arXiv subjects

Michaël Maex

Publications and source records attributed to Michaël Maex.

3 recordsLinked to original sources

arXiv2026

The integral (log) cotangent complex of extensions of valued fields

Let $(L, v_L) / (K, v_K)$ be a finite or purely transcendental extension of real valued fields. We construct the associated integral cotangent and log cotangent complexes in terms of a MacLane-Vaquié chain approximating $v_L$. This leads to explicit formulas for associated invariants such as the (absolute) (log) different, weight norm and Kähler norm. As a corollary of our methods we obtain strong control of the higher homology of the integral (log) cotangent complex, generalizing an important result of Gabber and Ramero to the logarithmic setting.

math.AG↗

arXiv2025

A synthetic proof of the spherical and hyperbolic Pythagorean theorem on models in Euclidean and Minkowski space

There are multiple generalisations of the Pythagorean theorem to spherical and hyperbolic geometry. A natural one, involving areas of disks with radii equal to the sides of a proper triangle, was discovered in the hyperbolic case by Maria Teresa Calapso and generalised to the spherical case by Paolo Maraner. All known proofs are analytic, and Maraner posed the question of whether there is a synthetic proof. In this paper, we explain the statement of the theorem in a way that is accessible to a wide audience. Next, we give an elementary geometric proof of this theorem using the sphere in Euclidean space and the hyperboloid in Minkowski space as models for spherical and hyperbolic geometry.

math.MG↗

arXiv2025

Jumps of Jacobians via orthogonal canonical forms

Given a smooth, proper curve $C$ over a discretely valued field $k$, we equip the $k$-vector space $H^{0}(C,ω_{C/k})$ with a canonical discrete valuation $v_{\mathrm{can}}$ which measures how canonical forms degenerate on regular integral models of $C$. More precisely, $v_{\mathrm{can}}$ maps a canonical form to the minimal value of its associated weight function, as introduced by Mustaţă--Nicaise. Our main result states that $v_{\mathrm{can}}$ computes Edixhoven's jumps of the Jacobian of $C$ when evaluated in an orthogonal basis. As a byproduct, we deduce a short proof for the rationality of the jumps of Jacobians. We also show how $v_{\mathrm{can}}$ and the jumps can be computed efficiently for the class of $Δ_v$-regular curves introduced by Dokchitser.

math.AG↗