Toda's local multiple cover conjecture for generalized Donaldson–Thomas invariants

Let $C\subset X$ be a reduced curve in a smooth projective Calabi–Yau $3$-fold, and let $n\in\mathbb{Z}$ and $\gamma\in H_2(C,\mathbb{Z})$. Let $N_{n,\gamma}\in\mathbb{Q}$ denote the local generalized Donaldson–Thomas invariant counting one-dimensional semistable sheaves with curve class $\gamma$ and Euler characteristic $n$. Write $k\mid(n,\gamma)$ when $k$ divides both $n$ and $\gamma$.

Toda's local multiple cover conjecture. We have

$$N_{n,\gamma}=\sum_{k\geq 1,\,k\mid(n,\gamma)}\frac{1}{k^2}N_{1,\gamma/k}.$$

This is the local version of the generalized DT multiple cover formula and was introduced to reduce curve-counting questions on Calabi–Yau threefolds to local geometry. The paper studies this formula for local curves with at worst nodal singularities and reduces it to the case of local trees of smooth rational curves, proving it in some cases.