Global Boundedness for Decorated Sheaves

An important classification problem in Algebraic Geometry deals with pairs $(E,phi)$, consisting of a torsion free sheaf $E$ and a non-trivial homomorphism $phicolon (E^{otimes a})^{oplus b}lradet(E)^{otimes c}otimes L$ on a polarized complex projective manifold $(X,O_X(1))$, the input data $a$, $b$, $c$, $L$ as well as the Hilbert polynomial of $E$ being fixed. The solution to the classification problem consists of a family of moduli spaces ${cal M}^delta:={cal M}^{delta-rm ss}_{a/b/c/L/P}$ for the $delta$-semistable objects, where $deltainQ[x]$ can be any positive polynomial of degree at most $dim X-1$. In this note we show that there are only finitely many distinct moduli spaces among the ${cal M}^delta$ and that they sit in a chain of GIT-flips. This property has been known and proved by ad hoc arguments in several special cases. In our paper, we apply refined information on the instability flag to solve this problem. This strategy is inspired by the fundamental paper of Ramanan and Ramanathan on the instability flag.