Fourier–Mukai and autoduality for compactified Jacobians, II

Abstract

To every reduced (projective) curve $X$ with planar singularities one can associate, following E Esteves, many fine compactified Jacobians, depending on the choice of a polarization on $X$, which are birational (possibly nonisomorphic) Calabi–Yau projective varieties with locally complete intersection singularities. We define a Poincaré sheaf on the product of any two (possibly equal) fine compactified Jacobians of $X$ and show that the integral transform with kernel the Poincaré sheaf is an equivalence of their derived categories, hence it defines a Fourier–Mukai transform. As a corollary of this result, we prove that there is a natural equivariant open embedding of the connected component of the scheme parametrizing rank-$1$ torsion-free sheaves on $X$ into the connected component of the algebraic space parametrizing rank-$1$ torsion-free sheaves on a given fine compactified Jacobian of $X$.

The main result can be interpreted in two ways. First of all, when the two fine compactified Jacobians are equal, the above Fourier–Mukai transform provides a natural autoequivalence of the derived category of any fine compactified Jacobian of $X$, which generalizes the classical result of S Mukai for Jacobians of smooth curves and the more recent result of D Arinkin for compactified Jacobians of integral curves with planar singularities. This provides further evidence for the classical limit of the geometric Langlands conjecture (as formulated by R Donagi and T Pantev). Second, when the two fine compactified Jacobians are different (and indeed possibly nonisomorphic), the above Fourier–Mukai transform provides a natural equivalence of their derived categories, thus it implies that any two fine compactified Jacobians of $X$ are derived equivalent. This is in line with Kawamata’s conjecture that birational Calabi–Yau (smooth) varieties should be derived equivalent and it seems to suggest an extension of this conjecture to (mildly) singular Calabi–Yau varieties.