[ edit ] Bridgeland stability conditions

The definition of Bridgeland Stability is essentially 
a derived category analogue of Mumford/Simpson stability
for coherent sheaves on smooth projective varieties.  Despite this,
it is geometrically non-intuitive and thus generating 
examples of Bridgeland stability conditions for general geometric triangulated categories is a difficult problem.

\begin{definition}
  A \emph{slicing} $\cl{P}$ on a triangulated category $\cl{D}$ is a 
  collection of full additive subcategories $\cl{P}(\phi)$
  parameterized by $\phi\in\RR$ and satisfying the following 
  axioms:
  \begin{enumerate}
     \item for all $\phi\in\RR$, $\cl{P}(\phi+1) = \cl{P}(\phi)[1]$,
     \item for all $\phi_1>\phi_2$ and $A_j\in\cl{P}(\phi_j)$, 
     $\Hom_D(A_1,A_2)=0$, and
     \item for every non-zero object $E\in \cl{D}$, there is a
     Harder-Narisimhan filtration of $E$.  That is, there
     is a finite sequence of real numbers $\phi_1>\ldots>\phi_n$
     and a sequence of exact triangles 
     \begin{equation}
        \xymatrixcolsep{1pc}\xymatrix{
           0=E_0 \ar[rr] && E_1 \ar[rr] \ar[dl] && E_2 \ar[r]
           \ar[dl] & \ldots \ar[r] & E_{n-1} \ar[rr] && E_n=E 
           \ar[dl] \\
           & A_1 \ar[ul]^{[1]} && A_2 \ar[ul]^{[1]} &&&&
           A_{n} \ar[ul]^{[1]}
        }
     \end{equation}
     with $A_j\in\cl{P}(\phi_j)$.  
  \end{enumerate}
\end{definition}

Let $K(\cl{D})$ be the Grothendieck group of $\cl{D}$.

\begin{definition}
   Given a triangulated category $\cl{D}$, a \emph{Bridgeland stability
   condition} on $\cl{D}$ is a pair $\sigma=(Z,\cl{P})$ consisting of 
   a slicing $\cl{P}$ and a group homomorphism $Z: K(\cl{D})\to \CC$, 
   called the \emph{central charge}, such that for all nonzero 
   $E\in\cl{P}(\phi)$, $Z(E)=m(E)e^{i\pi\phi}$ for some 
   $m(E)\in\RR_{>0}$.  We say an object $E\in\cl{P}(\phi)$
   is \emph{semi-stable of phase $\phi$}.
\end{definition}

These definitions are as given in {\bf Definition~3.1} of \cite{b}.

Let $X$ be a smooth projective variety over $\CC$, and let $\cl{C}oh(X)$ be the category 
of coherent sheaves on $X$, and $\rm{D^b}(X)$