From ed9adb9af62abace328cfa562b8d166e5d788e13 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Fri, 17 Mar 2023 00:25:47 +0000
Subject: [PATCH] Introduce Surface central charge
---
main.tex | 48 ++++++++++++++++++++++++++++++++++++++++++++++--
1 file changed, 46 insertions(+), 2 deletions(-)
diff --git a/main.tex b/main.tex
index d25a458..f7fe564 100644
--- a/main.tex
+++ b/main.tex
@@ -8,9 +8,9 @@
\usepackage{amsfonts}
\usepackage{mathtools}
\usepackage{sagetex}
-
\usepackage{ulem}
\usepackage{xcolor}
+\usepackage{tcolorbox}
\usetheme{edmaths}
\renewcommand\mathfamilydefault{\rmdefault}
@@ -31,6 +31,7 @@
\newcommand\imagpart{\mathfrak{Im}}
\newcommand\bigO{\mathcal{O}}
\newcommand\cohom{\mathcal{H}}
+\newcommand\chern{\mathrm{ch}}
\begin{document}
@@ -160,7 +161,50 @@
\end{itemize}
\end{frame}
-\section{Section 1}
+\section{Moving to Picard Rank 1 Surfaces}
+
+\begin{frame}{Moving to Surfaces}
+ \begin{itemize}
+ \item $\centralcharge(\bigO_x) = 0$ for Mumford stability
+ \\ ignores extra term in Chern character:
+ $(r, d \ell, \chi)$
+ \item Classically, Gieseker stability used
+ \\ \qquad slope comparison $\to$ lexicographic comparison
+ \end{itemize}
+\end{frame}
+
+\begin{frame}{New Central Charges for Surfaces}
+ Explicitly constructed for K3 - Bridgeland (2003)
+ \vfill
+
+ \begin{tcolorbox}[title=Picard Rank 1 with polarization $L$]
+ \begin{align*}
+ \centralcharge_{\alpha, \beta}(E) &:=
+ - \left<
+ \exp( \alpha \ell + \beta \ell i),
+ E
+ \right>
+ &\text{where}\:\:\ell := c_1(L)
+ \\ &= - \chern_{\mathrm{top}}(\exp( \alpha \ell + \beta \ell i)^{-1} \otimes E)
+ &\text{($\leftarrow$ abuse)}
+ \end{align*}
+ \end{tcolorbox}
+ \vfill
+ $\exp(a) = (1, a, \frac12 a^2)$ defined formally,
+ in particular: $\exp(n \ell) = L^{\otimes n}$
+ \vfill
+ {
+ \color{gray}
+ For Mumford stability on Curves:
+ \begin{align*}
+ \centralcharge(E) &= -\chern_1(E) + \chern_0(E) i
+ \\ &= - \left<\exp(-i), E\right>
+ \end{align*}
+ }
+
+
+
+\end{frame}
\section{Section 2}
--
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