Singularities: Geometric and Arithmetic Aspects

该建议涉及研究代数簇的奇异性。一个研究这些奇异性在两个设置：在零特征和积极的特征。在每一个设置中，定义测量奇点的不变量。尽管定义不同，但这两组不变量表现出许多共同的特征和微妙的联系。在特征零中，要研究的不变量（更准确地说，对数典型阈值和最小对数差异）是根据除数赋值定义的。一个基本的结果，存在的解决方案的奇异性，给出了一个强大的有限性声明时，计算这些不变量。对这种不变量感兴趣的动机来自于它们在双有理几何中的应用：这些不变量的某些几何性质意味着翻转的终止，这是高维中的一个悬而未决的开放问题。PI计划研究与对数典型阈值的主要猜想相关的问题，该猜想声称在固定维度中，没有这种不变量的递增序列。该提案的另一部分涉及通过Frobenius态射在正特征中定义的不变量。它们来自紧闭包理论，但与特征零中的不变量有许多微妙的联系。特别是，在该领域的一个基本猜想，在一个精确的方式连接的不变量在特征零的那些积极的特征，通过减少模p。PI建议使用D-模理论的想法，以研究这个猜想。奇异性是负责许多现象，无论是在几何和算术。例如，它们决定了模素数幂的同余的解的数量的渐近性，或者多项式的幂的可积性。在过去的几年里，人们已经认识到，在不同的背景下出现的奇点不变量之间有着深刻的联系。人们可以希望，通过从这些不同的观点来接近奇点，人们可以得到更丰富的画面，可以用来解决其他领域的应用问题。

The proposal concerns the study of singularities of algebraic varieties.One studies these singularities in two settings: in characteristic zero and in positive characteristic. In each setting one defines invariants that measure the singularities. Despite the different definitions, the two sets of invariants exhibit many common features and subtle connections. In characteristic zero, the invariants to be studied (more precisely, the log canonical threshold, and the minimal log discrepancies) are defined in terms of divisorial valuations. A fundamental result, the existence of resolution of singularities, gives a strong finiteness statement when computing these invariants. The motivation for the interest in such invariants comes from their applications to birational geometry: certain conjectural properties of these invariants imply the termination of flips, one of the outstanding open problems in higher dimensions. The PI plans to study questions related to the main conjecture on log canonical thresholds, that asserts that in fixed dimension, there are no increasing sequences of such invariants. Another part of the proposal concerns invariants defined in positive characteristic via the Frobenius morphism. They come out of tight closure theory, but have many subtle connections with the invariants in characteristic zero. In particular, a basic conjecture in the field connects in a precise way the invariants in characteristic zero to those in positive characteristic, via reduction mod p. The PI proposes to use D-module theoretic ideas in order to study this conjecture.Singularities are responsible for many phenomena, both in geometry and in arithmetic. For example, they govern the asymptotic of the number of solutions of congruences modulo prime powers, or the integrability of powers of polynomials. It has been realized in the past few years that invariants of singularities that show up in different contexts have deep connections amongst them. One can hope that by approaching singularities from these various points of view, one can get a richer picture, that can be used to attack problems with applications in other fields.