claimstandardinferred_dep

Theta height bounded by Faltings height

Bounds theta height by

C(1+\max\{0,h_{\mathrm F}\})

last link to the Faltings height

Claim·§4.4 Comparison of base height and Faltings height·inside lemBase height bounded by Faltings height

Statement (paper)

There is a constant

C

, independent of the point and its field of definition, such that

h_\Theta(\mathcal A_y)\le C\bigl(1+\max\{0,h_{\mathrm F}(\mathcal A_y)\}\bigr)

on the open where the theta data are defined. Here

h_\Theta

is Pazuki's exact theta height and

h_{\mathrm F}

is the stable Faltings height, which differs from the Deligne-normalized height

h_{\mathrm F}^{\mathrm D}

by the fixed constant

\log\pi

for these surfaces.

In plain English

Where the theta data are defined, Pazuki's theta height of the surface satisfies

h_\Theta(\mathcal A_y)\le C(1+\max\{0,h_{\mathrm F}(\mathcal A_y)\})

, with

C

independent of the point and its field of definition. Here

h_{\mathrm F}

is the stable Faltings height, which equals the Deligne-normalized

h_{\mathrm F}^{\mathrm D}

up to the fixed constant

\log\pi

for these surfaces.

Why it's here

This is the Pazuki comparison step that converts the theta height into the Faltings height. Chained with the previous claim, which bounds

\hgt(\pi(y))

C(1+h_\Theta(\mathcal A_y))

, it gives the inequality of the parent lemma. The

\log\pi

normalization shift is absorbed into the constant

C

, so the Deligne and stable conventions can be used interchangeably.

Classified standard:

Pazuki's comparison between the theta height and the stable Faltings height of principally polarized abelian varieties (Pazuki, 'Theta height and Faltings height', 2012), applied directly here; the

\log\pi

shift between the Deligne and stable normalizations is a fixed constant absorbed into

C

Uses

claimBase height bounded by theta height

Used by

Nothing in the paper

Sources

PaperOpen

Section: §4.4 Comparison of base height and Faltings height
Page: —
Label: —
Line: sections/04-height.tex:514

Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

info
inferred_dep
outline: "Pazuki comparison of theta and Faltings heights" rests on claim-hei-base-height-4