摘要翻译:
本文构造并研究了平面上哈密顿李代数的某些变形对曲线族${cal C}/s$的相对对称幂${cal C}^{[bullet]}$和相对Jacobian${cal J}$的Chow群(即上同调)的作用。作为应用之一,我们证明了在单曲线$C$的情况下,此作用在$C^{[N]}$的Chow群上诱导出一个Lefschetz$\operatorname{sl}_2$-作用的积分形式。另一个应用给出了关于$C$的Jacobian$J$(关于Pontryagin积)上的0-圈环的一个新的分级,并给出了线上向量场李代数的一个作用。我们还定义了$ch^*({\cal C}^{[\bullet]})$和$ch^*({\cal J})$中的重言类群,并证明了Beauville在Math.AG/0204188中关于单曲线雅可比的性质的类似性质。我们还证明了算子的our代数保持重言圈的子环,并通过一些显式的微分算子作用于它们。
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英文标题:
《Algebraic cycles on the relative symmetric powers and on the relative
Jacobian of a family of curves. I》
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作者:
Alexander Polishchuk
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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:
In this paper we construct and study the actions of certain deformations of the Lie algebra of Hamiltonians on the plane on the Chow groups (resp., cohomology) of the relative symmetric powers ${\cal C}^{[\bullet]}$ and the relative Jacobian ${\cal J}$ of a family of curves ${\cal C}/S$. As one of the applications, we show that in the case of a single curve $C$ this action induces a integral form of a Lefschetz $\operatorname{sl}_2$-action on the Chow groups of $C^{[N]}$. Another application gives a new grading on the ring of 0-cycles on the Jacobian $J$ of $C$ (with respect to the Pontryagin product) and equips it with an action of the Lie algebra of vector fields on the line. We also define the groups of tautological classes in $CH^*({\cal C}^{[\bullet]})$ and in $CH^*({\cal J})$ and prove for them analogs of the properties established in the case of the Jacobian of a single curve by Beauville in math.AG/0204188. We also show that the our algebras of operators preserve the subrings of tautological cycles and act on them via some explicit differential operators.
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PDF链接:
https://arxiv.org/pdf/0704.2848