拟阵构建集的切类
Tangent classes for matroid building sets
AI总结：
本文为任意拟阵构建集定义了切类，推广了de Concini-Procesi奇妙模型的切丛类，并证明了其满足拟阵Hirzebruch-Riemann-Roch性质，应用于Chern数不等式。
摘要：
设 \(M\) 是有限基集 \(E\) 上的无环拟阵，\(\G\) 是包含顶平坦 \(E\) 的构建集。我们在 \(K\)-环 \(K(M,\G)\) 中定义切类 \(T_{M,\G}\)，它将 de Concini-Procesi 奇妙模型的切丛类从可实现拟阵推广到任意带构建集的拟阵。类 \(T_{M,\G}\) 满足拟阵 Hirzebruch-Riemann-Roch 性质。更精确地，其 Hirzebruch 类 \[ \operatorname{ch}(\lambda_y T_{M,\G}^{\vee})\operatorname{td}(T_{M,\G}) \] 特化为 Todd 类并计算 \((M,\G)\) 的 Chow 多项式。在可实现情形，这些等式与相应奇妙模型上的通常切丛计算一致。作为应用，我们证明了 \(T_{M,\G}\) 的 Chern 数不等式，包括关于超平面类的 Miyaoka-Yau 型不等式。
英文摘要：
Significant combinatorial constraints and structure on matroids can come from notions in algebraic geometry, even without the matroids themselves being representable. Let \(M\) be a loopless matroid on a finite ground set \(E\), and let \(\G\) be a building set containing the top flat \(E\). We define a tangent class \(T_{M,\G}\) in the \(K\)-ring \(K(M,\G)\), which extends the tangent bundle class of the de Concini--Procesi wonderful model from realizable matroids to arbitrary matroids with building sets. The class \(T_{M,\G}\) satisfies a matroidal Hirzebruch--Riemann--Roch package. More precisely, its Hirzebruch class specializes to the Todd class and computes the Chow polynomial of \((M,\G)\). In the realizable case, these identities agree with the usual tangent-bundle computations on the corresponding wonderful model. As an application, we prove Chern-number inequalities for \(T_{M,\G}\), including a Miyaoka--Yau type inequality with respect to the hyperplane class.
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