Articles récemment publiés

Wahl's local Euler characteristic measures the local contributions of a singularity to the usual Euler characteristic of a sheaf. Using tools from toric geometry, we study the local Euler characteristic of sheaves of symmetric differentials for isolated surface singularities of type $A_n$. We prove an explicit formula for the local Euler characteristic of the $m$th symmetric power of the cotangent bundle; this is a quasi-polynomial in $m$ of period $n+1$. We also express the components of the local Euler characteristic as a count of lattice points in a non-convex polyhedron, again showing it is a quasi-polynomial. We apply our computations to obtain new examples of algebraic quasi-hyperbolic surfaces in $\mathbb{P}^3$ of low degree. We show that an explicit family of surfaces with many singularities constructed by Labs has no genus $0$ curves for the members of degree at least $8$ and no curves of genus $0$ or $1$ for degree at least $10$.

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An action of a complex reductive group $\mathrm G$ on a smooth projective variety $X$ is regular when all regular unipotent elements in $\mathrm G$ act with finitely many fixed points. Then the complex $\mathrm G$-equivariant cohomology ring of $X$ is isomorphic to the coordinate ring of a certain regular fixed point scheme. Examples include partial flag varieties, smooth Schubert varieties and Bott-Samelson varieties. We also show that a more general version of the fixed point scheme allows a generalisation to GKM spaces, such as toric varieties.

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For a natural class of cohomology theories with support (including \'etale or pro-\'etale cohomology with suitable coefficients), we prove a moving lemma for cohomology classes with support on smooth quasi-projective k-varieties that admit a smooth projective compactification (e.g. if char(k)=0). This has the following consequences for such k-varieties and cohomology theories: a local and global generalization of the effacement theorem of Quillen, Bloch--Ogus, and Gabber, a finite level version of the Gersten conjecture in characteristic zero, and a generalization of the injectivity property and the codimension 1 purity theorem for \'etale cohomology. Our results imply that the refined unramified cohomology groups from [Sch23] are motivic.

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Let $\Omega^d$ be the $d$-dimensional Drinfeld symmetric space for a finite extension $F$ of $\mathbb{Q}_p$. Let $\Sigma^1$ be a geometrically connected component of the first Drinfeld covering of $\Omega^d$ and let $\mathbb{F}$ be the residue field of the unique degree $d+1$ unramified extension of $F$. We show that the natural homomorphism determined by the second Drinfeld covering from the group of characters of $(\mathbb{F}, +)$ to $\text{Pic}(\Sigma^1)[p]$ is injective. In particular, $\text{Pic}(\Sigma^1)[p] \neq 0$. We also show that all vector bundles on $\Omega^1$ are trivial, which extends the classical result that $\text{Pic}(\Omega^1) = 0$.

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We partially extend to hyperk\"ahler fourfolds of Kummer type the results that we have proved regarding stable rigid vector bundles on hyperk\"ahler (HK) varieties of type $K3^{[n]}$. Let $(M,h)$ be a general polarized HK fourfold of Kummer type such that $q_M(h)\equiv -6\pmod{16}$ and the divisibility of $h$ is $2$, or $q_M(h)\equiv -6\pmod{144}$ and the divisibility of $h$ is $6$. We show that there exists a unique (up to isomorphism) slope stable vector bundle $\cal F$ on $M$ such that $r({\cal F})=4$, $ c_1({\cal F})=h$, $\Delta({\cal F})=c_2(M)$. Moreover $\cal F$ is rigid. One of our motivations is the desire to describe explicitly a locally complete family of polarized HK fourfolds of Kummer type.

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