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Volume 7
Septième volume de l'Épijournal de Géométrie Algébrique - 2023
1. Serre-invariant stability conditions and Ulrich bundles on cubic threefolds
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.
2. Smoothability of relative stable maps to stacky curves
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.
3. Quartic surfaces, their bitangents and rational points
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.
4. On the motive of O'Grady's six dimensional hyper-Kähler varieties
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.
5. Abundance for slc surfaces over arbitrary fields
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.
6. Families of stable 3-folds in positive characteristic
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.
7. Trace formalism for motivic cohomology
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.
8. Group-theoretic Johnson classes and a non-hyperelliptic curve with torsion Ceresa class
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.
9. Uniform K-stability of polarized spherical varieties
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.
10. Reduction of Kummer surfaces modulo 2 in the non-supersingular case
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.
11. Stably semiorthogonally indecomposable varieties
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.
12. Big Picard theorems and algebraic hyperbolicity for varieties admitting a variation of Hodge structures
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.
13. Algebraic subgroups of the group of birational transformations of ruled surfaces
We classify the maximal algebraic subgroups of Bir(CxPP^1), when C is a smooth projective curve of positive genus.
14. Triangular arrangements on the projective plane
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.
15. Curve counting and S-duality
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.
16. Affine Subspace Concentration Conditions
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.
17. On equations of fake projective planes with automorphism group of order $21$
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.
18. Torus Actions on Quotients of Affine Spaces
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.
19. On a decomposition of $p$-adic Coxeter orbits
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.
20. Cohomology of moduli spaces via a result of Chenevier and Lannes
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 $|\lambda| \leq 16$.
21. Finite $F$-representation type for homogeneous coordinate rings of non-Fano varieties
Finite $F$-representation type is an important notion in characteristic-$p$ commutative algebra, but explicit examples of varieties with or without this property are few. We prove that a large class of homogeneous coordinate rings in positive characteristic will fail to have finite $F$-representation type. To do so, we prove a connection between differential operators on the homogeneous coordinate ring of $X$ and the existence of global sections of a twist of $(\mathrm{Sym}^m \Omega_X)^\vee$. By results of Takagi and Takahashi, this allows us to rule out FFRT for coordinate rings of varieties with $(\mathrm{Sym}^m \Omega_X)^\vee$ not ``positive''. By using results positivity and semistability conditions for the (co)tangent sheaves, we show that several classes of varieties fail to have finite $F$-representation type, including abelian varieties, most Calabi--Yau varieties, and complete intersections of general type. Our work also provides examples of the structure of the ring of differential operators for non-$F$-pure varieties, which to this point have largely been unexplored.
22. On the minimal model program for projective varieties with pseudo-effective tangent sheaf
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.
23. Generalizations of quasielliptic curves
We generalize the notion of quasielliptic curves, which have infinitesimal symmetries and exist only in characteristic two and three, to a remarkable hierarchy of regular curves having infinitesimal symmetries, defined in all characteristics and having higher genera. This relies on the study of certain infinitesimal group schemes acting on the affine line and certain compactifications. The group schemes are defined in terms of invertible additive polynomials over rings with nilpotent elements, and the compactification is constructed with the theory of numerical semigroups. The existence of regular twisted forms relies on Brion's recent theory of equivariant normalization. Furthermore, extending results of Serre from the realm of group cohomology, we describe non-abelian cohomology for semidirect products, to compute in special cases the collection of all twisted forms.