Relative Stable Pairs and a Non-Calabi–Yau Wall Crossing

Abstract

Abstract Let $Y$ be a smooth projective threefold, and let $f:Y\to X$ be a birational map with $Rf_*\mathcal {O}_Y=\mathcal {O}_X$. When $Y$ is Calabi–Yau, Bryan–Steinberg (BS) defined enumerative invariants associated to such maps called $f$-relative stable (or BS) invariants. When $X$ has Gorenstein singularities and $f$ has relative dimension one, they compared these invariants to the Donaldson–Thomas, or equivalently Pandharipande–Thomas invariants (PT) of $Y$. We define BS invariants for maps $f$ as above without assuming that $Y$ is Calabi–Yau. For $X$ with Gorenstein and rational singularities, $f$ of relative dimension one, and for insertions from $X$ and arbitrary descendant levels, we conjecture a relation between the generating functions of BS and PT of $Y$. We check the conjecture for the contraction $f: Y\to X$ of a rational curve $C$ with normal bundle $N_{C/Y}\cong \mathcal {O}_C(-1)^{\oplus 2}$ using degeneration and localization techniques to reduce to a Calabi–Yau situation, which we then treat using Joyce’s motivic Hall algebra.