nLab Gieseker stability

Given an ample line bundle $H$ over a projective algebraic manifold, Gieseker $H$-stability conditions are the stability conditions for torsion-free coherent sheaves over projective algebraic manifolds generalizing say Mumford stability conditions for vector bundles over curves and used for similar reasons (building moduli spaces). These are in the setup of abelian categories rather than their derived categories where notions like Bridgeland stability are more appropriate.

Introduced in

• D. Gieseker, On the moduli of vector bundles on an algebraic surface, Ann. Math. 106 (1977) 45-60 doi

and in

• S. Kobayashi, Differential geometry of complex vector bundles, Iwanami Shoten Publishers and Princeton Univ. Press (1987)

The special case for algebraic surfaces has been introduced in

• F. Takemoto, Stable vector bundles on algebraic surfaces, Nagoya Math. J. 47 (1972) 29-48

category: algebraic geometry