67 research outputs found
Higher discriminants and the topology of algebraic maps
We show that the way in which Betti cohomology varies in a proper family of
complex algebraic varieties is controlled by certain "higher discriminants" in
the base. These discriminants are defined in terms of transversality
conditions, which in the case of a morphism between smooth varieties can be
checked by a tangent space calculation. They control the variation of
cohomology in the following two senses: (1) the support of any summand of the
pushforward of the IC sheaf along a projective map is a component of a higher
discriminant, and (2) any component of the characteristic cycle of the proper
pushforward of the constant function is a conormal variety to a component of a
higher discriminant.
The same would hold for the Whitney stratification of the family, but there
are vastly fewer higher discriminants than Whitney strata. For example, in the
case of the Hitchin fibration, the stratification by higher discriminants gives
exactly the {\delta} stratification introduced by Ngo.Comment: v2: proofs rewritten in the language of microsupport, and added
example of integrable system
The Hilbert scheme of a plane curve singularity and the HOMFLY polynomial of its link
The intersection of a complex plane curve with a small three-sphere
surrounding one of its singularities is a non-trivial link. The refined
punctual Hilbert schemes of the singularity parameterize subschemes supported
at the singular point of fixed length and whose defining ideals have a fixed
number of generators. We conjecture that the generating function of Euler
characteristics of refined punctual Hilbert schemes is the HOMFLY polynomial of
the link. The conjecture is verified for irreducible singularities y^k = x^n,
whose links are the k,n torus knots, and for the singularity y^4 = x^7 - x^6 +
4 x^5 y + 2 x^3 y^2, whose link is the 2,13 cable of the trefoil.Comment: 20 pages; some improvements are mad
A support theorem for Hilbert schemes of planar curves
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.Comment: 13 pages. (v2 updated to match published version.
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