Let $V,W$ be a pair of smooth varieties. We want to compare curve counts on $V\times W$ with those on $V$ and $W$. The product formula in Gromov–Witten theory compares the virtual fundamental classes of stable maps to a product ${\overline{M}}_{g,n}(V\times W)$ to the product of stable maps ${\overline{M}}_{g,n}(V)\times {\overline{M}}_{g,n}(W)$. We prove the analogous theorem for log stable maps to log smooth varieties $V,W$.

This extends results of Y.P. Lee and F. Qu, who introduced this formula after K. Behrend. We introduce “log normal cones” and “log virtual fundamental classes,” as well as modified versions of standard intersection-theoretic machinery adapted to log geometry.

让$V,W$是一对光滑的品种。我们想比较曲线计数$V\times W$与那些$V$和$W$. Gromov-Witten 理论中的乘积公式将稳定映射的虚拟基本类与乘积进行比较${\overline{米}}_{G,n}(V\times W)$稳定地图的产物${\overline{米}}_{G,n}(V)\times {\overline{米}}_{G,n}(W)$. 我们证明了对数稳定映射的类似定理来对数平滑变体$V,W$.

这扩展了 YP Lee 和 F. Qu 的结果，他们在 K. Behrend 之后引入了这个公式。我们介绍了“log normal cones”和“log virtual fundamental classes”，以及适用于对数几何的标准交集理论机制的修改版本。