A support theorem for Hilbert schemes of planar curves

Migliorini, Luca, and Shende, Vivek. "A support theorem for Hilbert schemes of planar curves." Journal of the European Mathematical Society 015.6 (2013): 2353-2367. <http://eudml.org/doc/277651>.

@article{Migliorini2013,
abstract = {Consider a family of integral complex locally planar curves whose relative Hilbert scheme of points is smooth. The decomposition theorem of Beilinson, Bernstein, and Deligne asserts that the pushforward of the constant sheaf on the relative Hilbert scheme splits as a direct sum of shifted semisimple perverse sheaves. We will show that no summand is supported in positive codimension. It follows that the perverse filtration on the cohomology of the compactified Jacobian of an integral plane curve encodes the cohomology of all Hilbert schemes of points on the curve. Globally, it follows that a family of such curves with smooth relative compactified Jacobian (“moduli space of $D$-branes”) in an irreducible curve class on a Calabi-Yau threefold will contribute equally to the BPS invariants in the formulation of Pandharipande and Thomas, and in the formulation of Hosono, Saito, and Takahashi.},
author = {Migliorini, Luca, Shende, Vivek},
journal = {Journal of the European Mathematical Society},
keywords = {locally planar curves; Hilbert scheme; compactified Jacobian; versal deformation; perverse cohomology; decomposition theorem; locally planar curves; Hilbert scheme; compactified Jacobian; versal deformation; perverse cohomology; decomposition theorem},
language = {eng},
number = {6},
pages = {2353-2367},
publisher = {European Mathematical Society Publishing House},
title = {A support theorem for Hilbert schemes of planar curves},
url = {http://eudml.org/doc/277651},
volume = {015},
year = {2013},
}

TY - JOUR
AU - Migliorini, Luca
AU - Shende, Vivek
TI - A support theorem for Hilbert schemes of planar curves
JO - Journal of the European Mathematical Society
PY - 2013
PB - European Mathematical Society Publishing House
VL - 015
IS - 6
SP - 2353
EP - 2367
AB - Consider a family of integral complex locally planar curves whose relative Hilbert scheme of points is smooth. The decomposition theorem of Beilinson, Bernstein, and Deligne asserts that the pushforward of the constant sheaf on the relative Hilbert scheme splits as a direct sum of shifted semisimple perverse sheaves. We will show that no summand is supported in positive codimension. It follows that the perverse filtration on the cohomology of the compactified Jacobian of an integral plane curve encodes the cohomology of all Hilbert schemes of points on the curve. Globally, it follows that a family of such curves with smooth relative compactified Jacobian (“moduli space of $D$-branes”) in an irreducible curve class on a Calabi-Yau threefold will contribute equally to the BPS invariants in the formulation of Pandharipande and Thomas, and in the formulation of Hosono, Saito, and Takahashi.
LA - eng
KW - locally planar curves; Hilbert scheme; compactified Jacobian; versal deformation; perverse cohomology; decomposition theorem; locally planar curves; Hilbert scheme; compactified Jacobian; versal deformation; perverse cohomology; decomposition theorem
UR - http://eudml.org/doc/277651
ER -