Cohomology of compactified Jacobians for locally planar integral curves

This article surveys some recent developments on the cohomology of the compactified Jacobian associated with a locally planar integral curve. Topics discussed here include the Ngô support theorem, the perverse filtration, connections to the Hilbert schemes, and cohomological structures induced by the Arinkin-Fourier-Mukai transform.

💡 Research Summary

This paper surveys recent progress on the cohomology of compactified Jacobians attached to locally planar integral curves. After recalling the classical Jacobian of a smooth projective curve, the author explains that for a singular integral curve C the ordinary Picard variety is no longer proper, and the natural modular compactification J̄(C) parametrizes rank‑1 torsion‑free sheaves of degree zero. When C is locally planar, J̄(C) is an integral, locally complete‑intersection variety and admits a flat family π : J̄ → B over a smooth base B whose general fiber is the Jacobian of a smooth deformation of C.

The first major topic is Ngô’s support theorem. Using the decomposition theorem for proper maps and Ngô’s δ‑regularity argument, the author shows that the derived direct image Rπ_*ℚ on B decomposes as a direct sum of intermediate extensions IC_B(∧^i R^1p_*ℚ)