发表机构
Louisiana State University; KTH Royal Institute of Technology（路易斯安那州立大学; 皇家理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究均匀拟阵相关环面簇的奇异上同调环，构造显式基，证明修改后的扇满足强Lefschetz性质，并证明精细Hodge--Poincaré多项式的实根性，解决猜想并推广已知定理。
AI 中文摘要
我们研究了与均匀拟阵产生的若干扇相关的环面簇的奇异上同调环。这些环推广了拟阵的Chow环和增广Chow环。对于由均匀拟阵的增广Bergman扇产生的奇异上同调环，我们基于第一作者引入的均匀拟阵奇异上同调环的retral基，构造了一个显式基。我们证明了增广Bergman扇不产生满足拟射影强Lefschetz性质的奇异上同调环，而对扇进行适当修改后则可以。接着，我们研究了相应的精细Hodge--Poincaré多项式的零点。对于均匀拟阵，我们证明了与奇异上同调环和修改后的增广奇异上同调环相关的精细Hodge--Poincaré多项式都是实根的。前一结果解决了第一作者的一个猜想。这些结果推广了Brändén和第二作者关于均匀拟阵Chow多项式的实根性定理。最后，我们将增广奇异上同调环实根性的失效与Lefschetz性质的失效联系起来。
英文摘要
We study the singular cohomology rings of toric varieties associated with several fans arising from uniform matroids. These rings generalize the Chow and augmented Chow rings of matroids. For the singular cohomology ring arising from the augmented Bergman fan of a uniform matroid, we construct an explicit basis derived from the retral basis for the singular cohomology ring of a uniform matroid introduced by the first author. We prove that the augmented Bergman fan does not yield a singular cohomology ring that satisfies the quasi-projective Strong Lefschetz property, whereas a suitable modification of the fan does. We then investigate the zeros of the corresponding refined Hodge--Poincaré polynomials. For uniform matroids, we prove that the refined Hodge--Poincaré polynomials associated with both the singular cohomology ring and the modified augmented singular cohomology ring are real-rooted. The former result resolves a conjecture of the first author. These results extend the real-rootedness theorem of Brändén and the second author for the Chow polynomials of uniform matroids. Finally, we relate the failure of real-rootedness for the augmented singular cohomology ring to the failure of Lefschetz properties.
Comments27 pages
