论文标题
复曲面束和条件环
Cohomology rings of toric bundles and the ring of conditions
论文作者
论文摘要
著名的bkk定理表达了牛顿多型相应系统的混合体的混合体的混合体系。Pukhlikov。Pukhlikov和第二作者注意到，$ \ Mathbb {c} $的平滑投射型折磨的共同体可以计算出bk {c} $可以计算出bkk theerem。这补充了关于曲曲面品种的共同描述，就像斯坦利 - 赖斯纳代数方面一样。 Sankaran和Uma将“ Stanley-Reisner描述”概括为曲折束的案例，即（不一定是代数）Torus主体捆绑包（不一定是代数）的束缚。我们提供了基于\ bkk定理的概括的复曲束的同种学环的描述，从而扩展了Pukhlikov和第二作者的方法。的确，对于福利束的底座的每个共同体学类别，我们都会获得BKK型定理。此外，我们的证明还依赖于满足庞加莱二元性的分级交换代数的描述。 根据曲曲面束的共同学环的计算，我们可以获得霍斯氏均匀空间的条件环以及Brion-kazarnovskii定理的版本。我们以许多例子结束了手稿。特别是，我们将结果应用于整个标志品种$ g/b $的曲折捆绑包。我们得到的描述概括了对福利品种的共同体戒指的相应描述，以及先前由Kaveh获得的完整国旗品种$ g/b $的描述。
The celebrated BKK Theorem expresses the number of roots of a system of generic Laurent polynomials in terms of the mixed volume of the corresponding system of Newton polytopes.Pukhlikov and the second author noticed that the cohomology ring of smooth projective toric varieties over $\mathbb{C}$ can be computed via the BKK Theorem. This complemented the known descriptions of the cohomology ring of toric varieties, like the one in terms of Stanley-Reisner algebras. Sankaran and Uma generalized the "Stanley-Reisner description" to the case of toric bundles, i.e. equivariant compactifications of (not necessarily algebraic) torus principal bundles. We provide a description of the cohomology ring of toric bundles which is based on a generalization of the \BKK Theorem, and thus extends the approach by Pukhlikov and the second author. Indeed, for every cohomology class of the base of the toric bundle, we obtain a BKK-type theorem. Furthermore, our proof relies on a description of graded-commutative algebras which satisfy Poincaré duality. From this computation of the cohomology ring of toric bundles, we obtain a description of the ring of conditions of horospherical homogeneous spaces as well as a version of Brion-Kazarnovskii theorem for them. We conclude the manuscript with a number of examples. In particular, we apply our results to toric bundles over a full flag variety $G/B$. The description that we get generalizes the corresponding description of the cohomology ring of toric varieties as well as the one of full flag varieties $G/B$ previously obtained by Kaveh.
