Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Harmonic covers of skeleta

Art Waeterschoot

TL;DR

This work develops a tropical, harmonic-analytic framework for skeleta arising from toroidal schemes over discretely valued fields. By establishing a balancing condition for covers of \mathbb{Z}-PL dual complexes, a Poincaré-Lelong slope formula, and a Riemann-Hurwitz type relation in arbitrary dimension, it connects combinatorial, metric, and algebro-geometric data via the different, canonical, and log-canonical structures. The results show that finite toroidal covers induce harmonic morphisms on skeleta and provide a tropical RH formula that ties the Laplacian of the different to the tropical relative canonical divisor, with a robust metrisation theory for \mathbb{Z}-PL complexes and a systematic treatment of specialisation of Cartier divisors. These tools yield a cohesive bridge between Berkovich geometry, tropical geometry, and log-geometry with potential applications to degeneration and reduction phenomena in higher dimensions.

Abstract

The geometry of a toroidal scheme over a DVR is encoded in a $\mathbb{Z}$-PL space known as the dual polyhedral complex. Any such dual complex is a skeleton, i.e. a nonarchimedean analytic retract, and admits a combinatorial divisor theory via specialisation. These structures on the dual complex interact via a Poincaré-Lelong slope formula, which interprets specialisations of divisors as Laplacians of PL functions. The main result presented here shows that finite covers of toroidal schemes give harmonic morphisms of dual complexes, i.e. morphisms that preserve the tropical Laplace equation. A crucial ingredient is a balancing condition which is a variant of the tropical multiplicity formula for dual complexes. We apply these results to obtain a Riemann-Hurwitz formula for covers of skeleta in any dimension: the Laplacian of the different function is the tropical relative canonical divisor.

Harmonic covers of skeleta

TL;DR

This work develops a tropical, harmonic-analytic framework for skeleta arising from toroidal schemes over discretely valued fields. By establishing a balancing condition for covers of \mathbb{Z}-PL dual complexes, a Poincaré-Lelong slope formula, and a Riemann-Hurwitz type relation in arbitrary dimension, it connects combinatorial, metric, and algebro-geometric data via the different, canonical, and log-canonical structures. The results show that finite toroidal covers induce harmonic morphisms on skeleta and provide a tropical RH formula that ties the Laplacian of the different to the tropical relative canonical divisor, with a robust metrisation theory for \mathbb{Z}-PL complexes and a systematic treatment of specialisation of Cartier divisors. These tools yield a cohesive bridge between Berkovich geometry, tropical geometry, and log-geometry with potential applications to degeneration and reduction phenomena in higher dimensions.

Abstract

The geometry of a toroidal scheme over a DVR is encoded in a -PL space known as the dual polyhedral complex. Any such dual complex is a skeleton, i.e. a nonarchimedean analytic retract, and admits a combinatorial divisor theory via specialisation. These structures on the dual complex interact via a Poincaré-Lelong slope formula, which interprets specialisations of divisors as Laplacians of PL functions. The main result presented here shows that finite covers of toroidal schemes give harmonic morphisms of dual complexes, i.e. morphisms that preserve the tropical Laplace equation. A crucial ingredient is a balancing condition which is a variant of the tropical multiplicity formula for dual complexes. We apply these results to obtain a Riemann-Hurwitz formula for covers of skeleta in any dimension: the Laplacian of the different function is the tropical relative canonical divisor.

Paper Structure

This paper contains 9 sections, 38 theorems, 192 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background on logarithmic regularity
  3. Balanced covers of dual complexes
  4. Metrisation of $\mathbb{Z}$-PL complexes
  5. Specialisation of Cartier divisors
  6. Harmonic morphism of $\mathbb{Z}$-PL tropical complexes
  7. Complements on determinants of log-differentials
  8. Riemann-Hurwitz for skeleta
  9. Conventions on PL geometry

Key Result

Theorem 1.4

Let $f:X'^\dagger\to {X}^\dagger$ be a finite morphism of toroidal $S^\dagger$-schemes. Then is a balanced cover of $\mathbb{Z}$-PL complexes and

Figures (3)

  • Figure 1.3: A finite cover of $\mathbb{Z}$-PL complexes is called balanced if the local degree $\deg_{\tau'}\varphi\coloneqq \sum_{(\sigma',\tau')\mapsto (\sigma,\tau)}[M_{\sigma'}:M_{\sigma}]$ is independent of $\sigma$ (Definition \ref{['balanced cover Z-pl spaces']}).
  • Figure 4.7: Picture of a bounded face $\sigma_x$ as the intersection of the dual cone $C(x)$ and the plane $H_{1,x}$.
  • Figure 4.8: Picture of an unbounded face$\sigma_x$ as the intersection of the dual cone $C(x)$ and the plane $H_{1,x}$. The number of vertices of $\sigma_x$ is $b(x)$, for some $1\le b(x)<s(x)$.

Theorems & Definitions (114)

  • Theorem 1.4: Balancing condition, See Theorem \ref{['thm: balanced']}
  • Theorem 1.7: Poincaré-Lelong slope formula, see Theorem \ref{['thm: pll']}
  • Theorem 1.9: Harmonic morphisms, see Theorem \ref{['thm: harmonic morphisms']}
  • Theorem 1.11: Riemann-Hurwitz formula for skeleta, see Theorem \ref{['thm:rh']}
  • Example 2.1: snc divisors
  • Lemma 2.2
  • proof
  • Remark 2.5
  • Proposition 2.6: Toroidal resolutions
  • proof
  • ...and 104 more