Sixth volume of Épijournal de Géométrie Algébrique - 2022
Javier Carvajal-Rojas.
We investigate finite torsors over big opens of spectra of strongly
$F$-regular germs that do not extend to torsors over the whole spectrum. Let
$(R,\mathfrak{m},k)$ be a strongly $F$-regular $k$-germ where $k$ is an
algebraically closed field of characteristic $p>0$. We prove the existence of a
finite local cover $R \subset R^{\star}$ so that $R^{\star}$ is a strongly
$F$-regular $k$-germ and: for all finite algebraic groups $G/k$ with solvable
neutral component, every $G$-torsor over a big open of $\mathrm{Spec}
R^{\star}$ extends to a $G$-torsor everywhere. To achieve this, we obtain a
generalized transformation rule for the $F$-signature under finite local
extensions. Such formula is used to show that that the torsion of $\mathrm{Cl}
R$ is bounded by $1/s(R)$. By taking cones, we conclude that the Picard group
of globally $F$-regular varieties is torsion-free. Likewise, it shows that
canonical covers of $\mathbb{Q}$-Gorenstein strongly $F$-regular singularities
are strongly $F$-regular.
Klaus Hulek ; Carsten Liese.
In this paper we study the Mori fan of the Dolgachev-Nikulin-Voisin family in
degree $2$ as well as the associated secondary fan. The main result is an
enumeration of all maximal dimensional cones of the two fans.
Grégoire Menet.
We describe the integral cohomology of $X/G$ where $X$ is a compact complex
manifold and $G$ a cyclic group of prime order with only isolated fixed points.
As a preliminary step, we investigate the integral cohomology of toric blow-ups
of quotients of $\mathbb{C}^n$. We also provide necessary and sufficient
conditions for the spectral sequence of equivariant cohomology of $(X,G)$ to
degenerate at the second page. As an application, we compute the
Beauville--Bogomolov form of $X/G$ when $X$ is a Hilbert scheme of points on a
K3 surface and $G$ a symplectic automorphism group of orders 5 or 7.
Emilio Franco ; Marcos Jardim.
The Dirac--Higgs bundle is a hyperholomorphic bundle over the moduli space of
stable Higgs bundles of coprime rank and degree. We provide an algebraic
generalization to the case of trivial degree and the rank higher than $1$. This
allow us to generalize to this case the Nahm transform defined by Frejlich and
the second named author, which, out of a stable Higgs bundle, produces a vector
bundle with connection over the moduli space of rank 1 Higgs bundles. By
performing the higher rank Nahm transform we obtain a hyperholomorphic bundle
with connection over the moduli space of stable Higgs bundles of rank $n$ and
degree 0, twisted by the gerbe of liftings of the projective universal bundle.
Such hyperholomorphic vector bundles over the moduli space of stable Higgs
bundles can be seen, in the physicist's language, as BBB-branes twisted by the
above mentioned gerbe. We refer to these objects as Nahm branes. Finally, we
study the behaviour of Nahm branes under Fourier--Mukai transform over the
smooth locus of the Hitchin fibration, checking that the resulting objects are
supported on a Lagrangian multisection of the Hitchin fibration, so they
describe partial data of BAA-branes.
Hélène Esnault ; Moritz Kerz.
We propose a conjecture on the density of arithmetic points in the
deformation space of representations of the étale fundamental group in
positive characteristic. This? conjecture has applications to étale
cohomology theory, for example it implies a Hard Lefschetz conjecture. We prove
the density conjecture in tame degree two for the curve $\mathbb{P}^1\setminus
\{0,1,\infty\}$. v2: very small typos corrected.v3: final. Publication in
Epiga.
Jim Bryan ; Ádám Gyenge.
Let $X$ be a complex $K3$ surface with an effective action of a group $G$
which preserves the holomorphic symplectic form. Let $$ Z_{X,G}(q) =
\sum_{n=0}^{\infty} e\left(\operatorname{Hilb}^{n}(X)^{G} \right)\, q^{n-1} $$
be the generating function for the Euler characteristics of the Hilbert schemes
of $G$-invariant length $n$ subschemes. We show that its reciprocal,
$Z_{X,G}(q)^{-1}$ is the Fourier expansion of a modular cusp form of weight
$\frac{1}{2} e(X/G)$ for the congruence subgroup $\Gamma_{0}(|G|)$. We give an
explicit formula for $Z_{X,G}$ in terms of the Dedekind eta function for all 82
possible $(X,G)$. The key intermediate result we prove is of independent
interest: it establishes an eta product identity for a certain shifted theta
function of the root lattice of a simply laced root system. We extend our
results to various refinements of the Euler characteristic, namely the Elliptic
genus, the Chi-$y$ genus, and the motivic class.
S. Lichtenbaum ; N. Ramachandran ; T. Suzuki.
We provide two proofs that the conjecture of Artin-Tate for a fibered surface
is equivalent to the conjecture of Birch-Swinnerton-Dyer for the Jacobian of
the generic fibre. As a byproduct, we obtain a new proof of a theorem of
Geisser relating the orders of the Brauer group and the Tate-Shafarevich group.
Alexandr Buryak ; Francisco Hernández Iglesias ; Sergey Shadrin.
We propose a conjectural formula for $DR_g(a,-a) \lambda_g$ and check all its
expected properties. Our formula refines the one point case of a similar
conjecture made by the first named author in collaboration with Guéré and
Rossi, and we prove that the two conjectures are in fact equivalent, though in
a quite non-trivial way.
Gebhard Martin.
Bielliptic and quasi-bielliptic surfaces form one of the four classes of
minimal smooth projective surfaces of Kodaira dimension $0$. In this article,
we determine the automorphism schemes of these surfaces over algebraically
closed fields of arbitrary characteristic, generalizing work of Bennett and
Miranda over the complex numbers; we also find some cases that are missing from
the classification of automorphism groups of bielliptic surfaces in
characteristic $0$.