标题： 可注入性定理和立方下降对于概形、堆栈和解析空间 显示英文标题

标题： Injectivity theorems and cubical descent for schemes, stacks, and analytic spaces

摘要： 我们证明了在概形、代数堆栈、形式概形、复解析空间的半解析芽、刚体解析空间、Berkovich空间以及在域上弱有限类型局部的adic空间中，广义法线交叉对的相对内射性、无挠性和消没定理。 我们在上述所有空间类别中给出了这些定理的统一证明，这些定理以前仅在代数簇和复解析空间中已知，这是Ambro和Fujino工作的结果。 Ambro和Fujino的结果在(半-)对数规范对的极小模型程序的基本定理和准对数结构理论的证明中是关键的。 我们的结果解决了将这些关于(半-)对数规范对和准对数结构的结果扩展到代数簇和复解析空间之外的重要障碍。 为了证明我们最一般的内射性定理，我们将Guillén和Navarro Aznar的一个准则推广到上述所有空间类别中，该准则用于表征在光滑代数簇上定义的函子何时能扩展到所有代数簇。 这个扩展结果使用了立方超解，我们在上述所有空间类别中构造了这些立方超解。 我们的扩展结果非常普遍，并且具有独立的兴趣。 我们使用这个扩展结果来证明广义法线交叉对的内射性定理。 我们还将这个扩展结果应用于建立这些空间类别中Deligne-Du Bois复形的理论基础，并在概形和刚体解析空间的(pro-)etale上同调上构造一个权重滤过。 这些结果在上述所有空间类别中确立了Deligne-Hodge理论的一些方面。 显示英文摘要

摘要： We prove relative injectivity, torsion-freeness, and vanishing theorems for generalized normal crossing pairs on schemes, algebraic stacks, formal schemes, semianalytic germs of complex analytic spaces, rigid analytic spaces, Berkovich spaces, and adic spaces locally of weakly finite type over a field, all in equal characteristic zero. We give a uniform proof for all these theorems in all the categories of spaces mentioned above, which were previously only known for varieties and complex analytic spaces due to work of Ambro and Fujino. Ambro and Fujino's results are integral in the proofs of the fundamental theorems of the minimal model program for (semi-)log canonical pairs and the theory of quasi-log structures. Our results resolve a significant barrier to extending these results on (semi-)log canonical pairs and quasi-log structures beyond the setting of varieties and complex analytic spaces. In order to prove our most general injectivity theorems, we generalize to all these categories of spaces a criterion due to Guill\'en and Navarro Aznar characterizing when functors defined on smooth varieties extend to all varieties. This extension result uses cubical hyperresolutions, which we construct in all categories of spaces mentioned above. Our extension result is very general and is of independent interest. We use this extension result to prove our injectivity theorems for generalized normal crossing pairs. We also apply our extension result to develop the theoretical foundations for the Deligne-Du Bois complex in these categories of spaces and to construct a weight filtration on the (pro-)\'etale cohomology of schemes and rigid analytic spaces. These results establish some aspects of Deligne-Hodge theory in all categories of spaces mentioned above.