簇的Grothendieck环与特征处的数值不变量
Grothendieck ring of varieties and at-the-characteristic numerical invariants
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中文总结 AI 辅助
本文在正特征弱奇点消解假设下，证明Hodge数等特征处数值不变量可由Grothendieck环同态给出，并回答了两个相关问题，附录中无条件证明低维双有理Calabi--Yau簇的Hodge数相同。
中文摘要 AI 辅助
假设正特征下存在弱形式的奇点消解，我们证明若干特征处的数值不变量（包括Hodge数）由簇的Grothendieck环的同态给出。在相同假设下，我们得到两个推论：（i）$\u005e K$-等价的簇具有相同的Hodge数，回答了de Fernex--Mere的问题；（ii）泛同胚的簇在Grothendieck环中不必具有相同的类，回答了Nicaise--Sebag的问题。在附录中，我们通过证明维数至多四的双有理Calabi--Yau簇具有相同的Hodge数，为（i）提供了无条件证据。
英文摘要
Assuming a weak form of resolution of singularities in positive characteristic, we show several at-the-characteristic numerical invariants, including Hodge numbers, are given by homomorphisms out of the Grothendieck ring of varieties. Under the same assumption, we obtain two consequences: (i) $\widehat K$-equivalent varieties have the same Hodge numbers, answering a question of de Fernex--Mere; and (ii) universally homeomorphic varieties need not have the same class in the Grothendieck ring of varieties, answering a question of Nicaise--Sebag. In the appendix, we provide unconditional evidence for (i) by proving that birational Calabi--Yau varieties of dimension at most four have the same Hodge numbers.
发表机构
- University of Toronto（多伦多大学）
- University of Waterloo（滑铁卢大学）
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