Introduce twisted Chern and associated condition

main.tex

@@ -12,6 +12,7 @@
 \newcommand{\chern}{\operatorname{ch}}
 \newcommand{\firsttilt}[1]{\mathcal{B}^{#1}}
 \newcommand{\bddderived}{\mathcal{D}^{b}}
+\newcommand{\centralcharge}{\mathcal{Z}}
 
 \begin{document}
 
@@ -67,9 +68,39 @@ used, followed by refinements on them which give explicit formulae for tighter
 bounds on $\chern_0(E)$ of potential destabilizers $E$ of $F$.
 
 
-\section{Section 1}
+\section{Characteristic Curves of Stability Conditions Associated to Chern
+Characters}
 
-\section{Section 2}
+\section{Twisted Chern Characters of Pseudo Destabilizers}
+
+For a given $\beta$, we can define a twisted Chern character
+$\chern^\beta(E) = \chern(E) \cdot \exp(-\beta \ell)$:
+
+\begin{align*}
+	\chern^\beta_0(E) &= \chern_0(E)
+\\
+	\chern^\beta_1(E) &= \chern_1(E) - \beta \chern_0(E)
+\\
+	\chern^\beta_2(E) &= \chern_2(E) - \beta \chern_1(E) + \frac{\beta^2}{2} \chern_0(E)
+\end{align*}
+
+$\chern^\beta_1(E)$ 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
+destabilizing sequence [ref]:
+\[
+	0 \leq \chern^\beta_1(E) \leq \chern^\beta_1(F)
+\]
+
+When finding Chern characters of potential destabilizers $E$ for some fixed
+Chern character $\chern(F)$, this bounds $\chern_1(E)$.
+
+The Bogomolov form applied to the twisted Chern character is the same as the
+normal one. So $0 \leq \Delta(E)$ yields:
+
+\[
+	\chern^\beta_0 \chern^\beta_2 \leq \left(\chern^\beta_1\right)^2
+\]
 
 \section{Section 3}