Postnikov-stability versus semistability of sheaves

The Euclid-Fourier-Mukai algorithm for elliptic surfaces

“…The main motivation for considering P -stability is that it is preserved by equivalences and covers the classical semistability of sheaves. As an example we have following comparison theorem from [6].…”

“…If E is a stable sheaf of given Hilbert polynomial, then there exists an integer N 0 such that H i (E(N )) = 0 for all i > 0 and N ≥ N 0 . The number N 0 depends only on the numerical invariants of E. In [6,Proposition 5] it is shown that for any Hilbert polynomial p there exists a finite number of vector bundles {L i } i=0,..,M and natural numbers N i such that we have an implication for all objects…”

“…The key point needed for this result is the construction of a stability condition in the derived category D b (X), called P -stability, defined in [5,6], which recovers the slope stability and is well behaved with respect to derived equivalences. We will describe how to recover the µ-stability with respect to a given polarization and we will then look at semistable sheaves as stable objects in the derived category.…”

Fourier-Mukai transforms of slope stable torsion-free sheaves on a product elliptic threefold

“…The second part of the question above, therefore, is where the real question lies. The above question has been discussed in different contexts, including: slope stability, Gieseker stability and Bridgeland stabiliy on Abelian surfaces [21,33,9,37,38,22], slope stability and Gieseker stability on K3 surfaces [3,2,16,12], twisted stability on Abelian and K3 surfaces [35,36,25], rank-one torsion-free sheaves on elliptic surfaces [37,38,8], rank-one torsionfree sheaves on elliptic threefolds [11], torsion sheaves on elliptic threefolds [14], torsion sheaves on K3-surface fibrations [1], and from the point of view of Postnikov stability [15], just to name a few. A comprehensive introduction to results of this type can be found in [4].…”