We prove a general criterion which ensures that a fractional Calabi--Yau
category of dimension $\leq 2$ admits a unique Serre-invariant stability
condition, up to the action of the universal cover of
$\text{GL}^+_2(\mathbb{R})$. We apply this result to the Kuznetsov component
$\text{Ku}(X)$ of a cubic threefold $X$. In particular, we show that all the
known stability conditions on $\text{Ku}(X)$ are invariant with respect to the
action of the Serre functor and thus lie in the same orbit with respect to the
action of the universal cover of $\text{GL}^+_2(\mathbb{R})$. As an
application, we show that the moduli space of Ulrich bundles of rank $\geq 2$
on $X$ is irreducible, answering a question asked by Lahoz, Macrì and
Stellari.
Using log geometry, we study smoothability of genus zero twisted stable maps
to stacky curves relative to a collection of marked points. One application is
to smoothing semi-log canonical fibered surfaces with marked singular fibers.
Let X be a smooth quartic surface not containing lines, defined over a number
field K. We prove that there are only finitely many bitangents to X which are
defined over K. This result can be interpreted as saying that a certain
surface, having vanishing irregularity, contains only finitely many rational
points. In our proof, we use the geometry of lines of the quartic double solid
associated to X. In a somewhat opposite direction, we show that on any quartic
surface X over a number field K, the set of algebraic points in X(\overeline K)
which are quadratic over a suitable finite extension K' of K is Zariski-dense.
We prove that the rational Chow motive of a six dimensional hyper-Kähler
variety obtained as symplectic resolution of O'Grady type of a singular moduli
space of semistable sheaves on an abelian surface $A$ belongs to the tensor
category of motives generated by the motive of $A$. We in fact give a formula
for the rational Chow motive of such a variety in terms of that of the surface.
As a consequence, the conjectures of Hodge and Tate hold for many
hyper-Kähler varieties of OG6-type.
We prove the abundance conjecture for projective slc surfaces over arbitrary
fields of positive characteristic. The proof relies on abundance for lc
surfaces over abritrary fields, proved by Tanaka, and on the technique of Hacon
and Xu to descend semi-ampleness from the normalization. We also present
applications to dlt threefold pairs, and to mixed characteristic families of
surfaces.
We show that flat families of stable 3-folds do not lead to proper moduli
spaces in any characteristic $p>0$. As a byproduct, we obtain log canonical
4-fold pairs, whose log canonical centers are not weakly normal.
The goal of this paper is to construct trace maps for the six functor
formalism of motivic cohomology after Voevodsky, Ayoub, and
Cisinski-Déglise. We also construct an $\infty$-enhancement of such a trace
formalism. In the course of the $\infty$-enhancement, we need to reinterpret
the trace formalism in a more functorial manner. This is done by using
Suslin-Voevodsky's relative cycle groups.
Let l be a prime and G a pro-l group with torsion-free abelianization. We
produce group-theoretic analogues of the Johnson/Morita cocycle for G -- in the
case of surface groups, these cocycles appear to refine existing constructions
when l=2. We apply this to the pro-l etale fundamental groups of smooth curves
to obtain Galois-cohomological analogues, and discuss their relationship to
work of Hain and Matsumoto in the case the curve is proper. We analyze many of
the fundamental properties of these classes and use them to give an example of
a non-hyperelliptic curve whose Ceresa class has torsion image under the l-adic
Abel-Jacobi map.
We express notions of K-stability of polarized spherical varieties in terms
of combinatorial data, vastly generalizing the case of toric varieties. We then
provide a combinatorial sufficient condition of G-uniform K-stability by
studying the corresponding convex geometric problem. Thanks to recent work of
Chi Li and a remark by Yuji Odaka, this provides an explicitly checkable
sufficient condition of existence of constant scalar curvature Kahler metrics.
As a side effect, we show that, on several families of spherical varieties,
G-uniform K-stability is equivalent to K-polystability with respect to
G-equivariant test configurations for polarizations close to the anticanonical
bundle.
We obtain necessary and sufficient conditions for the good reduction of
Kummer surfaces attached to abelian surfaces with non-supersingular reduction
when the residue field is perfect of characteristic 2. In this case, good
reduction with an algebraic space model is equivalent to good reduction with a
scheme model, which we explicitly construct.
A triangulated category is said to be indecomposable if it admits no
nontrivial semiorthogonal decompositions. We introduce a definition of a
noncommutatively stably semiorthogonally indecomposable (NSSI) variety. This
propery implies, among other things, that each smooth proper subvariety has
indecomposable derived category of coherent sheaves, and that if $Y$ is NSSI,
then for any variety $X$ all semiorthogonal decompositions of $X \times Y$ are
induced from decompositions of $X$. We prove that any variety whose Albanese
morphism is finite is NSSI, and that the total space of a fibration over NSSI
base with NSSI fibers is also NSSI. We apply this indecomposability to deduce
that there are no phantom subcategories in some varieties, including surfaces
$C \times \mathbb{P}^1$, where $C$ is any smooth proper curve of positive
genus.
In this paper, we study various hyperbolicity properties for a quasi-compact
Kähler manifold $U$ which admits a complex polarized variation of Hodge
structures so that each fiber of the period map is zero-dimensional. In the
first part, we prove that $U$ is algebraically hyperbolic and that the
generalized big Picard theorem holds for $U$. In the second part, we prove that
there is a finite étale cover $\tilde{U}$ of $U$ from a quasi-projective
manifold $\tilde{U}$ such that any projective compactification $X$ of
$\tilde{U}$ is Picard hyperbolic modulo the boundary $X-\tilde{U}$, and any
irreducible subvariety of $X$ not contained in $X-\tilde{U}$ is of general
type. This result coarsely incorporates previous works by Nadel, Rousseau,
Brunebarbe and Cadorel on the hyperbolicity of compactifications of quotients
of bounded symmetric domains by torsion-free lattices.
In this work we study line arrangements consisting in lines passing through
three non-aligned points. We call them triangular arrangements. We prove that
any combinatorics of a triangular arrangement is always realized by a
Roots-of-Unity-Arrangement, which is a particular class of triangular
arrangements. Among these Roots-of Unity-Arrangements, we provide conditions
that ensure their freeness. Finally, we give two triangular arrangements having
the same weak combinatorics, such that one is free but the other one is not.
We work on a projective threefold $X$ which satisfies the Bogomolov-Gieseker
conjecture of Bayer-Macrì-Toda, such as $\mathbb P^3$ or the quintic
threefold.
We prove certain moduli spaces of 2-dimensional torsion sheaves on $X$ are
smooth bundles over Hilbert schemes of ideal sheaves of curves and points in
$X$.
When $X$ is Calabi-Yau this gives a simple wall crossing formula expressing
curve counts (and so ultimately Gromov-Witten invariants) in terms of counts of
D4-D2-D0 branes. These latter invariants are predicted to have modular
properties which we discuss from the point of view of S-duality and
Noether-Lefschetz theory.
We define a new notion of affine subspace concentration conditions for
lattice polytopes, and prove that they hold for smooth and reflexive polytopes
with barycenter at the origin. Our proof involves considering the slope
stability of the canonical extension of the tangent bundle by the trivial line
bundle and with the extension class $c_1(\mathcal{T}_X)$ on Fano toric
varieties.
We study Dolgachev elliptic surfaces with a double and a triple fiber and
find explicit equations of two new pairs of fake projective plane with $21$
automorphisms, thus finishing the task of finding explicit equations of fake
projective planes with this automorphism group. This includes, in particular,
the fake projective plane discovered by J. Keum.
We study the locus of fixed points of a torus action on a GIT quotient of a
complex vector space by a reductive complex algebraic group which acts
linearly. We show that, under the assumption that $G$ acts freely on the stable
locus, the components of the fixed point locus are again GIT quotients of
linear subspaces by Levi subgroups.
We analyze the geometry of some $p$-adic Deligne--Lusztig spaces $X_w(b)$
introduced in [Iva21] attached to an unramified reductive group ${\bf G}$ over
a non-archimedean local field. We prove that when ${\bf G}$ is classical, $b$
basic and $w$ Coxeter, $X_w(b)$ decomposes as a disjoint union of translates of
a certain integral $p$-adic Deligne--Lusztig space. Along the way we extend
some observations of DeBacker and Reeder on rational conjugacy classes of
unramified tori to the case of extended pure inner forms, and prove a loop
version of Frobenius-twisted Steinberg's cross section.
We use a classification result of Chenevier and Lannes for algebraic
automorphic representations together with a conjectural correspondence with
$\ell$-adic absolute Galois representations to determine the Euler
characteristics (with values in the Grothendieck group of such representations)
of $\overline{\mathcal M}_{3,n}$ and $\mathcal M_{3,n}$ for $n \leq 14$ and of
local systems $\mathbb{V}_{\lambda}$ on $\mathcal{A}_3$ for $|\lambda| \leq
16$.
In this paper, we develop a theory of pseudo-effective sheaves on normal
projective varieties. As an application, by running the minimal model program,
we show that projective klt varieties with pseudo-effective tangent sheaf can
be decomposed into Fano varieties and Q-abelian varieties.