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Chaire de geometrie arithmetique

Chaire

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Domaines de recheche associés (1)

Alexandre Grothendieck, né Alexander Grothendieck (prononcé en allemand : ), est un mathématicien français, né le à Berlin et mort le à Saint-Lizier, près de Saint-Girons (Ariège). Il est resté long

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Publications associées (7)

Publications associées

Chargement

A geographical study of (M)over-bar(2)(P-2, 4)(main)

Francesca Carocci

We discuss criteria for a stable map of genus two and degree 4 to the projective plane to be smoothable, as an application of our modular desingularisation of (M) over bar (2,n)(P-r, d)(main) via logarithmic geometry and Gorenstein singularities.

WALTER DE GRUYTER GMBH2022

Chargement

Arithmetic and metric aspects of open de Rham spaces

Tamás Hausel, Michael Lennox Wong, Dimitri Stelio Wyss

In this paper we determine the motivic class---in particular, the weight polynomial and conjecturally the Poincaré polynomial---of the open de Rham space, defined and studied by Boalch, of certain moduli of irregular meromorphic connections on the trivial bundle on

\blackbb{P}^1

. The computation is by motivic Fourier transform. We show that the result satisfies the purity conjecture, that is, it agrees with the pure part of the conjectured mixed Hodge polynomial of the corresponding wild character variety. We also identify the open de Rham spaces with quiver varieties with multiplicities of Yamakawa and Geiss--Leclerc--Schröer. We finish with constructing natural complete hyperkähler metrics on them, which in the

4

-dimensional cases are expected to be of type ALF.

Chargement

We prove the Topological Mirror Symmetry Conjecture by Hausel-Thaddeus for smooth moduli spaces of Higgs bundles of type

SL_n

and

PGL_n

. More precisely, we establish an equality of stringy Hodge numbers for certain pairs of algebraic orbifolds generically fibred into dual abelian varieties. Our proof utilises

p

-adic integration relative to the fibres, and interprets canonical gerbes present on these moduli spaces as characters on the Hitchin fibres using Tate duality. Furthermore we prove for

d

coprime to

n

, that the number of rank

n

Higgs bundles of degree

d

over a fixed curve defined over a finite field, is independent of

d

. This proves a conjecture by Mozgovoy--Schiffman in the coprime case.

2017

Chargement

Arithmetic and metric aspects of open de Rham spaces

Tamás Hausel, Michael Lennox Wong, Dimitri Stelio Wyss

\blackbb{P}^1

4

-dimensional cases are expected to be of type ALF.

We prove the Topological Mirror Symmetry Conjecture by Hausel-Thaddeus for smooth moduli spaces of Higgs bundles of type

SL_n

and

PGL_n

. More precisely, we establish an equality of stringy Hodge numbers for certain pairs of algebraic orbifolds generically fibred into dual abelian varieties. Our proof utilises

p

-adic integration relative to the fibres, and interprets canonical gerbes present on these moduli spaces as characters on the Hitchin fibres using Tate duality. Furthermore we prove for

d

coprime to

n

, that the number of rank

n

Higgs bundles of degree

d

over a fixed curve defined over a finite field, is independent of

d

. This proves a conjecture by Mozgovoy--Schiffman in the coprime case.

2017