| Résume | The theory of weight functions by Mustaţă-Nicaise on Berkovich spaces hints at an intrinsic approach to studying skeleta without the limitations of the usual tools from algebraic geometry, i.e., semistable reduction or resolution of singularities. In this talk we discuss how Mustaţă-Nicaise's construction can be generalised to define weight functions intrinsically on any quasi-smooth Berkovich space as a sort of normalised log-discrepancy of a canonical form.
Next, we turn our attention to computing weight functions on the Berkovich affine line over any non-Archimedean field K. This question reduces to the entirely value-theoretic problem of describing the integral log differentials $\Omega^{\log}_{L^\circ / K^\circ}$ for a simple extension of real valued fields $L / K$ , which we describe using the data in MacLane-Vaquié chains.
We find that on the affine line, weight functions are completely determined by the integral piecewise linear (Z-PL) structure the affine line and vice versa, yielding new insight into the Z-PL structure.
Finally, we discuss how this can be used to define and compute the essential skeleton of certain Berkovich spaces.
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