Slide on tilting

main.tex

@@ -5,6 +5,7 @@
 \usepackage[T1]{fontenc}
 \usepackage{lmodern}
 \usepackage{amsmath}
+\usepackage{amssymb}
 \usepackage{amsfonts}
 \usepackage{mathtools}
 \usepackage{sagetex}
@@ -25,6 +26,7 @@
 \newcommand\CC{\mathbb{C}}
 \newcommand\centralcharge{\mathcal{Z}}
 \newcommand\coh{\operatorname{Coh}}
+\newcommand\hom{\mathrm{hom}}
 \newcommand\rank{\operatorname{rk}}
 \newcommand\degree{\operatorname{deg}}
 \newcommand\realpart{\mathfrak{Re}}
@@ -32,6 +34,10 @@
 \newcommand\bigO{\mathcal{O}}
 \newcommand\cohom{\mathcal{H}}
 \newcommand\chern{\mathrm{ch}}
+\newcommand\Torsion{\mathcal{T}}
+\newcommand\Free{\mathcal{F}}
+\newcommand\firsttilt[1]{\mathcal{B}^{#1}}
+\newcommand\derived{\mathcal{D}}
 
 \begin{document}
 
@@ -226,5 +232,49 @@
 	\end{center}
 \end{frame}
 
+\begin{frame}{New Heart - Tilting}
+	Role of $\coh(X)$ as heart of $\derived^b(X)$ replaced:
+	\vfill
+	\begin{tcolorbox}[title=First Tilt of $\coh(X)$]
+		\begin{align*}
+			\firsttilt\beta :=
+			\left\{
+				E \in \derived^b(X) \colon \quad
+				\cohom^{0}(E) \in \Torsion_\beta, \quad
+				\cohom^{-1}(E) \in \Free_\beta, \quad
+				\cohom^i(E) = 0 \:\: \text{o.w.}
+			\right\}
+		\end{align*}
+		where $\beta \in \RR$ and:
+		\begin{align*}
+			\Torsion_\beta &:=
+			\left\{\:
+				E \in \coh(X) \colon \qquad
+				\mu(G) > \beta \quad \text{whenever} \: E \twoheadrightarrow G \not=0,E
+			\:\right\}
+		\\
+			\Free_\beta &:=
+			\left\{\:
+				E \in \coh(X) \colon \qquad
+				\mu(G) \leq \beta \quad \text{whenever} \: 0 \not= G \hookrightarrow E
+			\:\right\}
+		\end{align*}
+	\end{tcolorbox}
+	\vfill
+	\begin{itemize}
+		\item $\Torsion_\beta \subset \firsttilt\beta$ includes Mumford semistable
+			$E \in Coh(X)$ s.t. $\mu(E) \geq \beta$
+		\item As $\beta \to - \infty$, \: $\firsttilt\beta \rightsquigarrow \coh(X)$
+			\begin{itemize}
+				\item $\Torsion_\beta \rightsquigarrow \coh(X)$
+				\item $\Free_\beta \rightsquigarrow 0$
+			\end{itemize}
+		{\color{gray}
+		\item $\hom(T, F) = 0$ for $T \in \Torsion_\beta, F \in \Free_\beta$
+			makes this a torsion theory
+		}
+	\end{itemize}
+\end{frame}
+
 
 \end{document}