Volume 6

Sixth volume of Épijournal de Géométrie Algébrique - 2022

1. Finite torsors over strongly $F$-regular singularities

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 analgebraically closed field of characteristic $p>0$. We prove the existence of afinite 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 solvableneutral component, every $G$-torsor over a big open of $\mathrm{Spec}R^{\star}$ extends to a $G$-torsor everywhere. To achieve this, we obtain ageneralized transformation rule for the $F$-signature under finite localextensions. 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 groupof globally $F$-regular varieties is torsion-free. Likewise, it shows thatcanonical covers of $\mathbb{Q}$-Gorenstein strongly $F$-regular singularitiesare strongly $F$-regular.

2. The Mori fan of the Dolgachev-Nikulin-Voisin family in genus $2$

Klaus Hulek ; Carsten Liese.
In this paper we study the Mori fan of the Dolgachev-Nikulin-Voisin family indegree $2$ as well as the associated secondary fan. The main result is anenumeration of all maximal dimensional cones of the two fans.

3. Integral cohomology of quotients via toric geometry

Grégoire Menet.
We describe the integral cohomology of $X/G$ where $X$ is a compact complexmanifold 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-upsof quotients of $\mathbb{C}^n$. We also provide necessary and sufficientconditions for the spectral sequence of equivariant cohomology of $(X,G)$ todegenerate at the second page. As an application, we compute theBeauville--Bogomolov form of $X/G$ when $X$ is a Hilbert scheme of points on aK3 surface and $G$ a symplectic automorphism group of orders 5 or 7.

4. Mirror symmetry for Nahm branes

Emilio Franco ; Marcos Jardim.
The Dirac--Higgs bundle is a hyperholomorphic bundle over the moduli space ofstable Higgs bundles of coprime rank and degree. We provide an algebraicgeneralization to the case of trivial degree and the rank higher than $1$. Thisallow us to generalize to this case the Nahm transform defined by Frejlich andthe second named author, which, out of a stable Higgs bundle, produces a vectorbundle with connection over the moduli space of rank 1 Higgs bundles. Byperforming the higher rank Nahm transform we obtain a hyperholomorphic bundlewith connection over the moduli space of stable Higgs bundles of rank $n$ anddegree 0, twisted by the gerbe of liftings of the projective universal bundle. Such hyperholomorphic vector bundles over the moduli space of stable Higgsbundles can be seen, in the physicist's language, as BBB-branes twisted by theabove mentioned gerbe. We refer to these objects as Nahm branes. Finally, westudy the behaviour of Nahm branes under Fourier--Mukai transform over thesmooth locus of the Hitchin fibration, checking that the resulting objects aresupported on a Lagrangian multisection of the Hitchin fibration, so theydescribe partial data of BAA-branes.

5. Density of Arithmetic Representations of Function Fields

Hélène Esnault ; Moritz Kerz.
We propose a conjecture on the density of arithmetic points in thedeformation space of representations of the étale fundamental group inpositive characteristic. This? conjecture has applications to étalecohomology theory, for example it implies a Hard Lefschetz conjecture. We provethe density conjecture in tame degree two for the curve $\mathbb{P}^1\setminus\{0,1,\infty\}$. v2: very small typos corrected.v3: final. Publication inEpiga.

6. $G$-fixed Hilbert schemes on $K3$ surfaces, modular forms, and eta products

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 schemesof $G$-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 anexplicit formula for $Z_{X,G}$ in terms of the Dedekind eta function for all 82possible $(X,G)$. The key intermediate result we prove is of independentinterest: it establishes an eta product identity for a certain shifted thetafunction of the root lattice of a simply laced root system. We extend ourresults to various refinements of the Euler characteristic, namely the Ellipticgenus, the Chi-$y$ genus, and the motivic class.

7. The conjectures of Artin-Tate and Birch-Swinnerton-Dyer

S. Lichtenbaum ; N. Ramachandran ; T. Suzuki.
We provide two proofs that the conjecture of Artin-Tate for a fibered surfaceis equivalent to the conjecture of Birch-Swinnerton-Dyer for the Jacobian ofthe generic fibre. As a byproduct, we obtain a new proof of a theorem ofGeisser relating the orders of the Brauer group and the Tate-Shafarevich group.

8. A conjectural formula for $DR_g(a,-a) \lambda_g$

Alexandr Buryak ; Francisco Hernández Iglesias ; Sergey Shadrin.
We propose a conjectural formula for $DR_g(a,-a) \lambda_g$ and check all itsexpected properties. Our formula refines the one point case of a similarconjecture made by the first named author in collaboration with Guéré andRossi, and we prove that the two conjectures are in fact equivalent, though ina quite non-trivial way.

9. Automorphism group schemes of bielliptic and quasi-bielliptic surfaces

Gebhard Martin.
Bielliptic and quasi-bielliptic surfaces form one of the four classes ofminimal smooth projective surfaces of Kodaira dimension $0$. In this article,we determine the automorphism schemes of these surfaces over algebraicallyclosed fields of arbitrary characteristic, generalizing work of Bennett andMiranda over the complex numbers; we also find some cases that are missing fromthe classification of automorphism groups of bielliptic surfaces incharacteristic $0$.

10. Examples of surfaces with canonical map of degree 4

Carlos Rito.
We give two examples of surfaces with canonical map of degree 4 onto acanonical surface.

11. Autour de la conjecture de Tate entière pour certains produits de dimension $3$ sur un corps fini

Federico Scavia.
Let $X$ be the product of a surface satisfying $b_2=\rho$ and of a curve overa finite field. We study a strong form of the integral Tate conjecture for$1$-cycles on $X$. We generalize and give unconditional proofs of severalresults of our previous paper with J.-L. Colliot-Thélène.

12. A gluing construction of projective K3 surfaces

Takayuki Koike ; Takato Uehara.
We construct a non-Kummer projective K3 surface $X$ which admits compactLevi-flats by holomorphically patching two open complex surfaces obtained asthe complements of tubular neighborhoods of elliptic curves embedded inblow-ups of the projective plane at nine general points.

13. Moduli spaces on the Kuznetsov component of Fano threefolds of index 2

Matteo Altavilla ; Marin Petkovic ; Franco Rota.
General hyperplane sections of a Fano threefold $Y$ of index 2 and Picardrank 1 are del Pezzo surfaces, and their Picard group is related to a rootsystem. To the corresponding roots, we associate objects in the Kuznetsovcomponent of $Y$ and investigate their moduli spaces, using the stabilitycondition constructed by Bayer, Lahoz, Macrì, and Stellari, and theAbel--Jacobi map. We identify a subvariety of the moduli space isomorphic to$Y$ itself, and as an application we prove a (refined) categorical Torellitheorem for general quartic double solids.

14. Chern currents of coherent sheaves

Richard Lärkäng ; Elizabeth Wulcan.
Given a finite locally free resolution of a coherent analytic sheaf $\mathcalF$, equipped with Hermitian metrics and connections, we construct an explicitcurrent, obtained as the limit of certain smooth Chern forms of $\mathcal F$,that represents the Chern class of $\mathcal F$ and has support on the supportof $\mathcal F$. If the connections are $(1,0)$-connections and $\mathcal F$has pure dimension, then the first nontrivial component of this Chern currentcoincides with (a constant times) the fundamental cycle of $\mathcal F$. Theproof of this goes through a generalized Poincaré-Lelong formula, previouslyobtained by the authors, and a result that relates the Chern current to theresidue current associated with the locally free resolution.

15. Tropicalization of the universal Jacobian

Margarida Melo ; Samouil Molcho ; Martin Ulirsch ; Filippo Viviani.
In this article we provide a stack-theoretic framework to study the universaltropical Jacobian over the moduli space of tropical curves. We develop twoapproaches to the process of tropicalization of the universal compactifiedJacobian over the moduli space of curves -- one from a logarithmic and theother from a non-Archimedean analytic point of view. The central result fromboth points of view is that the tropicalization of the universal compactifiedJacobian is the universal tropical Jacobian and that the tropicalization mapsin each of the two contexts are compatible with the tautological morphisms. Ina sequel we will use the techniques developed here to provide explicitpolyhedral models for the logarithmic Picard variety.

16. Higher Hida and Coleman theories on the modular curve

George Boxer ; Vincent Pilloni.
We construct Hida and Coleman theories for the degree 0 and 1 cohomology ofautomorphic line bundles on the modular curve and we define a p-adic dualitypairing between the theories in degree 0 and 1.

17. Néron models of Jacobians over bases of arbitrary dimension

Thibault Poiret.
We work with a smooth relative curve $X_U/U$ with nodal reduction over anexcellent and locally factorial scheme $S$. We show that blowing up a nodalmodel of $X_U$ in the ideal sheaf of a section yields a new nodal model, anddescribe how these models relate to each other. We construct a Néron modelfor the Jacobian of $X_U$, and describe it locally on $S$ as a quotient of thePicard space of a well-chosen nodal model. We provide a combinatorial criterionfor the Néron model to be separated.

18. A characterization of finite étale morphisms in tensor triangular geometry

Beren Sanders.
We provide a characterization of finite étale morphisms in tensortriangular geometry. They are precisely those functors which have aconservative right adjoint, satisfy Grothendieck--Neeman duality, and for whichthe relative dualizing object is trivial (via a canonically-defined map).

19. An atlas of K3 surfaces with finite automorphism group

Xavier Roulleau.
We study the geometry of the K3 surfaces $X$ with a finite numberautomorphisms and Picard number $\geq 3$. We describe these surfaces classifiedby Nikulin and Vinberg as double covers of simpler surfaces or embedded in aprojective space. We study moreover the configurations of their finite set of$(-2)$-curves.

20. Etale and crystalline companions, I

Kiran S. Kedlaya.
Let $X$ be a smooth scheme over a finite field of characteristic $p$.Consider the coefficient objects of locally constant rank on $X$ in $\ell$-adicWeil cohomology: these are lisse Weil sheaves in étale cohomology when $\ell\neq p$, and overconvergent $F$-isocrystals in rigid cohomology when $\ell=p$.Using the Langlands correspondence for global function fields in both theétale and crystalline settings (work of Lafforgue and Abe, respectively), onesees that on a curve, any coefficient object in one category has "companions"in the other categories with matching characteristic polynomials of Frobeniusat closed points. A similar statement is expected for general $X$; building onwork of Deligne, Drinfeld showed that any étale coefficient object hasétale companions. We adapt Drinfeld's method to show that any crystallinecoefficient object has étale companions; this has been shown independently byAbe--Esnault. We also prove some auxiliary results relevant for theconstruction of crystalline companions of étale coefficient objects; thissubject will be pursued in a subsequent paper.

21. The behavior of essential dimension under specialization

Zinovy Reichstein ; Federico Scavia.
Let $A$ be a discrete valuation ring with generic point $\eta$ and closedpoint $s$. We show that in a family of torsors over $\operatorname{Spec}(A)$,the essential dimension of the torsor above $s$ is less than or equal to theessential dimension of the torsor above $\eta$. We give two applications ofthis result, one in mixed characteristic, the other in equal characteristic.

22. Walls and asymptotics for Bridgeland stability conditions on 3-folds

Marcos Jardim ; Antony Maciocia.
We consider Bridgeland stability conditions for three-folds conjectured byBayer-Macrì-Toda in the case of Picard rank one. We study the differentialgeometry of numerical walls, characterizing when they are bounded, discussingpossible intersections, and showing that they are essentially regular. Next, weprove that walls within a certain region of the upper half plane thatparametrizes geometric stability conditions must always intersect the curvegiven by the vanishing of the slope function and, for a fixed value of s, havea maximum turning point there. We then use all of these facts to prove thatGieseker semistability is equivalent to asymptotic semistability along a classof paths in the upper half plane, and to show how to find large families ofwalls. We illustrate how to compute all of the walls and describe theBridgeland moduli spaces for the Chern character (2,0,-1,0) on complexprojective 3-space in a suitable region of the upper half plane.

23. Automorphisms of $\mathbb{P}^1$-bundles over rational surfaces

Jérémy Blanc ; Andrea Fanelli ; Ronan Terpereau.
In this paper we provide the complete classification of$\mathbb{P}^1$-bundles over smooth projective rational surfaces whose neutralcomponent of the automorphism group is maximal. Our results hold over anyalgebraically closed field of characteristic zero.

24. Maximality of moduli spaces of vector bundles on curves

Erwan Brugallé ; Florent Schaffhauser.
We prove that moduli spaces of semistable vector bundles of coprime rank anddegree over a non-singular real projective curve are maximal real algebraicvarieties if and only if the base curve itself is maximal. This provides a newfamily of maximal varieties, with members of arbitrarily large dimension. Weprove the result by comparing the Betti numbers of the real locus to the Hodgenumbers of the complex locus and showing that moduli spaces of vector bundlesover a maximal curve actually satisfy a property which is stronger thanmaximality and that we call Hodge-expressivity. We also give a brief account onother varieties for which this property was already known.