Moduli spaces of multi-polarised projective varieties

模空间在代数几何和微分几何的分类问题中自然出现，并在许多不同的领域发挥着重要作用。一个模问题，例如非奇异复射影曲线到同构的分类，或等效紧致黎曼曲面到生物全纯的分类，通常可以分解成一些基本步骤。第一步是找到尽可能多的待分类对象的离散不变量（在非奇异复射影曲线的情况下，属是唯一的离散不变量）。第二步是固定离散不变量并尝试构造一个模空间；也就是说，一个代数变量，它的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好地工作，尽管包含奇异曲线需要更加小心。具有非常轻微奇点的复杂投影曲线（所谓的稳定曲线）可以毫不困难地包含在内；不同属的稳定曲线的模空间本身就是射影变体，其枚举几何在过去几十年中得到了广泛的研究。复射影曲线的分类是代数几何中最基本的分类问题之一：复射影变（固定维数）的分类问题。通常使用偏振射影变体（X，L），其中L是射影变体X上的一个充足的线束，并试图施加适当的稳定性条件，以便可以构造（半）稳定的偏振复射影变体的模空间。（当X是至少有2个属的非奇异复射影曲线时，我们可以选择一个合适的标准线束的幂作为偏振）。近年来，通过结合代数、微分和辛格几何的方法，将所谓的（X，L）的k稳定性与X上特殊Kahler度量的存在性联系起来，在这个方向上取得了非常重大的进展。本研究项目的目的是研究复射光变数X的模空间，它不仅具有一个充足的线束L，而是用有限多个足够的线束来表示X的Neron-Severi群的一个基（a的子集）。给定X上的一个足够的线束L，我们可以使用L的足够大幂的部分来将X嵌入射影空间中。然后，人们可以希望应用Mumford的几何不变理论（GIT）的思想，该理论在20世纪60年代发展起来，通过约化群作用来构造和研究代数变量的商，来定义相关的特殊线性群在希尔伯特方案上的作用（半）稳定性的概念，希尔伯特方案表示该射影空间的射影子方案，具有与x相同的希尔伯特多项式。然而，这些依赖于所选L的幂，并没有明显的几何解释；定义K-（半）稳定性背后的动机是在线束的幂趋于无穷时，提供这种GIT（半）稳定性的某种渐近版本。给定X上的几个不同的线束，我们可以取这些线束幂的张量积的部分来将X嵌入到射影环变中。然后，我们可以研究在这些情况下相应的环型上相应的群作用，以及k -稳定性的类似物。该项目的目的是在dimX=2的情况下研究这一点，dimX=2是最低的维度，传统情况下只有一个充足的线束。该项目属于EPSRC几何和拓扑研究领域。没有公司或合作者参与其中。

Moduli spaces arise naturally in classification problems in algebraic and differential geometry, and play important roles in many different areas. 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 discrete invariants and try to construct a moduli space; that is, an algebraic variety whose points correspond in a natural way to the equivalence classes of the objects to be classified. This works nicely for nonsingular curves, though to include singular curves 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. The classification of complex projective curves is part of one of the most fundamental classification problems in algebraic geometry: that of classifying complex projective varieties (of fixed dimension). It is usual to work with polarised projective varieties (X,L) where L is an ample line bundle over the projective variety X, and to try to impose suitable stability conditions so that moduli spaces of (semi)stable polarised complex projective varieties can be constructed. (In the case when X is a nonsingular complex projective curve of genus at least two then we can choose a suitable power of the canonical line bundle as the polarisation). Very significant advances in this direction have been made in recent years, by combining methods from algebraic, differential and symplectic geometry, relating the so-called K-stability of (X,L) to the existence of special Kahler metrics on X. The aim of this research project is to study moduli spaces of complex projective varieties X equipped not just with one ample line bundle L, but instead with finitely many ample line bundles representing a (subset of a) basis of the Neron-Severi group of X. Given one ample line bundle L on X, we can use the sections of a sufficiently large power of L to embed X in a projective space. Then one can hope to apply ideas coming from Mumford's geometric invariant theory (GIT), developed in the 1960s to construct and study quotients of algebraic varieties by reductive group actions, to define notions of (semi)stability for the action of the associated special linear group on the Hilbert scheme representing projective subschemes of this projective space with the same Hilbert polynomial as X. However these depend on the power of L chosen, and do not have obvious geometric interpretation; the motivation behind the definition of K-(semi)stability is to provide some sort of asymptotic version of this GIT (semi)stability as the power of the line bundle tends to infinity. Given several different ample line bundles on X we can take sections of tensor products of powers of these line bundles to embed X in projective toric varieties. We can then study the corresponding group actions on the corresponding toric varieties, and analogues of K-stability in these situations. The project aims to investigate this in the case when dimX=2, which is the lowest dimension which is not already covered by the traditional situation with just one ample line bundle.This project falls within the EPSRC Geometry and Topology research area. No companies or collaborators are involved.