Moduli spaces of stable varieties and applications

Descripción del proyecto

La construcción de espacios modulares para variedades algebraicas de muchas dimensiones

Las variedades estables —introducidas por Kollár y Shepherd-Barron— son generalizaciones para más dimensiones de curvas estables en geometría algebraica. Su espacio modular hipotético clasifica las variedades proyectivas suaves hasta la equivalencia birracional, además de ofrecer también una compactificación proyectiva. Esto último es esencial para aplicar la geometría algebraica al propio espacio modular. El proyecto MODSTABVAR, financiado con fondos europeos, construirá el espacio modular grueso de superficies estables con un volumen fijo sobre los enteros. Esto implica mostrar el programa modelo mínimo para una variedad algebraica de tres dimensiones que se proyecte sobre una base característica mezclada de una dimensión. Los resultados del proyecto serán muy importantes para los campos de la geometría algebraica y la aritmética de variedades de más dimensiones.

Objetivo

Stable varieties, originally introduced by Kollár and Shepherd-Barron, are higher dimensional generalizations of the algebro-geometric notion of stable curves from many perspectives. Their partially conjectural moduli space classifies smooth projective varieties of general type up to birational equivalence, and it also provides a projective compactification for this classifying space. The latter is essential for applying algebraic geometry to the moduli space itself. Furthermore, over the complex numbers, stable varieties can be also defined surprisingly as the projective varieties admitting a negative curvature (singular) Kähler-Einstein metric by the work of Berman and Guenancia, or as the canonically polarized K-stable varieties by Odaka.

The fundamental objective of the project is to construct the coarse moduli space of stable surfaces with fixed volume over the integers (possibly excluding finitely many primes, not depending on the volume). In particular this involves showing the Minimal Model Program for 3-folds that are projective over a 1 dimensional mixed characteristic base. The main motivations are applications to the general algebraic geometry and arithmetic of higher dimensional varieties.

The above fundamental goal is also an incarnation of Grothendieck's philosophy that algebraic geometry statements should be proved in a relative setting. This was implemented right at the beginning for stable curves, but it has not been possible to attain for stable varieties of higher dimensions, due to the lack of technology. Hence, the project aims to establish new technology in mixed and positive characteristic geometry based on recent developments, such as modern Minimal Model Program, the vanishings given by balanced big Cohen-Macaulay algebras (the existence of which was shown by André using Scholze's perfectoid theory), trace method for lifting sections, p-torsion cohomology killing via alterations (by Bhatt), torsor method on singular varieties, etc.