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Slope boundedness and Equidistribution theorem

Wenbin Luo

Abstract

In this article, we prove the boundedness of minimal slopes of adelic line bundles over function fields of characteristic 0. This can be applied to prove the equidistribution of generic and small points with respect to a big and semipositive adelic line bundle. Our methods can be applied to the finite places of number fields as well. We also show the continuity of $χ$-volumes over function fields.

Slope boundedness and Equidistribution theorem

Abstract

In this article, we prove the boundedness of minimal slopes of adelic line bundles over function fields of characteristic 0. This can be applied to prove the equidistribution of generic and small points with respect to a big and semipositive adelic line bundle. Our methods can be applied to the finite places of number fields as well. We also show the continuity of -volumes over function fields.
Paper Structure (14 sections, 2 theorems, 92 equations)

This paper contains 14 sections, 2 theorems, 92 equations.

Table of Contents

  1. Review of Arakelov geometry over adelic curves
  2. Adelic curves and adelic vector bundles
  3. Adelic line bundles
  4. Adelic Cartier divisors
  5. Equidistribution theorem over adelic curves
  6. Differentiability and concavity of $\chi$-volume
  7. Heights and measures
  8. Boundedness of minimal slopes
  9. Normed graded linear series of bounded type
  10. Proof of Theorem \ref{['theo_bounded_slope']}
  11. Applications to function fields
  12. Algebro-geometric setting vs adelic setting
  13. Slope boundedness
  14. Continuity of $\chi$-volume over function fields

Key Result

Theorem A

Let $\overline L=(L,\phi)$ be an adelic line bundle such that $L$ is big and nef, $\phi$ is semipositive and $\widehat{\mu}_{\min}^{\mathrm{asy}}(\overline L)> -\infty$. Let $\{x_\iota\in X(\overline K)\}_{\iota\in I}$ be a generic net of algebraic points on $X$ such that that is, for any $\epsilon>0$, there exists $\iota_0\in I$ such that for any $\iota\geq \iota_0$, we have $\lvert h_{\overline

Theorems & Definitions (17)

  • Theorem A: cf. Theorem \ref{['theo_equidistribution_big']}
  • Theorem B: cf. Theorem \ref{['theo_equi_fun']} and Remark \ref{['rema_num_corps']}
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 7 more