The Euler Characteristic of Parabolic Sheaves

Objectives: The primary aim of this study is to explicitly determine the Euler characteristic of the parabolic sheaves with rank 2 on a smooth projective algebraic surface defined over complex numbers with the smooth irreducible parabolic divisor . Methods: The computation of the parabolic Hilbert polynomial involves the use of -filtered sheaves on a smooth projective surface , with weights corresponding to the points where the filtration jumps. The Riemann-Roch theorem and Chern class computation have also been used. Findings: The study provides explicit computations of the parabolic Hilbert polynomial as well as the parabolic Chern classes for parabolic rank 2 bundles. Novelty: This work contributes to the understanding of parabolic sheaves on smooth projective surfaces, bridging the gap between different constructions of stable bundles. The explicit computation of the parabolic Hilbert polynomial for rank 2 bundles adds valuable insights to the study of moduli spaces of parabolic bundles. Keywords: Euler characteristic, Hilbert polynomial, Chern class, Parabolic sheaves, Smooth projective algebraic surface