Equivalence of Mirror Families Constructed from Toric Degenerations of Flag Varieties
Equivalence of Mirror Families Constructed from Toric Degenerations of Flag Varieties
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旗品种环面退化构造的镜像族的等价性
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
Joseph P. Rusinko
中科院分区：
文献类型：
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作者：
Joseph P. Rusinko
Batyrev等人利用全旗品种G/B的小环向退化构建了一个Calabi-Yau品种家族。他们推测这个家族是G/B中一般反正则超曲面的镜像。最近，Alexeev和Brion，作为他们研究球形变种的环向退化的一部分，构造了许多G/B的退化。对于任何这样的退化，我们都构造了一个变种族，并证明它在小情况下与巴特列夫的变种族是一致的。我们证明任何两个这样的家族都是出生的，从而证明镜像家族与退化的选择无关。所涉及的两国地图与Berenstein和Zelevinsky将热带地图几何提升为完全正变异之间的地图密切相关。
Batyrev et al. constructed a family of Calabi–Yau varieties using small toric degenerations of the full flag variety G/B. They conjecture this family to be mirror to generic anticanonical hypersurfaces in G/B. Recently, Alexeev and Brion, as a part of their work on toric degenerations of spherical varieties, have constructed many degenerations of G/B. For any such degeneration we construct a family of varieties, which we prove coincides with Batyrev’s in the small case. We prove that any two such families are birational, thus proving that mirror families are independent of the choice of degeneration. The birational maps involved are closely related to Berenstein and Zelevinsky’s geometric lifting of tropical maps to maps between totally positive varieties.