Asymptotic geometry of moduli spaces of curves

几种重要的几何结构可以通过它们的扭量空间来构造和研究，即作为复流形中（真实的）有理曲线的参数空间。自然产生的例子，这些几何，其中包括hyperkaehler度量，是非常重要的意义，在几个分支的数学和数学物理：例如在表示论的多个品种，希钦的模空间在代数几何和可积系统理论，规范理论的模空间的单极子和瞬子在数学物理。上述例子中的许多也可以被构造为具有反全纯对合的复3重高阶亏格曲线的模空间，本项目的目的是研究这类模空间的几何性质;更准确地说，本文研究了真实的代数曲线的Hilbert格式的光滑轨迹的整体和渐近微分几何（满足一定的稳定性条件）（非紧）流形，特别是在3-folds中。主要研究目标如下：1）获得了几个有趣的微分几何结构的新例子，包括超Kaehler度量和四元数Kaehler度量以及复结构。2）研究了这些新例子的整体性质，特别是它们的完备性。3）通过对相应的3 $-fold的紧化来研究作为曲线的模空间而产生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开放的问题（如森和瓦法-威滕代数）有关的渐近行为的物理相关的流形，如希钦或的模空间。

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.