Fifth volume of Épijournal de Géométrie Algébrique - 2021
Bruno Kahn ; Hiroyasu Miyazaki ; Shuji Saito ; Takao Yamazaki.
We develop a theory of modulus sheaves with transfers, which generalizes
Voevodsky's theory of sheaves with transfers. This paper and its sequel are
foundational for the theory of motives with modulus, which is developed in
[KMSY20].
Bruno Kahn ; Hiroyasu Miyazaki ; Shuji Saito ; Takao Yamazaki.
We develop a theory of sheaves and cohomology on the category of proper
modulus pairs. This complements [KMSY21], where a theory of sheaves and
cohomology on the category of non-proper modulus pairs has been developed.
Cédric Bonnafé ; Alessandra Sarti.
We construct here many families of K3 surfaces that one can obtain as
quotients of algebraic surfaces by some subgroups of the rank four complex
reflection groups. We find in total 15 families with at worst
$ADE$--singularities. In particular we classify all the K3 surfaces that can be
obtained as quotients by the derived subgroup of the previous complex
reflection groups. We prove our results by using the geometry of the weighted
projective spaces where these surfaces are embedded and the theory of Springer
and Lehrer-Springer on properties of complex reflection groups. This
construction generalizes a previous construction by W. Barth and the second
author.
Alastair Craw ; Liana Heuberger ; Jesus Tapia Amador.
Reid's recipe for a finite abelian subgroup $G\subset
\text{SL}(3,\mathbb{C})$ is a combinatorial procedure that marks the toric fan
of the $G$-Hilbert scheme with irreducible representations of $G$. The
geometric McKay correspondence conjecture of Cautis--Logvinenko that describes
certain objects in the derived category of $G\text{-Hilb}$ in terms of Reid's
recipe was later proved by Logvinenko et al. We generalise Reid's recipe to any
consistent dimer model by marking the toric fan of a crepant resolution of the
vaccuum moduli space in a manner that is compatible with the geometric
correspondence of Bocklandt--Craw--Quintero-Vélez. Our main tool
generalises the jigsaw transformations of Nakamura to consistent dimer models.
Sébastien Boucksom ; Simone Diverio.
It was conjectured by Lang that a complex projective manifold is Kobayashi
hyperbolic if and only if it is of general type together with all of its
subvarieties. We verify this conjecture for projective manifolds whose
universal cover carries a bounded, strictly plurisubharmonic function. This
includes in particular compact free quotients of bounded domains.
Nicolas Tholozan ; Jérémy Toulisse.
We prove that some relative character varieties of the fundamental group of a
punctured sphere into the Hermitian Lie groups $\mathrm{SU}(p,q)$ admit compact
connected components. The representations in these components have several
counter-intuitive properties. For instance, the image of any simple closed
curve is an elliptic element. These results extend a recent work of Deroin and
the first author, which treated the case of $\textrm{PU}(1,1) =
\mathrm{PSL}(2,\mathbb{R})$. Our proof relies on the non-Abelian Hodge
correspondance between relative character varieties and parabolic Higgs
bundles. The examples we construct admit a rather explicit description as
projective varieties obtained via Geometric Invariant Theory.
Laurent Evain ; Mathias Lederer.
The Bialynicki-Birula strata on the Hilbert scheme $H^n(\mathbb{A}^d)$ are
smooth in dimension $d=2$. We prove that there is a schematic structure in
higher dimensions, the Bialynicki-Birula scheme, which is natural in the sense
that it represents a functor. Let $\rho_i:H^n(\mathbb{A}^d)\rightarrow {\rm
Sym}^n(\mathbb{A}^1)$ be the Hilbert-Chow morphism of the ${i}^{th}$
coordinate. We prove that a Bialynicki-Birula scheme associated with an action
of a torus $T$ is schematically included in the fiber $\rho_i^{-1}(0)$ if the
${i}^{th}$ weight of $T$ is non-positive. We prove that the monic functors
parametrizing families of ideals with a prescribed initial ideal are
representable.
Daniele Faenzi ; Joan Pons-Llopis.
We show that all reduced closed subschemes of projective space that have a
Cohen-Macaulay graded coordinate ring are of wild Cohen-Macaulay type, except
for a few cases which we completely classify.
Tom Bachmann ; Elden Elmanto ; Marc Hoyois ; Adeel A. Khan ; Vladimir Sosnilo ; Maria Yakerson.
We obtain geometric models for the infinite loop spaces of the motivic
spectra $\mathrm{MGL}$, $\mathrm{MSL}$, and $\mathbf{1}$ over a field. They are
motivically equivalent to $\mathbb{Z}\times
\mathrm{Hilb}_\infty^\mathrm{lci}(\mathbb{A}^\infty)^+$, $\mathbb{Z}\times
\mathrm{Hilb}_\infty^\mathrm{or}(\mathbb{A}^\infty)^+$, and $\mathbb{Z}\times
\mathrm{Hilb}_\infty^\mathrm{fr}(\mathbb{A}^\infty)^+$, respectively, where
$\mathrm{Hilb}_d^\mathrm{lci}(\mathbb{A}^n)$ (resp.
$\mathrm{Hilb}_d^\mathrm{or}(\mathbb{A}^n)$,
$\mathrm{Hilb}_d^\mathrm{fr}(\mathbb{A}^n)$) is the Hilbert scheme of lci
points (resp. oriented points, framed points) of degree $d$ in $\mathbb{A}^n$,
and $+$ is Quillen's plus construction. Moreover, we show that the plus
construction is redundant in positive characteristic.
Katharina Heinrich ; Roy Skjelnes ; Jan Stevens.
We consider the Cohen-Macaulay compactification of the space of twisted
cubics in projective n-space. This compactification is the fine moduli scheme
representing the functor of CM-curves with Hilbert polynomial 3t+1. We show
that the moduli scheme of CM-curves in projective 3-space is isomorphic to the
twisted cubic component of the Hilbert scheme. We also describe the
compactification for twisted cubics in n-space.
Samuel Boissière ; Enrica Floris.
Let $G$ be a connected algebraic group. We study $G$-equivariant extremal
contractions whose centre is a codimension three $G$-simply connected orbit. In
the spirit of an important result by Kawakita in 2001, we prove that those
contractions are weighted blow-ups.
Burt Totaro.
We formulate a conjecture on actions of the multiplicative group in motivic
homotopy theory. In short, if the multiplicative group G_m acts on a
quasi-projective scheme U such that U is attracted as t approaches 0 in G_m to
a closed subset Y in U, then the inclusion from Y to U should be an
A^1-homotopy equivalence.
We prove several partial results. In particular, over the complex numbers,
the inclusion is a homotopy equivalence on complex points. The proofs use an
analog of Morse theory for singular varieties. Application: the Hilbert scheme
of points on affine n-space is homotopy equivalent to the subspace consisting
of schemes supported at the origin.
Sébastien Boucksom ; Walter Gubler ; Florent Martin.
Let $L$ be a line bundle on a proper, geometrically reduced scheme $X$ over a
non-trivially valued non-Archimedean field $K$. Roughly speaking, the
non-Archimedean volume of a continuous metric on the Berkovich analytification
of $L$ measures the asymptotic growth of the space of small sections of tensor
powers of $L$. For a continuous semipositive metric on $L$ in the sense of
Zhang, we show first that the non-Archimedean volume agrees with the energy.
The existence of such a semipositive metric yields that $L$ is nef. A second
result is that the non-Archimedean volume is differentiable at any semipositive
continuous metric. These results are known when $L$ is ample, and the purpose
of this paper is to generalize them to the nef case. The method is based on a
detailed study of the content and the volume of a finitely presented torsion
module over the (possibly non-noetherian) valuation ring of $K$.
Julia Schneider ; Susanna Zimmermann.
We show that any infinite algebraic subgroup of the plane Cremona group over
a perfect field is contained in a maximal algebraic subgroup of the plane
Cremona group. We classify the maximal groups, and their subgroups of rational
points, up to conjugacy by a birational map.
Rodolfo Aguilar.
We provide a description of the fundamental group of the quotient of a product of topological spaces X i, each admitting a universal cover, by a finite group G, provided that there is only a finite number of path-connected components in X g i for every g ∈ G. This generalizes previous work of Bauer-Catanese-Grunewald-Pignatelli and Dedieu-Perroni.
Indranil Biswas ; Peter O'Sullivan.
Let H be a complex Lie group acting holomorphically on a complex analytic
space X such that the restriction to X_{\mathrm{red}} of every H-invariant
regular function on X is constant. We prove that an H-equivariant holomorphic
vector bundle E over X is $H$-finite, meaning f_1(E)= f_2(E) as H-equivariant
bundles for two distinct polynomials f_1 and f_2 whose coefficients are
nonnegative integers, if and only if the pullback of E along some H-equivariant
finite étale covering of X is trivial as an H-equivariant bundle.
Claudia Stadlmayr.
We determine all configurations of rational double points that occur on RDP
del Pezzo surfaces of arbitrary degree and Picard rank over an algebraically
closed field $k$ of arbitrary characteristic ${\rm char}(k)=p \geq 0$,
generalizing classical work of Du Val to positive characteristic. Moreover, we
give simplified equations for all RDP del Pezzo surfaces of degree $1$
containing non-taut rational double points.
Kenta Hashizume.
We prove the existence of a crepant sdlt model for slc pairs whose
irreducible components are normal in codimension one.
Svetlana Makarova.
The main result of the present paper is a construction of relative moduli
spaces of stable sheaves over the stack of quasipolarized projective surfaces.
For this, we use the theory of good moduli spaces, whose study was initiated by
Alper. As a corollary, we extend the relative Strange Duality morphism to the
locus of quasipolarized K3 surfaces.
Fumiaki Suzuki.
As an application of the theory of Lawson homology and morphic cohomology,
Walker proved that the Abel-Jacobi map factors through another regular
homomorphism. In this note, we give a direct proof of the theorem.
Yoshinori Hashimoto ; Julien Keller.
For a holomorphic vector bundle $E$ over a polarised Kähler manifold, we
establish a direct link between the slope stability of $E$ and the asymptotic
behaviour of Donaldson's functional, by defining the Quot-scheme limit of
Fubini-Study metrics. In particular, we provide an explicit estimate which
proves that Donaldson's functional is coercive on the set of Fubini-Study
metrics if $E$ is slope stable, and give a new proof of Hermitian-Einstein
metrics implying slope stability.