Volume 2

Second volume de l'Épijournal de Géométrie Algébrique - 2018.

1. Abundance for varieties with many differential forms

Lazić, Vladimir ; Peternell, Thomas.
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.

2. On the Prym variety of genus 3 covers of genus 1 curves

Ritzenthaler, Christophe ; Romagny, Matthieu.
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.

3. Prime Fano threefolds of genus 12 with a $G_m$-action

Kuznetsov, Alexander ; Prokhorov, Yuri.
We give an explicit construction of prime Fano threefolds of genus 12 with a $G_m$-action, describe their isomorphism classes and automorphism groups.

4. Crepant resolutions and open strings II

Brini, Andrea ; Cavalieri, Renzo.
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.

5. Stable rationality of higher dimensional conic bundles

Ahmadinezhad, Hamid ; Okada, Takuzo.
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 $n\geq 3$.

6. A characterization of finite vector bundles on Gauduchon astheno-Kahler manifolds

Biswas, Indranil ; Pingali, Vamsi Pritham.
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.

7. Hyper-K\"ahler Fourfolds Fibered by Elliptic Products

Kamenova, Ljudmila.
Every fibration of a projective hyper-K\"ahler 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.

8. The parabolic exotic t-structure

Achar , Pramod ,  ; Cooney , Nicholas ; Riche , Simon , .
Let G be a connected reductive algebraic group over an algebraically closed eld k, with simply connected derived subgroup. The exotic t-structure on the cotangent bundle of its ag 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-Mirkovi´cMirkovi´c conjecture. In this paper, we study (under mild technical assumptions) an analogous t-structure on the cotangent bundle of a partial ag 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-Mirkovi´cMirkovi´c conjecture for some singular blocks. [Français] Titre. La t-structure exotique parabolique. Soit G un groupe algébrique réductif connexe sur un corps k algébriquement clos. La t-structure exotique sur le bré cotangent de sa […]

9. Limits of the trivial bundle on a curve

Beauville, Arnaud.
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.

10. Hamiltonian actions of unipotent groups on compact K\"ahler manifolds

Greb, Daniel ; Miebach, Christian.
We study meromorphic actions of unipotent complex Lie groups on compact K\"ahler 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 K\"ahler 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.

11. Lefschetz (1,1)-theorem in tropical geometry

Jell, Philipp ; Rau, Johannes ; Shaw, Kristin.
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 Poincar\'e duality for integral tropical homology.

12. Troisi\`eme groupe de cohomologie non ramifi\'ee des torseurs universels sur les surfaces rationnelles

Cao, Yang.
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 Ch\^atelet surface and that this group is reduced to its $2$-primary part if $X$ is a del Pezzo surface of degree at least 2.

13. Algebraic models of the Euclidean plane

Blanc, Jérémy ; Dubouloz, Adrien.
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.