An update on semisimple quantum cohomology and F-manifolds
An update on semisimple quantum cohomology and F-manifolds
复制标题
半简单量子上同调和 F 流形的更新
DOI：
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
C. Teleman
中科院分区：
文献类型：
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作者：
C. Hertling;Y. Manin;C. Teleman
摘要在本说明的第一部分中，我们证明拜耳-马宁定理 1.8.1 可以通过以下方式得到加强：如果射影代数流形 V 的偶量子上同调一般是半单的，则 V 没有奇上同调并且是 Hodge-Tate 型。特别是，这回答了 G. Ciolli 讨论的问题。在第二部分中，我们证明具有给定超交换关联$$的解析（或形式）超流形M
数学{O}_M
$$-其切线束上的双线性乘法$$
数学{T}_M
$$ 是赫特林-马宁意义上的 F 流形，当且仅当它的谱覆盖（作为余切丛 TM* 的解析子空间）是最大维数各向同性时。这回答了 V. Ginzburg 的问题。最后，我们在镜像对称和 Fano 变种的 Landau-Ginzburg 模型的背景下讨论这些结果。
AbstractIn the first section of this note, we show that Theorem 1.8.1 of Bayer-Manin can be strengthened in the following way: If the even quantum cohomology of a projective algebraic manifold V is generically semisimple, then V has no odd cohomology and is of Hodge-Tate type. In particular, this answers a question discussed by G. Ciolli. In the second section, we prove that an analytic (or formal ) supermanifold M with a given supercommutative associative$$
mathcal{O}_M
$$-bilinear multiplication on its tangent sheaf$$
mathcal{T}_M
$$is an F-manifold in the sense of Hertling-Manin if and only if its spectral cover, as an analytic subspace of the cotangent bundle TM*, is coisotropic of maximal dimension. This answers a question of V. Ginzburg. Finally, we discuss these results in the context of mirror symmetry and Landau-Ginzburg models for Fano varieties.