Volume 9 (2009), Number 3. Abstracts

H. Esnault and O. Wittenberg. Remarks on Cycle Classes of Sections of the Arithmetic Fundamental Group [PDF]

Given a smooth and separated K(π,1) variety X over a field k, we associate a “cycle class” in étale cohomology with compact supports to any continuous section of the natural map from the arithmetic fundamental group of X to the absolute Galois group of k. We discuss the algebraicity of this class in the case of curves over p-adic fields. Finally, an étale adaptation of Beilinson's geometrization of the pronilpotent completion of the topological fundamental group allows us to lift this cycle class in suitable cohomology groups.

Keywords. Étale fundamental group, cycle class map, pronilpotent completion.

2000 Mathematics Subject Classification. Primary: 14F35; Secondary: 14C25, 14F20.

M. Green, Ph. Griffiths, and M. Kerr. Some Enumerative Global Properties of Variations of Hodge Structures [PDF]

The global enumerative invariants of a variation of polarized Hodge structures over a smooth quasi-projective curve reflect the global twisting of the family and numerical measures of the complexity of the limiting mixed Hodge structures that arise when the family degenerates. We study several of these global enumerative invariants and give applications to questions such as: Give conditions under which a non-isotrivial family of Calabi–Yau threefolds must have singular fibres? Determine the correction terms arising from the limiting mixed Hodge structures that are required to turn the classical Arakelov inequalities into exact equalities.

Keywords. Variation of Hodge structure, isotrivial family, elliptic surface, Calabi–Yau threefold, Arakelov equalities, Hodge bundles, Hodge metric, positivity, Grothendieck–Riemann–Roch, limiting mixed Hodge structure, semistable degeneration, relative minimality.

2000 Mathematics Subject Classification. 14C17, 14D05, 14D06, 14D07, 14J27, 14J28, 14J32, 32G20.

E. Hellmann. On the Structure of Some Moduli Spaces of Finite Flat Group Schemes [PDF]

We consider the moduli space, in the sense of Kisin, of finite flat models of a 2-dimensional representation with values in a finite field of the absolute Galois group of a totally ramified extension of Qp. We determine the connected components of this space and describe its irreducible components in the case of an irreducible Galois representation. These results prove a modified version of a conjecture of Kisin.

Keywords. Affine Grassmannian, φ-module, finite flat group scheme.

2000 Mathematics Subject Classification. Primary: 14M99; Secondary: 20G25.

E. Hrushovski and D. Kazhdan. Motivic Poisson Summation [PDF]

We develop a “motivic integration” version of the Poisson summation formula for function fields, with values in the Grothendieck ring of definable exponential sums. We also study division algebras over the function field, and show (under some assumptions) that the Fourier transform of a conjugation-invariant test function does not depend on the form of the division algebra. This yields a motivic-integration analog of certain theorems of Deligne–Kazhdan–Vigneras.

Keywords. Motivic integration, Poisson summation, division algebras, Grothendieck ring.

2000 Mathematics Subject Classification. 03C60, 11R56, 22E55.

G. Pappas and M. Rapoport. Φ-Modules and Coefficient Spaces [PDF]

We define and study certain moduli stacks of modules endowed with a Frobenius semi-linear endomorphism. These stacks can be thought of as parametrizing the coefficients of a variable Galois representation and are global variants of the spaces of Kisin–Breuil Φ-modules used by Kisin in his study of deformation spaces of local Galois representations. A version of a rigid analytic period map is defined for these spaces, and it is shown how their local structure can be described in terms of “local models”. We also show how Bruhat–Tits buildings can be used to study their special fibers.

Keywords. Frobenius module, Galois representation, local model, affine Grassmannian.

2000 Mathematics Subject Classification. Primary: 14G22, 11S20; Secondary: 14M15.

C. Simpson. Geometricity of the Hodge Filtration on the ∞-stack of perfect complexes over XDR [PDF] [Abstract]

We construct a locally geometric ∞-stack MHod(X,Perf) of perfect complexes with λ-connection structure on a smooth projective variety X. This maps to A1/Gm, so it can be considered as the Hodge filtration of its fiber over 1 which is MHod(X,Perf), parametrizing complexes of DX-modules which are OX-perfect. We apply the result of Toen–Vaquié that Perf(X) is locally geometric. The proof of geometricity of the map MHod(X,Perf) → Perf(X) uses a Hochschild-like notion of weak complexes of modules over a sheaf of rings of differential operators. We prove a strictification result for these weak complexes, and also a strictification result for complexes of sheaves of O-modules over the big crystalline site.

Keywords. Hodge filtration, λ-connection, perfect complex, D-module, Higgs bundle, twistor space, Hochschild complex, Dold–Puppe, Maurer–Cartan equation.

2000 Mathematics Subject Classification. Primary: 14D20; Secondary: 32G34, 32S35.

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