From d3819e943d911d140ad2a73f99a8590fddccc157 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Wed, 12 Apr 2023 12:31:33 +0100
Subject: [PATCH] Bound for rank
---
main.tex | 21 ++++++++++++++++++++-
1 file changed, 20 insertions(+), 1 deletion(-)
diff --git a/main.tex b/main.tex
index 2b3b217..00d3d23 100644
--- a/main.tex
+++ b/main.tex
@@ -9,7 +9,10 @@
\usepackage{color}
\newcommand{\QQ}{\mathbb{Q}}
+\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\chern}{\operatorname{ch}}
+\newcommand{\lcm}{\operatorname{lcm}}
+\newcommand{\gcd}{\operatorname{gcd}}
\newcommand{\firsttilt}[1]{\mathcal{B}^{#1}}
\newcommand{\bddderived}{\mathcal{D}^{b}}
\newcommand{\centralcharge}{\mathcal{Z}}
@@ -84,6 +87,8 @@ $\chern^\beta(E) = \chern(E) \cdot \exp(-\beta \ell)$:
\chern^\beta_2(E) &= \chern_2(E) - \beta \chern_1(E) + \frac{\beta^2}{2} \chern_0(E)
\end{align*}
+% TODO I think this^ needs adjusting for general Surface with $\ell$
+
$\chern^\beta_1(E)$ is the imaginary component of the central charge
$\centralcharge_{\alpha,\beta}(E)$ and any element of $\firsttilt\beta$
satisfies $\chern^\beta_1 \geq 0$. This, along with additivity gives us, for any
@@ -101,7 +106,7 @@ normal one. So $0 \leq \Delta(E)$ yields:
\begin{equation}
\label{eqn-bgmlv-on-E}
- \chern^\beta_0(E) \chern^\beta_2(E) \leq \left(\chern^\beta_1(E)\right)^2
+ 2\chern^\beta_0(E) \chern^\beta_2(E) \leq \left(\chern^\beta_1(E)\right)^2
\end{equation}
The restrictions on $\chern^\beta_0(E)$ and $\chern^\beta_2(E)$
@@ -109,6 +114,20 @@ is best seen with the following graph:
% TODO: hyperbola restriction graph (shaded)
+This is where the $\beta_{-}$ criterion comes in. If $\beta_{-} = \frac{*}{n}$
+for some $*,n \in \ZZ$.
+Then $\chern^\beta_2(E) \in \frac{1}{\lcm(m,2n^2)}\ZZ$ where $m$ is the integer
+which guarantees $\chern_2(E) \in \frac{1}{m}\ZZ$ (determined by the variety).
+In particular, since $\chern_2(E) > 0$ we must also have
+$\chern^\beta_2(E) \geq \frac{1}{\lcm(m,2n^2)}$, which then in turn gives a bound
+for the rank of $E$:
+
+\begin{align}
+ \chern_0(E) &= \chern^\beta_0(E) \\
+ &\leq \frac{\lcm(m,2n^2) \chern^\beta_1(E)^2}{2} \\
+ &\leq \frac{mn^2 \chern^\beta_1(F)^2}{\gcd(m,2n^2)}
+\end{align}
+
\section{Section 3}
\section{Conclusion}
--
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