-commutative deformation algebra of the curve \(C\) in \(X\). The non-commutative width and the commutative width are then defined, respectively, as \[ \mathrm{wid}(C):=\dim_{\mathbb C} A_{\mathrm{con}}, \; \mathrm{cwid}(C):=\dim_{\mathbb C} A_{\mathrm{con}}^{\mathrm{ab}}. \] The paper's main result is Theorem 1.1, where the author shows the following formulas \[ \mathrm{wid}(C)=\sum_{j=1}^l j^2 \cdot n_j, \; \mathrm{cwid}(C)=n_1, \] where the \(n_j\)'s are the genus zero Gopakumar-Vafa invariants defined by Katz as virtual numbers of one dimensional stable sheaves on \(X\) and \(l\) is the scheme theoretic length of \(f^{-1}(p)\) at \(C\). The proof of the above theorem combines results from Donovan-Wemyss work with deformation theoretic arguments.
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