Moduli spaces of multi-polarised projective varieties

模空间自然而然地出现在代数和微分几何的分类问题中，并在许多不同的领域中扮演着重要的角色。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)。第二步是固定离散不变量，并试图构造一个模空间，即一个代数簇，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有效，尽管要包含奇异曲线需要更多的注意。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。复射影曲线的分类是代数几何中最基本的分类问题之一：(定维)复射影簇的分类问题。对于极化射影簇(X，L)，其中L是射影簇X上的一个充分线丛，并试图施加适当的稳定性条件，以构造(半)稳定的极化复射影簇的模空间是很常见的。(在X是亏格至少为2的非奇异复射影曲线的情况下，我们可以选择标准线丛的合适的幂作为极化)。近年来，通过结合代数、微分和辛几何的方法，在这个方向上取得了非常显著的进展，将(X，L)的所谓K-稳定性与X上特殊的Kahler度量的存在性联系起来。本研究的目的是研究复射影变种X的模空间，它不仅具有一个充分的线丛L，而且具有有限多个表示X的Nelon-Severi群的基的(子集)的充分线丛。给定X上的一个充分线丛L，我们可以用L的一个足够大的幂的截面将X嵌入到射影空间中。然后，人们可以希望应用来自20世纪60年代发展起来的Mumford几何不变理论(GIT)的思想，通过约化群作用来构造和研究代数簇的商，以定义关联的特殊线性群在表示该投影空间的投影子方案的与X相同的希尔伯特多项式上的(半)稳定性的概念。然而，这些概念取决于所选择的L的幂，并且没有明显的几何解释；K-(半)稳定性定义背后的动机是当线丛的幂趋于无穷时，提供这种GIT(半)稳定性的某种渐近版本。给定X上的几个不同的充分线丛，我们可以取这些线丛的幂的张量积的截面来将X嵌入到射影环簇中。然后，我们可以研究在这些情况下相应的环簇上的相应的群作用，以及K-稳定性的类似物。该项目旨在研究dimX=2时的情况，这是传统情况下仅有一个充值线束尚未覆盖的最低维度。该项目属于EPSRC几何和拓扑研究领域。没有任何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.