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课题：不可约（全纯）辛流形上的代数循环。它主要研究两个密切相关的问题：一类是代数上的共同性子变异的存在性问题；另一类是投影不可约辛流形的Chow环上Bloch-Beilinson型滤波的分裂问题。我们的项目灵感来自于Claire Voisin最近关于一个射影不可约辛流形的Chow环结构的一系列非凡的猜想。Voisin提出了一个影响深远的猜想程序，其中共同性亚种和零环起着特殊的作用。这些都是美丽的猜想，它们的证明将成为理解不可约辛流形上的代数循环的基石。此外，Voisin的框架相当明确，近年来在hyperkähler理论中有许多适用于此类应用的进展，因此利用这些方法在代数循环领域取得进展的时机似乎是正确的。这些猜想已经在实例中得到了部分证实，但得到的一般结果却很少。目的是通过一般变形理论方法对hyperkähler型的代数循环的构造作出实质性的贡献，并解决关于不可约辛流形的具体极大代数族的猜想Bloch-Beilinson过滤的分裂的更深远的问题。对沃辛猜想的完整证明是遥不可及的。因此，长期目标应该是证明Voisin对已知变形类型的推测。

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.