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We present a notion of $\Delta$-stability and stability filtration inarbitrary categories which is equivalent to the existence of Harder-Narasimhan(HN) sequences on objects. Indeed it is equivalent to the existence of a zeromorphism, a partial order on objects, and a collection of some universalsequences. In additive categories embedded in an ambient triangulated category,we could obtain a numerical polynomial or central charge of objects bycalculating the Euler characteristic of slope sequences and objects, inducing apartial order and HN sequences. In the case of normal projective surfaces overan arbitrary perfect field $k$ we show the existence of $\Delta$-stabilities ofdegree one on the relevant bounded derived categories which is equivalent tothe existence of Bridgeland's stabilities on normal surfaces. This result alsoleads to new effective restriction theorem of slope semistable sheave on normalprojective varieties over perfect fields. Our approach also gives alternativeproofs of Hodge Index Theorem and Bogomolov Inequality.

Stability in Categories and Normal Projective Varieties over Perfect Fields