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I am trying to understand what exactly Ngo's support theorem says about how the cohomology of compactified Jacobians varies over the versal deformation of a curve with plane singularities.

That is, let $C$ be a compact complex curve with singularities of embedding dimension 2. Let $\mathcal{C} \to \mathbf{V}$ be a versal deformation, i.e., there is a point $0 \in \mathbf{V}$ over which the fibre is $C$, and any deformation of $C$ pulls back from this family. Let $\pi: \overline{\mathcal{J}} \to \mathbf{V}$ be the relative compactified Jacobian of the family; I want to understand the family of cohomologies $\mathrm{R}\pi_* \mathbb{Q}$. As pointed out in the few references I've browsed, this is a situtation to which Ngo's support theorem applies: there is a relative action of the (uncompactified but generalized) Jacobian $\mathcal{J}$, and the locus $\mathcal{V}_h$ where the affine part of $\mathcal{J}$ is of dimension $h$ is of precisely codimension $h$ (by an old result of Teissier). However, said references go on to treat a more famous example instead -- the Hitchin fibration.

Unfortunately my competence with the machinery of perverse sheaves is so poor that I can barely understand the statement of the support theorem. So, I apologize if the following questions are overly trivial or confused.

0.) Is the application of the support theorem to the situation I describe above treated explicitly in the literature?

1.) Does the support theorem imply that $\mathrm{R} \pi_* \mathbb{Q}$ is the IC-sheaf on $\mathbf{V}$ determined by the local system of cohomologies of Jacobians on the locus $\mathbf{V}_{\mathrm{reg}} \subset \mathbf{V}$ of smooth curves? If not, does it specify what other local systems on what other loci must be added?

2.) Does the knowledge of the Picard-Lefshetz monodromy around nodal degenerations determine (explicitly) the local system of cohomologies of Jacobians on $\mathbf{V}_{\mathrm{reg}}$, and, if so, does it allow an in principle calculation of the cohomology of the central fibre? Can it be done in practice?

3.) Does it follow that the cohomology of the compactified Jacobian of the central fibre is determined by the topology of the singularity of the discriminant (the complement of the locus of smooth curves)? Or -- perhaps it is necessary to know how monodromies around different nodes interact -- the stratification of the discriminant by the (closures of the) loci with a fixed number of nodes?

4.) Any other comments on how this (or anything else) would be used to compute the cohomology of the compactified Jacobian are welcome.

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1 Answer 1

up vote 4 down vote accepted

Very partial answer - I don't think I can comment yet...

I found it helpful to rephrase the statement of the support theorem like this:

Let $R\pi_\ast \mathbb Q = \bigoplus _i IC_{Z_i}(L_i)[d_i]$ be the decomposition of the pushforward sheaf. If Z is one of the $Z_i$ appearing in the sum above, then $Z$ is the support of a direct summand of $R^{2d}\pi_\ast \mathbb Q$ (i.e. there is extra stuff appearing in the top cohomology of the fibres). Such a summand only will occur when there are extra irreducible components in the fibres of $\pi$.

In particular, if the fibres of $\pi$ are irreducible, then the only possible support appearing in the direct sum is the whole of V. This means that the pushforward is the IC extension of the local system over the locus where $\pi$ is smooth.

I think this means that in your situation, the answer to (1) is yes (as compactified Jacobians are irreducible for such singularities). I would be interested to find out the answers to (2) and (3).

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Thanks, that's quite helpful... –  Vivek Shende Sep 13 '10 at 13:01

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