Objective
In the proposed research we will focus our attention on certain orthogonal Shimura varieties and their relation to K3 surfaces. Over the complex numbers, using periods of K3 surfaces, one can identify the moduli space of polarized K3 surfaces with a certain level K-structure, for a suitable group K, with an open subvariety of Sh(SO(2,19),X)-K.
This fact boils down to two very important results in complex algebraic geometry, namely the Torelli theorem for K3 surfaces and Kulikov's degeneration theorem. Further analysis of this immersion shows that it is defined over Q. We want to carry out these ideas in mixed characteristic and prove similar arithmetic results. An integral canonical model of Sh(SO(2,19),X)-K is a scheme over the completion of the ring of integers Z at a prime p, having Sh(SO(2,19),X)/K as the general fiber and satisfying a certain extension property.
A way to construct those models is to use moduli schemes of polarized abelian varieties. We want to use this strategy to show the existence o f a period map from the space of polarized K3 surfaces with level K-structure to the integral canonical model of Sh(SO(2,19),X)/K. We propose to study this morphism and show that it is etale and, even stronger, that it is an open immersion. This is a gener alization of the global Torelli theorem for K3 surfaces in mixed characteristic. To do this, we plan to use some recent developments in integral p-adic Hodge theory and p-adic periods, which is an analogue of the complex case in positive characteristic.
This result will give a modular interpretation of the integral canonical model of Sh(SO(2,19),X)/K. Carrying out this program would provide us with strong tools for studying the irreducibility of the moduli spaces of polarized K3 surface in mixed characteristic. Such a result, being very important on its own, would also imply the validity of some other open problems in the theory of K3 surfaces such as Artin's conjecture on supersingular K3 surfaces.
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.
- engineering and technology materials engineering fibers
- natural sciences mathematics pure mathematics arithmetics
- natural sciences mathematics pure mathematics geometry
- natural sciences mathematics pure mathematics algebra algebraic geometry
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Programme(s)
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Multi-annual funding programmes that define the EU’s priorities for research and innovation.
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.
Call for proposal
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Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
FP6-2004-MOBILITY-5
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Funding Scheme
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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.
Coordinator
BELLATERRA
Spain
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