To every reduced (projective) curve
with planar singularities one can associate, following E Esteves, many fine
compactified Jacobians, depending on the choice of a polarization on
,
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
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- torsion-free
sheaves on
into the connected component of the algebraic space parametrizing
rank-
torsion-free sheaves on a given fine compactified Jacobian of
.
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
,
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
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.
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