Moduli spaces of multi-polarised projective varieties

多极化射影簇的模空间

• 批准号: 2580832
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2580832
• 关键词: Moduli; spaces; multi; polarised; projective

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.

模空间在代数和微分几何的分类问题中自然出现，并在许多不同领域发挥着重要作用。模问题，例如非奇异复射影曲线直至同构的分类，或等效紧致黎曼曲面直至双全纯的分类，通常可以解析为一些基本步骤。第一步是找到尽可能多的待分类对象的离散不变量（在非奇异复射影曲线的情况下，属是唯一的离散不变量）。第二步是修复离散不变量并尝试构造模空间；也就是说，一种代数簇，其点以自然方式对应于要分类的对象的等价类。这对于非奇异曲线非常有效，但要包含奇异曲线则需要更加小心。具有非常轻微奇点的复杂投影曲线（所谓的稳定曲线）可以毫无困难地包含在内；不同属的稳定曲线的模空间本身就是射影簇，其枚举几何在过去几十年中得到了深入研究。复射影曲线的分类是代数几何中最基本的分类问题之一的一部分：对复射影簇（固定维数）进行分类。通常使用偏振射影簇 (X,L)，其中 L 是射影簇 X 上的充足线丛，并尝试施加合适的稳定性条件，以便（半）稳定偏振复射影簇的模空间可以建造的。 （当X是至少为2的非奇异复射影曲线时，我们可以选择一个合适的正则线束幂作为偏振）。近年来，通过结合代数、微分和辛几何的方法，将 (X,L) 的所谓 K 稳定性与 X 上特殊卡勒度量的存在联系起来，在这个方向上取得了非常重大的进展。该研究项目的目的是研究复射影簇 X 的模空间，不仅配备一个充足的线丛 L，而且配备有限多个充足的线丛，代表 X 的 Neron-Severi 群（a 的子集）基。给定 X 上的一个充足的线束 L，我们可以使用 L 的足够大的幂的部分将 X 嵌入射影空间中。然后，人们可以希望应用 Mumford 几何不变量理论 (GIT) 的思想，该理论于 20 世纪 60 年代发展起来，通过还原群作用来构造和研究代数簇的商，定义相关特殊线性作用的（半）稳定性概念希尔伯特方案上的群表示具有与 X 相同的希尔伯特多项式的射影空间的射影子方案。然而，这些取决于所选 L 的幂，并且不具有明显的几何解释； K-（半）稳定性定义背后的动机是提供这种 GIT（半）稳定性的某种渐近版本，因为线束的幂趋于无穷大。给定 X 上的几个不同的充足线束，我们可以采用这些线束幂的张量积的一部分，将 X 嵌入射影环面簇中。然后我们可以研究相应的环面变体上相应的群体行为，以及这些情况下 K 稳定性的类似物。该项目旨在在dimX=2 的情况下研究这一问题，这是最低维度，只有一个充足的线束的传统情况尚未覆盖。该项目属于 EPSRC 几何和拓扑研究领域。不涉及任何公司或合作者。