For a nonsingular threefold \(X\) the stable pair invariants are defined by virtual integration over the moduli space of stable pairs \((F,s)\) where \(F\) is a pure 1-dimensional sheaf on \(X\) and \(s:\mathcal{O}_X\to F\) is a section of \(F\) with 0-dimensional cokernel. Denote by \(P_n(X,\beta)\) this moduli space, where \(\beta \in H_2(X,\mathbb{Z})\) is the class of the support of \(F\) and \(n\) is the Euler characteristic of \(F\). For fixed \(\beta\) and \(\gamma_i\in H^*(X,\mathbb{Z})\) let \(Z_\beta^X(\prod_{j=1}^k \tau_{i_j}(\gamma_j))\) be the partition function of the stable pair invariants with \(k\) descendents \(\tau_{i_j}(\gamma_j)\). It is a Laurent series in a variable \(q\). It was conjectured in [R. Pandharipande andR. P. Thomas, Geom. Topol. 13, No. 4, 1835–1876 (2009;Zbl 1195.14073)] that \(Z_\beta^X(\prod_{j=1}^k \tau_{i_j}(\gamma_j))\) is a rational function in \(q\). The conjecture is known for the insertions of the form \(\tau_0(\gamma_i)\) when \(X\) is a toric or a Calabi-Yau threefold. If \(X\) is a local curve the conjecture is known for the insertions \(\tau_{>0}(\gamma_i)\). Rationality plays a crucial role in the study of Gromov-Witten/stable pair correspondence.
The paper under review proves a stronger equivariant version of this conjecture in the case that \(X\) is a toric threefold. The proof is based on the analysis of the capped 3-leg descendent vertex. The capped vertex was used in proving Gromov-Witten/Donaldson-Thomas correspondence by Maulik et al. The paper under review uses two geometric constraints involving \(\mathcal{A}_n\)-surfaces and Hirzebruch surfaces to reduce to the 1-leg case studied before in the local curve theory of stable pairs by the authors. It is proven in the paper under review that the partition function of capped 3-leg descendant vertex in the stable pair theory is a rational function in \(q\) and the equivariant variables. The rationality of the capped descendent vertex does not hold in Gromov-Witten and Donaldson-Thomas theories.
Suppose that \(X\) is a toric nonsingular projective threefold and \(S \subset X\) is a nonsingular anticanonical divisor isomorphic to a \(K3\) surface. As another consequence of the rationality result above, the paper under review proves that a certain partition function of stable pair invariants for the log Calabi-Yau geometry \((X,S)\) is also a rational function.

Reviewer: Amin Gholampour (College Park)

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