Extending meromorphic connections to coadmissible ̑𝒟-modules

We investigate when a meromorphic connection on a smooth rigid analytic variety 𝑋 gives rise to a coadmissible D ⏜ X \overparen{\mathcal{D}}_{X} -module, and show that this is always the case when the roots of the corresponding 𝑏-functions are all of positive type. We also use this theory to give an example of an integrable connection on the punctured unit disk whose pushforward is not a coadmissible module.

Mixed Bruce–Roberts numbers

We extend the notions of μ*-sequences and Tjurina numbers of functions to the framework of Bruce–Roberts numbers, that is, to pairs formed by the germ at 0 of a complex analytic variety X ⊆ ℂn and a finitely ${\mathcal R}(X)$-determined analytic function germ f : (ℂn, 0) → (ℂ, 0). We analyze some fundamental properties of these numbers.

LOGARITHMIC DE RHAM COMPARISON FOR OPEN RIGID SPACES

In this note, we prove the logarithmic $p$ -adic comparison theorem for open rigid analytic varieties. We prove that a smooth rigid analytic variety with a strict simple normal crossing divisor is locally $K(\unicode[STIX]{x1D70B},1)$ (in a certain sense) with respect to $\mathbb{F}_{p}$ -local systems and ramified coverings along the divisor. We follow Scholze’s method to produce a pro-version of the Faltings site and use this site to prove a primitive comparison theorem in our setting. After introducing period sheaves in our setting, we prove aforesaid comparison theorem.

Constancy of generalized Hodge–Tate weights of a local system

Sen attached to each $p$-adic Galois representation of a $p$-adic field a multiset of numbers called generalized Hodge–Tate weights. In this paper, we discuss a rigidity of these numbers in a geometric family. More precisely, we consider a $p$-adic local system on a rigid analytic variety over a $p$-adic field and show that the multiset of generalized Hodge–Tate weights of the local system is constant. The proof uses the $p$-adic Riemann–Hilbert correspondence by Liu and Zhu, a Sen–Fontaine decompletion theory in the relative setting, and the theory of formal connections. We also discuss basic properties of Hodge–Tate sheaves on a rigid analytic variety.

Goresky–Pardon lifts of Chern classes and associated Tate extensions

Let $X$ be an irreducible complex-analytic variety, ${\mathcal{S}}$ a stratification of $X$ and ${\mathcal{F}}$ a holomorphic vector bundle on the open stratum ${X\unicode[STIX]{x0030A}}$. We give geometric conditions on ${\mathcal{S}}$ and ${\mathcal{F}}$ that produce a natural lift of the Chern class $\operatorname{c}_{k}({\mathcal{F}})\in H^{2k}({X\unicode[STIX]{x0030A}};\mathbb{C})$ to $H^{2k}(X;\mathbb{C})$, which, in the algebraic setting, is of Hodge level ${\geqslant}k$. When applied to the Baily–Borel compactification $X$ of a locally symmetric variety ${X\unicode[STIX]{x0030A}}$ and an automorphic vector bundle ${\mathcal{F}}$ on ${X\unicode[STIX]{x0030A}}$, this refines a theorem of Goresky–Pardon. In passing we define a class of simplicial resolutions of the Baily–Borel compactification that can be used to define its mixed Hodge structure. We use this to show that the stable cohomology of the Satake ($=$ Baily–Borel) compactification of ${\mathcal{A}}_{g}$ contains nontrivial Tate extensions.

NON-NEGATIVE DEFORMATIONS OF WEIGHTED HOMOGENEOUS SINGULARITIES

We consider a weighted homogeneous germ of complex analytic variety (X, 0) ⊂ (ℂn, 0) and a function germ f : (ℂn, 0) → (ℂ, 0). We derive necessary and sufficient conditions for some deformations to have non-negative degree (i.e., for any additional term in the deformation, the weighted degree is not smaller) in terms of an adapted version of the relative Milnor number. We study the cases where (X, 0) is an isolated hypersurface singularity and the invariant is the Bruce-Roberts number of f with respect to (X, 0), and where (X, 0) is an isolated complete intersection or a curve singularity and the invariant is the Milnor number of the germ f: (X, 0) → ℂ. In the last part, we give some formulas for the invariants in terms of the weights and the degrees of the polynomials.

QUATERNIONIC MODULAR FORMS OF ANY WEIGHT

In this work we give a geometric definition, as sections of line bundles, of p-adic analytic families of overconvergent modular forms attached to an indefinite quaternion algebra over ℚ. As a consequence of this, we obtain the existence of an eigencurve in this context. Our theory includes the interpretation of a modular form as a rule on test objects. We introduce the Hecke operators U and T l, both in families and for a single weight. We show that the U -operator acts compactly on the space of overconvergent modular forms. We finally construct the eigencurve, a rigid analytic variety whose points correspond to systems of eigenvalues associated to overconvergent eigenforms of finite slope with respect to the U -operator.

Invariants of topological relative right equivalences

Let (V,0) be the germ of an analytic variety in $\mathbb{C}^n$ and f an analytic function germ defined on V. For functions with isolated singularity on V, Bruce and Roberts introduced a generalization of the Milnor number of f, which we call Bruce–Roberts number, μBR(V,f). Like the Milnor number of f, this number shows some properties of f and V. In this paper we investigate algebraic and geometric characterizations of the constancy of the Bruce–Roberts number for families of functions with isolated singularities on V. We also discuss the topological invariance of the Bruce–Roberts number for families of quasihomogeneous functions defined on quasihomogeneous varieties. As application of the results, we prove a relative version of the Zariski multiplicity conjecture for quasihomogeneous varieties.