Volume 3

Third volume of Épijournal de Géométrie Algébrique - 2019

1. Infinite families of inequivalent real circle actions on affine four-space

Moser-Jauslin Lucy.
The main result of this article is to construct infinite families of non-equivalent equivariant real forms of linear C*-actions on affine four-space. We consider the real form of $\mathbb{C}^*$ whose fixed point is a circle. In [F-MJ] one example of a non-linearizable circle action was constructed. Here, this result is generalized by developing a new approach which allows us to compare different real forms. The constructions of these forms are based on the structure of equivariant $\mathrm{O}_2(\mathbb{C})$-vector bundles.

2. Double spinor Calabi-Yau varieties

Manivel, Laurent.
Consider the ten-dimensional spinor variety in the projectivization of a half-spin representation of dimension sixteen. The intersection X of two general translates of this variety is a smooth Calabi-Yau fivefold, as well as the intersection Y of their projective duals. We prove that although X and Y are not birationally equivalent, they are derived equivalent and L-equivalent in the sense of Kuznetsov and Shinder.

3. $\overline{M}_{1,n}$ is usually not uniruled in characteristic $p$

Sawin, Will.
Using etale cohomology, we define a birational invariant for varieties in characteristic $p$ that serves as an obstruction to uniruledness - a variant on an obstruction to unirationality due to Ekedahl. We apply this to $\overline{M}_{1,n}$ and show that $\overline{M}_{1,n}$ is not uniruled in characteristic $p$ as long as $n \geq p \geq 11$. To do this, we use Deligne's description of the etale cohomology of $\overline{M}_{1,n}$ and apply the theory of congruences between modular forms.

4. Socle pairings on tautological rings

Janda, Felix ; Pixton, Aaron.
We study some aspects of the $\lambda_g$ pairing on the tautological ring of $M_g^c$, the moduli space of genus $g$ stable curves of compact type. We consider pairing kappa classes with pure boundary strata, all tautological classes supported on the boundary, or the full tautological ring. We prove that the rank of this restricted pairing is equal in the first two cases and has an explicit formula in terms of partitions, while in the last case the rank increases by precisely the rank of the $\lambda_g\lambda_{g - 1}$ pairing on the tautological ring of $M_g$.

5. Smooth affine group schemes over the dual numbers

ROMAGNY, Matthieu ; Tossici, Dajano.
We provide an equivalence between the category of affine, smooth group schemes over the ring of generalized dual numbers $k[I]$, and the category of extensions of the form $1 \to \text{Lie}(G, I) \to E \to G \to 1$ where G is an affine, smooth group scheme over k. Here k is an arbitrary commutative ring and $k[I] = k \oplus I$ with $I^2 = 0$. The equivalence is given by Weil restriction, and we provide a quasi-inverse which we call Weil extension. It is compatible with the exact structures and the $\mathbb{O}_k$-module stack structures on both categories. Our constructions rely on the use of the group algebra scheme of an affine group scheme; we introduce this object and establish its main properties. As an application, we establish a Dieudonné classification for smooth, commutative, unipotent group schemes over $k[I]$.

6. P-adic lattices are not Kähler groups

Klingler, Bruno.
In this note we show that any lattice in a simple p-adic Lie group is not the fundamental group of a compact Ka\"hler manifold, as well as some variants of this result.

7. Irregular Hodge numbers of confluent hypergeometric differential equations

Sabbah, Claude ; Yu, Jeng-Daw.
We give a formula computing the irregular Hodge numbers for a confluent hypergeometric differential equation.

8. Equivariant perverse sheaves on Coxeter arrangements and buildings

Weissman, Martin H..
When $W$ is a finite Coxeter group acting by its reflection representation on $E$, we describe the category ${\mathsf{Perv}}_W(E_{\mathbb C}, {\mathcal{H}}_{\mathbb C})$ of $W$-equivariant perverse sheaves on $E_{\mathbb C}$, smooth with respect to the stratification by reflection hyperplanes. By using Kapranov and Schechtman's recent analysis of perverse sheaves on hyperplane arrangements, we find an equivalence of categories from ${\mathsf{Perv}}_W(E_{\mathbb C}, {\mathcal{H}}_{\mathbb C})$ to a category of finite-dimensional modules over an algebra given by explicit generators and relations. We also define categories of equivariant perverse sheaves on affine buildings, e.g., $G$-equivariant perverse sheaves on the Bruhat--Tits building of a $p$-adic group $G$. In this setting, we find that a construction of Schneider and Stuhler gives equivariant perverse sheaves associated to depth zero representations.

9. Pluricomplex Green's functions and Fano manifolds

McCleerey, Nicholas ; Tosatti, Valentino.
We show that if a Fano manifold does not admit Kahler-Einstein metrics then the Kahler potentials along the continuity method subconverge to a function with analytic singularities along a subvariety which solves the homogeneous complex Monge-Ampere equation on its complement, confirming an expectation of Tian-Yau.

10. Q_l-cohomology projective planes and singular Enriques surfaces in characteristic two

Schütt, Matthias.
We classify singular Enriques surfaces in characteristic two supporting a rank nine configuration of smooth rational curves. They come in one-dimensional families defined over the prime field, paralleling the situation in other characteristics, but featuring novel aspects. Contracting the given rational curves, one can derive algebraic surfaces with isolated ADE-singularities and trivial canonical bundle whose Q_l-cohomology equals that of a projective plane. Similar existence results are developed for classical Enriques surfaces. We also work out an application to integral models of Enriques surfaces (and K3 surfaces).

11. Sur l'hyperbolicité de graphes associés au groupe de Cremona

Lonjou, Anne.
To reinforce the analogy between the mapping class group and the Cremona group of rank $2$ over an algebraic closed field, we look for a graph analoguous to the curve graph and such that the Cremona group acts on it non-trivially. A candidate is a graph introduced by D. Wright. However, we demonstrate that it is not Gromov-hyperbolic. This answers a question of A. Minasyan and D. Osin. Then, we construct two graphs associated to a Vorono\"i tesselation of the Cremona group introduced in a previous work of the autor. We show that one is quasi-isometric to the Wright graph. We prove that the second one is Gromov-hyperbolic.

12. On the B-Semiampleness Conjecture

Floris, Enrica ; Lazić, Vladimir.
The B-Semiampleness Conjecture of Prokhorov and Shokurov predicts that the moduli part in a canonical bundle formula is semiample on a birational modification. We prove that the restriction of the moduli part to any sufficiently high divisorial valuation is semiample, assuming the conjecture in lower dimensions.

13. Chern classes of automorphic vector bundles, II

Esnault, Hélène ; Harris, Michel.
We prove that the $\ell$-adic Chern classes of canonical extensions of automorphic vector bundles, over toroidal compactifications of Shimura varieties of Hodge type over $\bar{ \mathbb{Q}}_p$, descend to classes in the $\ell$-adic cohomology of the minimal compactifications. These are invariant under the Galois group of the $p$-adic field above which the variety and the bundle are defined. Epiga, Vol. 3 (2019), Art. Nr. 14

14. Coincidence of two Swan conductors of abelian characters

Kato, Kazuya ; Saito, Takeshi.
There are two ways to define the Swan conductor of an abelian character of the absolute Galois group of a complete discrete valuation field. We prove that these two Swan conductors coincide.

15. Isomorphisms between complements of projective plane curves

Hemmig, Mattias.
In this article, we study isomorphisms between complements of irreducible curves in the projective plane $\mathbb{P}^2$, over an arbitrary algebraically closed field. Of particular interest are rational unicuspidal curves. We prove that if there exists a line that intersects a unicuspidal curve $C \subset \mathbb{P}^2$ only in its singular point, then any other curve whose complement is isomorphic to $\mathbb{P}^2 \setminus C$ must be projectively equivalent to $C$. This generalizes a result of H. Yoshihara who proved this result over the complex numbers. Moreover, we study properties of multiplicity sequences of irreducible curves that imply that any isomorphism between the complements of these curves extends to an automorphism of $\mathbb{P}^2$. Using these results, we show that two irreducible curves of degree $\leq 7$ have isomorphic complements if and only if they are projectively equivalent. Finally, we describe new examples of irreducible projectively non-equivalent curves […]

16. Higher rank sheaves on threefolds and functional equations

Gholampour, Amin ; Kool, Martijn.
We consider the moduli space of stable torsion free sheaves of any rank on a smooth projective threefold. The singularity set of a torsion free sheaf is the locus where the sheaf is not locally free. On a threefold it has dimension $\leq 1$. We consider the open subset of moduli space consisting of sheaves with empty or 0-dimensional singularity set. For fixed Chern classes $c_1,c_2$ and summing over $c_3$, we show that the generating function of topological Euler characteristics of these open subsets equals a power of the MacMahon function times a Laurent polynomial. This Laurent polynomial is invariant under $q \leftrightarrow q^{-1}$ (upon replacing $c_1 \leftrightarrow -c_1$). For some choices of $c_1,c_2$ these open subsets equal the entire moduli space. The proof involves wall-crossing from Quot schemes of a higher rank reflexive sheaf to a sublocus of the space of Pandharipande-Thomas pairs. We interpret this sublocus in terms of the singularities of the reflexive sheaf.

17. Rigidity properties of holomorphic Legendrian singularities

Hwang, Jun-Muk.
We study the singularities of Legendrian subvarieties of contact manifolds in the complex-analytic category and prove two rigidity results. The first one is that Legendrian singularities with reduced tangent cones are contactomorphically biholomorphic to their tangent cones. This result is partly motivated by a problem on Fano contact manifolds. The second result is the deformation-rigidity of normal Legendrian singularities, meaning that any holomorphic family of normal Legendrian singularities is trivial, up to contactomorphic biholomorphisms of germs. Both results are proved by exploiting the relation between infinitesimal contactomorphisms and holomorphic sections of the natural line bundle on the contact manifold.

18. CARTAN GEOMETRIES ON COMPLEX MANIFOLDS OF ALGEBRAIC DIMENSION ZERO

Biswas, Indranil ; Dumitrescu, Sorin ; McKay, Benjamin.
We show that compact complex manifolds of algebraic dimension zero bearing a holomorphic Cartan geometry of algebraic type have infinite fundamental group. This generalizes the main Theorem in [DM] where the same result was proved for the special cases of holomorphic affine connections and holomorphic conformal structures.