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摘要翻译：
在极小Fano三重的情况下，我们证明了Dubrovin在1998年柏林ICM上提出的一个猜想。该猜想预言了以例外对象的类为基础表示的具有例外集合的Fano变体在$K_0$上的对称/交替欧拉特征配对与Dubrovin第二连接中消失圈的交配对一致。利用秩为1的Fano三倍的模性结果，我们证明了该猜想对于$v_{22}$是反协调度为22的最小Fano三倍，对于$v_5$是反协调度为40的最小Fano三倍是成立的，而对于v_{22}$是反协调度为22的最小Fano三倍，对于v_{22}$是反协调度为40的最小Fano三倍，对于v_{22}$是成立的。关于$\p^3$和三维二次曲面的猜想的真值已知；为了完整起见，我们考虑这些情况。
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英文标题：
《A remark on minimal Fano threefolds》
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作者：
V. Golyshev
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最新提交年份：
2008
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分类信息：
一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
We prove in the case of minimal Fano threefolds a conjecture stated by Dubrovin at the ICM 1998 in Berlin. The conjecture predicts that the symmetrized/alternated Euler characteristic pairing on $K_0$ of a Fano variety with an exceptional collection expressed in the basis of the classes of the exceptional objects coincides with the intersection pairing of the vanishing cycles in Dubrovin's second connection. We show that the conjecture holds for $V_{22}$, a minimal Fano threefold of anticanonical degree~22, and for $V_5$, the minimal Fano threefold of anticanonical degree~40, by applying the modularity result for rank 1 Fano threefolds. The truth of the conjecture for $\P ^3$ and the three--dimensional quadric is known; we consider these cases for the sake of completeness.
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PDF链接：
https://arxiv.org/pdf/0803.0031
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