Application (ongoing research with R. Pandharipande, S. Molcho)
On there exists a vector bundle , called the Hodge bundle, with fibres The Chern classes are called the -classes. They are tautological, with an explicit formula first computed by Mumford, and can be computed in terms of generators using
The formula following by Mumford's computation is reasonably nice, but it does feature some slightly complicated terms. However, we could hope for something better! Consider the open subset of curves such that the stable graph of is a tree (i.e. contains no circular path). Then there exists a class such that So, up to a scalar factor, the class is a power of a divisor class! However, it turns out that this equality does not hold in general on the whole of .
What we can show now is that such a formula cannot work at all if is sufficiently large! To be slightly more precise, denote by the sub--algebra of generated by elements of cohomological degree at most . In particular, is the set of classes which can be written as linear combinations of products of divisor classes.
Then we have the following:
Theorem (Molcho, S, Pandharipande - last Friday)
For we have and for we have assuming that the generalized Faber-Zagier relations give all the relations in the spaces Idea of proof Use
admcycles to check the statement in and . The assumption tells us that in the corresponding spaces
toTautbasis() really does express as well as elements of in a basis of , so the statement is just linear algebra. The case of larger can then be shown using a small argument applying boundary gluing maps like using the fact that .
With the strategy discussed before, we can hopefully get rid of the assumption of the Faber-Zagier relations. The only problem is that the expected dimension of is , so the matrix we have to compute is a matrix of size at least of intersection numbers on the space of complex dimension . This will take a bit of time and effort, but doesn't seem impossible.
Thanks for your attention!