A note on stability conditions on product-quotient surfaces

Source abstract

Let $X$ be a product-quotient surface, which is by definition a minimal resolution of $(C_1\times C_2)/G$, where $C_i$ are smooth curves of genus $\geq 2$ and $G$ is a finite group. When $G$ acts freely, we show that every product Bridgeland stability condition on $C_1\times C_2$ induces a tilt stability condition on $X$, and that, if $p_g(X)=q(X)=0$, every tilt stability condition on $X$ arises this way. When $(C_1\times C_2)/G$ has only $A_1$-singularities, we use induction \cite{MMS} and derived McKay correspondence \cite{BKR} to produce non-geometric stability conditions $σ_X$ on $X$, which satisfy the full support property and lie in the closure of the geometric chamber. Whether $σ_X$ also lies in the closure of the $\widetilde{\mathrm{GL}}^+(2,\mathbb{R})$-orbit of the tilt stability conditions depends on a single inequality, $ω_X^2 \ge n/8$, where $n$ is the number of $A_1$-singularities. When it fails, the geometric chamber is strictly larger than this orbit.