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.
Fields of science (EuroSciVoc)
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
- natural sciences mathematics pure mathematics algebra
- natural sciences mathematics pure mathematics geometry
- natural sciences mathematics pure mathematics arithmetics prime numbers
- humanities philosophy, ethics and religion philosophy
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Keywords
Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
Project’s keywords as indicated by the project coordinator. Not to be confused with the EuroSciVoc taxonomy (Fields of science)
Programme(s)
Multi-annual funding programmes that define the EU’s priorities for research and innovation.
Multi-annual funding programmes that define the EU’s priorities for research and innovation.
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H2020-EU.1.1. - EXCELLENT SCIENCE - European Research Council (ERC)
MAIN PROGRAMME
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Topic(s)
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Calls for proposals are divided into topics. A topic defines a specific subject or area for which applicants can submit proposals. The description of a topic comprises its specific scope and the expected impact of the funded project.
Funding Scheme
Funding scheme (or “Type of Action”) inside a programme with common features. It specifies: the scope of what is funded; the reimbursement rate; specific evaluation criteria to qualify for funding; and the use of simplified forms of costs like lump sums.
Funding scheme (or “Type of Action”) inside a programme with common features. It specifies: the scope of what is funded; the reimbursement rate; specific evaluation criteria to qualify for funding; and the use of simplified forms of costs like lump sums.
ERC-STG - Starting Grant
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Call for proposal
Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
(opens in new window) ERC-2018-STG
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Net EU financial contribution. The sum of money that the participant receives, deducted by the EU contribution to its linked third party. It considers the distribution of the EU financial contribution between direct beneficiaries of the project and other types of participants, like third-party participants.
1015 LAUSANNE
Switzerland
The total costs incurred by this organisation to participate in the project, including direct and indirect costs. This amount is a subset of the overall project budget.