# A support theorem for Hilbert schemes of planar curves

Journal of the European Mathematical Society (2013)

- Volume: 015, Issue: 6, page 2353-2367
- ISSN: 1435-9855

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topMigliorini, 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 -

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