Algebraic cycles and coisotropic subvarieties on irreducible symplectic manifolds

该研究项目涉及代数几何中一个中心和非常活跃的研究课题：不可约（全纯）辛流形上的代数圈。本文主要研究两个密切相关的问题：一方面是ISM中代数上各向同性的子簇的存在性，另一方面是投射不可约辛流形的Chow环的Bloch-Beilinson型滤子的分裂性. Voisin提出了一个影响深远的代数计划，其中共各向同性子簇和零圈起着特殊的作用。这些都是美丽的几何图形，它们的证明将是理解不可约辛流形上代数循环的基石。此外，Voisin的框架是相当明确的，近年来有许多进展hyperkähler理论适用于这种应用，所以现在似乎是正确的利用这些方法取得进展领域的代数周期。这些结构已在实例中得到部分验证，本文的目的是通过一般的变形理论方法，对超kähler簇的代数圈的构造作出实质性的贡献，并解决更深远的关于空间Bloch簇分裂的问题。关于不可约辛流形的具体极大代数族的Beilinson滤子，Voisin的滤子的完整证明是遥不可及的。因此，长期的目标应该是证明Voisin的已知变形类型的假设。

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.