A New Functorial Invariant for Moduli Spaces of Sheaves on Algebraic Surfaces

We introduce a new functorial invariant for moduli spaces of coherent sheaves on smooth projective algebraic surfaces over an algebraically closed field of characteristic zero. This invariant, called the Derived Chern–Perverse Invariant (DCPI), is constructed using derived categories, perverse filtrations, and motivic integration. We show that DCPI is stable under derived equivalences, deformations, and birational transformations respecting Bridgeland stability conditions. Key properties include factorization through wall-crossing morphisms in Bridgeland moduli spaces. Applications include Hilbert schemes of points, rank-two stable sheaves, and abelian surfaces, showing that DCPI distinguishes spaces which classical invariants such as Betti numbers, Hodge structures, or intersection forms cannot. This opens new directions in the study of moduli spaces, Donaldson–Thomas theory, mirror symmetry, and categorification. Keywords: Derived Categories, Moduli Spaces of Sheaves, Functorial Invariants, Perverse Filtrations, Hilbert Schemes, Bridgeland Stability Conditions, Donaldson– Thomas Theory, Mirror Symmetry, Categorification