Algebraic cycles and coisotropic subvarieties on irreducible symplectic manifolds

该研究项目涉及代数几何中的一个中心和非常活跃的研究主题：不可约(全纯)辛流形上的代数圈。它主要研究两个密切相关的问题：一方面，ISM中代数余迷子簇的存在性；另一方面，Bloch-Beilinson型分裂渗入射影不可约辛流形的Chow环。我们的计划是受到Claire Voisin最近关于射影不可约辛流形的Chow环结构的一组引人注目的猜想的启发。Visin提出了一个影响深远的猜想方案，在该方案中，上各向同性亚簇和零环扮演着特殊的角色。这些都是美丽的猜想，它们的证明将是理解不可约辛流形上的代数圈的基石。此外，Voisin的框架是相当明确的，近年来在Hyperkähler理论中已经有了许多适合于这类应用的进展，所以现在似乎是利用这些方法来在代数圈领域取得进展的时候了。这些猜想已经在实例中得到了部分验证，但可得到的一般性结果很少。其目的是用一般形变理论方法对Hyperkähler簇的代数圈的构造做出实质性贡献，并解决关于不可约辛流形的具体极大代数族的猜想Bloch-Beilinson滤子分裂的更深远的问题。因此，长期目标应该是证明Voisin关于已知变形类型的猜想。

The research project is concerned with a central and very active research topic in algebraic geometry: algebraic cycles on irreducible (holomorphic) symplectic manifolds. It mainly pursues two closely related questions: on the one hand, the existence of algebraically coisotropic subvarieties in ISMs and on the other hand, the splitting of Bloch-Beilinson type filtrations the Chow ring of a projective irreducible symplectic manifold.Our project is inspired by a remarkable recent set of conjectures by Claire Voisin concerning the structure of the Chow ring of a projective irreducible symplectic manifold. Voisin presented a far reaching conjectural program where coisotropic subvarieties as well as zero-cycles play a special role. These are beautiful conjectures and their proof would be a cornerstone in the understanding of algebraic cycles on irreducible symplectic manifolds. Moreover, Voisin's framework is rather explicit and in recent years there have been many advances in hyperkähler theory suitable to applications of this kind, so the moment seems to be right to exploit these methods to make progress on the field of algebraic cycles. These conjectures have partially been verified in examples, but there are only very few general results available.The goal is to substantially contribute to the construction of algebraic cycles for hyperkähler varieties by general deformation theoretic methods and to tackle more far reaching problems concerning the splitting of the conjectural Bloch-Beilinson filtration for a concrete maximal algebraic family of irreducible symplectic manifolds.A complete proof of Voisin's conjectures is out of reach. So the long-term goal should be to prove Voisin's conjectures for the known deformation types.