Moduli of products of stable varieties

Abstract We study the moduli space of a product of stable varieties over the field of complex numbers, as defined via the minimal model program. Our main results are: (a) taking products gives a well-defined morphism from the product of moduli spaces of stable varieties to the moduli space of a product of stable varieties; (b) this map is always finite étale; and (c) this map very often is an isomorphism. Our results generalize and complete the work of Van Opstall in dimension $1$. The local results rely on a study of the cotangent complex using some derived algebro-geometric methods, while the global ones use some differential-geometric input.

摘要：我们研究在复数域上稳定簇的乘积的模空间，这是通过极小模型纲领来定义的。我们的主要结果如下：(a) 取乘积给出了从稳定簇的模空间的乘积到稳定簇乘积的模空间的一个定义良好的态射；(b) 这个映射总是有限平展的；(c) 这个映射经常是一个同构。我们的结果推广并完善了范·奥普斯塔尔在一维情形下的工作。局部结果依赖于使用一些导出代数几何方法对余切复形的研究，而整体结果使用了一些微分几何的输入。