Algebraic Geometry Inspired by Physics

本文讨论了代数几何中模理论的三个研究课题。最近弦理论对代数几何的巨大影响导致了新的模空间研究的引入，研究它们的新技术，以及根据它们定义的有趣的“量子”不变量。这一建议受到物理学的启发，但仍然坚定地以代数几何为基础。模空间是将几何对象参数化的空间。第一个项目涉及空间参数化的全纯映射，从黎曼球到一个复杂的投影流形，以及“Gromov-Witten”不变量，可以在这样的空间上成为积分。第二部分讨论了全纯对应的空间参数化（即黎曼曲面与复射影流形的全纯映射一起位于黎曼球面上）作为正属黎曼曲面的Gromov-Witten不变量的替代来源。最后一个课题是对模空间的研究，推广了代数曲面上全纯向量束的空间。

Three projects in the theory of moduli in algebraic geometryare addressed in this proposal. The recent considerable influence ofstring theory on algebraic geometry has led to the introduction ofnew moduli spaces to study, new techniques for studying them, andinteresting ``quantum'' invariants defined in terms of them.This proposal is inspired by the physics, but remains firmly groundedin algebraic geometry.Moduli spaces are spaces that parametrize geometric objects.The first project deals with the spaces parametrizingholomorphic maps from the Riemann sphere to a complex projectivemanifold, and the ``Gromov-Witten'' invariants that can becomputed as integrals on such spaces. The second deals withthe spaces parametrizing holomorphic correspondences (i.e.Riemann surfaces lying over the Riemann sphere together witha holomorphic map to a complex projective manifold) as analternative source of Gromov-Witten invariants for Riemann surfacesof positive genus. The last project is an investigation ofmoduli spaces generalizing the spaces of holomorphicvector bundles on an algebraic surface.