拟阵的格罗滕迪克权与K-理论正性
Grothendieck weights and K-theoretic positivity for matroids
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中文总结 AI 辅助
该研究通过格罗滕迪克权理论，证明拟阵相关重言丛欧拉示性数正性、推广外部活动复形并推导其K-多项式公式，还证明其Cohen-Macaulay性，部分回答了相关学者的问题。
中文摘要 AI 辅助
我们引入一种通过热带几何中产生的空间的拓扑来研究排列体环簇上K-理论正性的方法，关键要素是作者发展的格罗滕迪克权理论。我们用该方法证明两个正性结果：第一个结果是，与任意拟阵关联的重言丛被丰沛线丛扭曲后的欧拉示性数具有正性，这为一个猜想的消失定理提供了数值证据；第二个结果将Berget-Fink的外部活动复形（该复形最初针对一对拟阵定义）推广到无公共环的任意拟阵元组的情形，我们据此推导了其分次K-多项式的公式，该公式由拟阵的对偶重言商类的外幂表示，经变量替换后其系数符号交替。我们还利用Eur-Fink-Larson发展的组合几何的消失定理证明了每个此类复形的Cohen-Macaulay性，该证明即使在一对拟阵的情形下也是新的。作为应用，我们将重言商类的某些陈数解释为面对的计数，部分回答了Berget-Eur-Spink-Tseng提出的一个问题。
英文摘要
We introduce a method for studying $K$-theoretic positivity on permutohedral toric varieties through the topology of spaces arising in tropical geometry. The key ingredient is the theory of Grothendieck weights developed by the author. We prove two positivity results using this method. The first result is the positivity of the Euler characteristics of tautological bundles associated with an arbitrary matroid and twisted by a nef line bundle. This gives numerical evidence for a conjectural vanishing theorem. The second result generalizes the external activity complex of Berget--Fink, originally defined for a pair of matroids, to the case of any tuple of matroids with no common loop. We deduce a formula for its graded $K$-polynomial in terms of exterior powers of the dual tautological quotient classes of the matroids. After a change of variables, its coefficients alternate in sign. We also prove the Cohen--Macaulayness of each such complex using the vanishing theorems for combinatorial geometries developed by Eur--Fink--Larson. This proof is new even in the case of a pair of matroids. As an application, we interpret certain Chern numbers of tautological quotient classes as counts of facets, partially answering a question of Berget--Eur--Spink--Tseng.
发表机构
- The Ohio State University（俄亥俄州立大学）
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