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Sharp bounds on the height of K-semistable Fano varieties II, the log case

Rolf Andreasson, Robert J. Berman

Abstract

In our previous work we conjectured - inspired by an algebro-geometric result of Fujita - that the height of an arithmetic Fano variety X of relative dimension $n$ is maximal when X is the projective space $\mathbb{P}^n_{\mathbb{Z}}$ over the integers, endowed with the Fubini-Study metric, if the corresponding complex Fano variety is K-semistable. In this work the conjecture is settled for diagonal hypersurfaces in $\mathbb{P}^{n+1}_{\mathbb{Z}}$. The proof is based on a logarithmic extension of our previous conjecture, of independent interest, which is established for toric log Fano varieties of relative dimension at most three, hyperplane arrangements on $\mathbb{P}^n_{\mathbb{Z}}$, as well as for general arithmetic orbifold Fano surfaces.

Sharp bounds on the height of K-semistable Fano varieties II, the log case

Abstract

In our previous work we conjectured - inspired by an algebro-geometric result of Fujita - that the height of an arithmetic Fano variety X of relative dimension is maximal when X is the projective space over the integers, endowed with the Fubini-Study metric, if the corresponding complex Fano variety is K-semistable. In this work the conjecture is settled for diagonal hypersurfaces in . The proof is based on a logarithmic extension of our previous conjecture, of independent interest, which is established for toric log Fano varieties of relative dimension at most three, hyperplane arrangements on , as well as for general arithmetic orbifold Fano surfaces.
Paper Structure (31 sections, 24 theorems, 162 equations)

This paper contains 31 sections, 24 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Toric log Fano varieties
  4. Hyperplane arrangements
  5. Application to diagonal hypersurfaces
  6. Arithmetic log surfaces
  7. Outlook on the case of $\Bbb{P}_{ \Bbb{Z}}^{1}$ endowed with a divisor with three components
  8. Acknowledgments
  9. General setup
  10. Log Fano varieties over $\Bbb{C}$ and volume-normalized metrics on $-(K_{X}+\Delta)$
  11. Local representations of metrics and measures
  12. Log Kähler-Einstein metrics
  13. K-semistability
  14. Arithmetic log Fano varieties
  15. Arithmetic intersection numbers and heights
  16. ...and 16 more sections

Key Result

Theorem 1.2

Let $(\mathcal{X},\mathcal{D})$ be the canonical integral model of a K-semistable toric log Fano variety $(X,\Delta).$ Conjecture conj:height log intro holds for $(\mathcal{X},\mathcal{D})$ under anyone of the following conditions:

Theorems & Definitions (41)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Remark 2.1
  • Theorem 3.1
  • Lemma 3.2
  • Lemma 3.3
  • ...and 31 more