Given a stability condition on a smooth projective variety X, we construct a family of stability conditions on $X\times C$, where C is a smooth projective curve. In particular, this gives the existence of stability conditions on arbitrary products of curves. The proof uses, by following an idea of Toda, the positivity lemma established by Bayer and Macrì and weak stability conditions on the Abramovich-Polishchuk heart of a bounded t-structure in $D\left(X\times C\right)$.

中文翻译：

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产品品种的稳定性条件

给定一个光滑射影变种X的稳定条件，我们构造了一个稳定条件族$X\times C$，其中C是平滑的投影曲线。特别地，这给出了在任意曲线产品上的稳定性条件的存在。该证明遵循Toda的思想，使用了由Bayer和Macrì建立的正定性引理，以及在有限t结构的Abramovich-Polishchuk心脏上的弱稳定性条件。$d（X\times C）$。