Asymptotic geometry of moduli spaces of curves

一些重要的几何结构可以通过它们的扭转空间来构造和研究，即作为复流形中(实)有理曲线的参数空间。自然产生的这些几何的例子，包括超Kaehler度量，在数学和数学物理的几个分支中具有重要意义：例如，表示理论中的箭形变分，代数几何和可积系统理论中的Hitchin模空间，数学物理中的单极和瞬子的规范理论模空间。许多上述例子也可以构造为具有反全纯对合复3重中的高亏格曲线的模空间。本课题的目的是研究这种模空间的几何性质；更确切地说，我们建议研究复(非紧)流形中实代数曲线(满足一定稳定性条件)的Hilbert格式的光滑轨迹的全局和渐近微分几何。主要研究目标如下：1)获得几个有趣的微分几何结构的新例子，包括Hyperkaehler和四元数Kaehler度量以及多复结构。2)研究这些新例子的全局性质，特别是它们的完备性。3)通过紧化相关的$$折叠来研究作为此类曲线模空间的流形的渐近几何和几何紧化。我们希望这种方法将允许回答与物理相关流形的渐近行为有关的公开问题(例如Sen猜想和Vafa-Witten猜想)，例如单极子或Hitchin模空间。

Several important geometric structures can be constructed and studied via their twistor space, i.e. as a parameter space of (real) rational curves in a complex manifold. Naturally arising examples of these geometries, which include hyperkaehler metrics, are of great significance in several branches of mathematics and mathematical physics: e.g. quiver varieties in representation theory, Hitchin's moduli spaces in algebraic geometry and integrable systems theory, gauge-theoretic moduli spaces of monopoles and instantons in mathematical physics. Many of the above-mentioned examples can also be constructed as moduli spaces of higher genus curves in a complex 3-fold equipped with an antiholomorphic involution.The aim of this project is to investigate the geometry of such moduli spaces; more precisely, we propose to study the global and asymptotic differential geometry of smooth loci of Hilbert schemes of real algebraic curves (satisfying certain stability conditions) in complex (non-compact) manifolds, particularly in 3-folds.The main research goals are as follows:1) to obtain new examples of several interesting differential-geometric structures, including hyperkaehler and quaternion-Kaehler metrics and pluricomplex structures.2) to investigate global properties of these new examples, in particular their completeness.3) to study the asymptotic geometry and geometric compactifications of manifolds arising as such moduli spaces of curves via compactification of the relevant $3$-fold. We hope that this approach will allow to answer open questions (e.g. the Sen and the Vafa-Witten conjectures) related to the asymptotic behaviour of physically relevant manifolds, such as monopole or Hitchin's moduli spaces.