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Moduli spaces of stable varieties and applications

Project description

Constructing moduli spaces of high dimensional algebraic varieties

Introduced by Kollár and Shepherd-Barron, stable varieties are higher-dimensional generalisations of stable curves in algebraic geometry. Their conjectural moduli space classifies smooth projective varieties up to birational equivalence, while also providing a projective compactification. The latter is essential for applying algebraic geometry to the moduli space itself. The EU-funded MODSTABVAR project will construct the coarse moduli space of stable surfaces with fixed volume over the integers. This involves showing the minimal model program for three-dimensional algebraic variety that is projective over a one-dimensional mixed characteristic base. Project results will be very important to the fields of algebraic geometry and the arithmetic of higher-dimensional varieties.

Objective

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.

Host institution

ECOLE POLYTECHNIQUE FEDERALE DE LAUSANNE
Net EU contribution
€ 1 201 370,00
Address
BATIMENT CE 3316 STATION 1
1015 Lausanne
Switzerland

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Region
Schweiz/Suisse/Svizzera Région lémanique Vaud
Activity type
Higher or Secondary Education Establishments
Links
Total cost
€ 1 201 370,00

Beneficiaries (1)