Moduli spaces of stable and unstable maps to curves and surfaces

模空间自然而然地出现在几何中的分类问题中，并在许多不同的领域中扮演着重要的角色，特别是在代数、微分和辛几何中。这项研究的目的是利用代数几何、微分几何和辛拓扑的思想和方法，利用三位导师的互补专长，研究从复射影曲线到低维复射影簇的全纯映射的模空间，允许其域和目标中的奇点。模问题，例如同构的非奇异复射影曲线的分类，或双全纯的等价紧致Riemann曲面的分类，通常可以分解为一些基本步骤。第一步是找到尽可能多的待分类对象的离散不变量(对于非奇异复杂射影曲线，亏格是唯一的离散不变量)。第二步是确定所有离散不变量的值，并试图构造一个模空间；即，一个代数簇(或更一般的几何对象)，其点以一种自然的方式对应于待分类对象的等价类。这对于非奇异曲线很有效(至少如果我们愿意使用“粗糙的”模空间)，但如果我们想要包含奇异曲线，则需要更多的注意。具有非常弱奇点的复射影曲线(所谓的稳定曲线)可以很容易地包括在内；不同属的稳定曲线的模空间本身就是射影变种，其计数几何在过去几十年中得到了深入的研究。然而，如果我们包含具有更严重奇点的曲线，那么模空间的存在就太不可能了，除非我们愿意用更复杂、更不容易处理的几何对象来取代空间，这些几何对象被称为堆栈。模堆栈可以具有相关联的模空间，但通常的情况是甚至不存在粗模空间。传统上，为了构造模空间，不稳定的对象被忽略，因此分类问题的最后一步是了解哪些对象是(半)稳定的并且将出现在模空间中。然而，近年来发展了一些新的方法(使用“非约化几何不变量理论”)，这种方法可以构造不稳定物体的模空间。更准确地说，分类问题的最后一步变成了给每个不稳定的物体分配一个精化版本的‘Hard-Narasimhan型’，以及构造固定的Hard-Narasimhan型的模空间…弗朗西斯·基尔万的学生约书亚·杰克逊2019年的论文研究了不稳定复杂射影曲线的模空间。过去30年来，一个卓有成效的研究领域不是仅仅考虑曲线的模空间，而是考虑曲线到固定目标射影变种X的映射，通常被认为是非奇异的。同样，如果我们限制到区域曲线只有非常温和的奇点的‘稳定映射’，那么我们可以希望构造粗模空间并定义和计算X的计数不变量：‘Gromov-Witten不变量’，它们在代数几何和辛拓扑(以及弦理论)中已经并继续发挥着如此重要的作用。当X是单点时，我们恢复了稳定曲线的模空间。这个项目的目的是将约书亚·杰克逊的工作从0维扩展到1维和2维，构造和研究不稳定映射的模空间到低维射影变种X(其中X本身也可能变化)，并引入辛方法和微分方法以及代数方法来完成这项工作。本项目属于EPSRC数学科学研究领域

Moduli spaces arise naturally in classification problems in geometry, and play important roles in many different areas, in particular algebraic, differential and symplectic geometry. The aim of this research project is to use ideas and methods from algebraic geometry, differential geometry and symplectic topology, exploiting the complementary areas of expertise of the three supervisors, to study moduli spaces of holomorphic maps from complex projective curves to low-dimensional complex projective varieties, allowing singularities in their domains and targets. A moduli problem, for example the classification of nonsingular complex projective curves up to isomorphism, or equivalently compact Riemann surfaces up to biholomorphism, can usually be resolved into some basic steps. The first step is to find as many discrete invariants of the objects to be classified as possible (in the case of nonsingular complex projective curves the genus is the only discrete invariant). The second step is to fix the values of all the discrete invariants and try to construct a moduli space; that is, an algebraic variety (or some more general geometric object) whose points correspond in a natural way to the equivalence classes of the objects to be classified. This works nicely for nonsingular curves (at least if we are willing to work with 'coarse' moduli spaces), but if we want to include singular curves then much more care is needed. Complex projective curves with very mild singularities (so-called stable curves) can be included without difficulty; the moduli spaces of stable curves of different genera are themselves projective varieties whose enumerative geometry has been intensively studied over the last decades. However if we include curves with more serious singularities, then the existence of a moduli space is too much to hope for, unless we are willing to replace spaces by much more sophisticated and less tractable geometric objects called stacks. A moduli stack may have an associated moduli space, but it is often the case that not even a coarse moduli space will exist. Traditionally, 'unstable' objects are left out in order to construct a moduli space, so the final step in the classification problem is to understand which objects are (semi)stable and will appear in the moduli space. However in recent years new methods (using 'non-reductive geometric invariant theory') have been developed which allow the construction of moduli spaces of unstable objects ... more precisely, the final step in the classification problem becomes the assignment of a refined version of 'Harder-Narasimhan type' to each unstable object, and the construction of moduli spaces of fixed Harder-Narasimhan type ... and Frances Kirwan's student Joshua Jackson's 2019 thesis studied moduli spaces of unstable complex projective curves.Instead of just considering moduli of curves, a hugely fruitful area of research over the last three decades has been to consider maps of curves to a fixed target projective variety X, usually taken to be nonsingular. Again if we restrict to 'stable maps' where the domain curve has only very mild singularities, then we can hope to construct coarse moduli spaces and define and calculate enumerative invariants of X: the 'Gromov-Witten invariants' which have played and continue to play such important roles in algebraic geometry and symplectic topology (as well as in string theory). When X is a single point we recover the moduli spaces of stable curves. This project aims to extend Joshua Jackson's work from the case when X has dimension 0 to dimensions 1 and 2, constructing and studying moduli spaces of unstable maps to low-dimensional projective varieties X (where X itself may also vary), and to bring in symplectic and differential methods as well as algebraic ones to do this.This project falls within the EPSRC Mathematical Sciences research area