We prove that the abundance conjecture holds on a variety $X$ with mild
singularities if $X$ has many reflexive differential forms with coefficients in
pluricanonical bundles, assuming the Minimal Model Program in lower dimensions.
This implies, for instance, that under this condition, hermitian semipositive
canonical divisors are almost always semiample, and that klt pairs whose
underlying variety is uniruled have good models in many circumstances. When the
numerical dimension of $K_X$ is $1$, our results hold unconditionally in every
dimension. We also treat a related problem on the semiampleness of nef line
bundles on Calabi-Yau varieties.
Given a generic degree-2 cover of a genus 1 curve D by a non hyperelliptic
genus 3 curve C over a field k of characteristic different from 2, we produce
an explicit genus 2 curve X such that Jac(C) is isogenous to the product of
Jac(D) and Jac(X). This construction can be seen as a degenerate case of a
result by Nils Bruin.
We recently formulated a number of Crepant Resolution Conjectures (CRC) for
open Gromov-Witten invariants of Aganagic-Vafa Lagrangian branes and verified
them for the family of threefold type A-singularities. In this paper we enlarge
the body of evidence in favor of our open CRCs, along two different strands. In
one direction, we consider non-hard Lefschetz targets and verify the disk CRC
for local weighted projective planes. In the other, we complete the proof of
the quantized (all-genus) open CRC for hard Lefschetz toric Calabi-Yau three
dimensional representations by a detailed study of the G-Hilb resolution of
$[C^3/G]$ for $G=\mathbb{Z}_2 \times \mathbb{Z}_2$. Our results have
implications for closed-string CRCs of Coates-Iritani-Tseng, Iritani, and Ruan
for this class of examples.
We prove that a very general nonsingular conic bundle
$X\rightarrow\mathbb{P}^{n-1}$ embedded in a projective vector bundle of rank
$3$ over $\mathbb{P}^{n-1}$ is not stably rational if the anti-canonical
divisor of $X$ is not ample and $n\geq 3$.
A vector bundle E on a projective variety X is called finite if it satisfies
a nontrivial polynomial equation with integral coefficients. A theorem of Nori
implies that E is finite if and only if the pullback of E to some finite etale
Galois covering of X is trivial. We prove the same statement when X is a
compact complex manifold admitting a Gauduchon astheno-Kahler metric.
Every fibration of a projective hyper-Kähler fourfold has fibers which are
Abelian surfaces. In case the Abelian surface is a Jacobian of a genus two
curve, these have been classified by Markushevich. We study those cases where
the Abelian surface is a product of two elliptic curves, under some mild
genericity hypotheses.
Let G be a connected reductive algebraic group over an algebraically closed field k, with simply connected derived subgroup. The exotic t-structure on the cotangent bundle of its flag variety T^*(G/B), originally introduced by Bezrukavnikov, has been a key tool for a number of major results in geometric representation theory, including the proof of the graded Finkelberg-Mirkovic conjecture. In this paper, we study (under mild technical assumptions) an analogous t-structure on the cotangent bundle of a partial flag variety T^*(G/P). As an application, we prove a parabolic analogue of the Arkhipov-Bezrukavnikov-Ginzburg equivalence. When the characteristic of k is larger than the Coxeter number, we deduce an analogue of the graded Finkelberg-Mirkovic conjecture for some singular blocks.
We attempt to describe the rank 2 vector bundles on a curve C which are
specializations of the trivial bundle. We get a complete classifications when C
is Brill-Noether generic, or when it is hyperelliptic; in both cases all limit
vector bundles are decomposable. We give examples of indecomposable limit
bundles for some special curves.
We study meromorphic actions of unipotent complex Lie groups on compact
Kähler manifolds using moment map techniques. We introduce natural stability
conditions and show that sets of semistable points are Zariski-open and admit
geometric quotients that carry compactifiable Kähler structures obtained by
symplectic reduction. The relation of our complex-analytic theory to the work
of Doran--Kirwan regarding the Geometric Invariant Theory of unipotent group
actions on projective varieties is discussed in detail.
For a tropical manifold of dimension n we show that the tropical homology
classes of degree (n-1, n-1) which arise as fundamental classes of tropical
cycles are precisely those in the kernel of the eigenwave map. To prove this we
establish a tropical version of the Lefschetz (1, 1)-theorem for rational
polyhedral spaces that relates tropical line bundles to the kernel of the wave
homomorphism on cohomology. Our result for tropical manifolds then follows by
combining this with Poincaré duality for integral tropical homology.
Let $k$ a field of characteristic zero. Let $X$ be a smooth, projective,
geometrically rational $k$-surface. Let $\mathcal{T}$ be a universal torsor
over $X$ with a $k$-point et $\mathcal{T}^c$ a smooth compactification of
$\mathcal{T}$. There is an open question: is $\mathcal{T}^c$ $k$-birationally
equivalent to a projective space? We know that the unramified cohomology groups
of degree 1 and 2 of $\mathcal{T}$ and $\mathcal{T}^c$ are reduced to their
constant part. For the analogue of the third cohomology groups, we give a
sufficient condition using the Galois structure of the geometrical Picard group
of $X$. This enables us to show that
$H^{3}_{nr}(\mathcal{T}^{c},\mathbb{Q}/\mathbb{Z}(2))/H^3(k,\mathbb{Q}/\mathbb{Z}(2))$
vanishes if $X$ is a generalised Châtelet surface and that this group is
reduced to its $2$-primary part if $X$ is a del Pezzo surface of degree at
least 2.
En combinant une méthode de C. Voisin avec la descente galoisienne sur le
groupe de Chow en codimension $2$, nous montrons que le troisième groupe de
cohomologie non ramifiée d'un solide cubique lisse défini sur le corps des
fonctions d'une courbe complexe est nul. Ceci implique que la conjecture de
Hodge entière pour les classes de degré 4 vaut pour les variétés
projectives et lisses de dimension 4 fibrées en solides cubiques au-dessus
d'une courbe, sans restriction sur les fibres singulières.
---------------
We prove that the third unramified cohomology group of a smooth cubic
threefold over the function field of a complex curve vanishes. For this, we
combine a method of C. Voisin with Galois descent on the codimension $2$ Chow
group. As a corollary, we show that the integral Hodge conjecture holds for
degree $4$ classes on smooth projective fourfolds equipped with a fibration
over a curve, the generic fibre of which is a smooth cubic threefold, with
arbitrary singularities on the special fibres.
We introduce a new invariant, the real (logarithmic)-Kodaira dimension, that
allows to distinguish smooth real algebraic surfaces up to birational
diffeomorphism. As an application, we construct infinite families of smooth
rational real algebraic surfaces with trivial homology groups, whose real loci
are diffeomorphic to $\mathbb{R}^2$, but which are pairwise not birationally
diffeomorphic. There are thus infinitely many non-trivial models of the
euclidean plane, contrary to the compact case.
We prove that the affine cone over a general primitively polarised K3 surface of genus g is smoothable if and only if g ≤ 10 or g = 12. We also give several examples of singularities with special behaviour, such as surfaces whose affine cone is smoothable even though the projective cone is not.
