- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:09
Journal of the European Mathematical Society
J. Eur. Math. Soc.
JEMS
1435-9855
1435-9863
General
10.4171/JEMS
http://www.ems-ph.org/doi/10.4171/JEMS
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
15
2013
6
A support theorem for Hilbert schemes of planar curves
Luca
Migliorini
Università di Bologna, BOLOGNA, ITALY
Vivek
Shende
Massachusetts Institute of Technology, CAMBRIDGE, UNITED STATES
locally planar curves, Hilbert scheme, compactified Jacobian, versal deformation, perverse cohomology, decomposition theorem
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 {\em 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.
Algebraic geometry
2353
2367
10.4171/JEMS/423
http://www.ems-ph.org/doi/10.4171/JEMS/423