Can Chern class/character be categorified? - MathOverflow

Can Chern class/character be categorified?

2011-04-03T03:00:26Z

The Chern character sends the class of a locally free sheaf to the cohomology ring of the underlying variety X. And it is a ring homomorphism from K to H^*. I saw people write its source as the bounded derived category too, which make sense if the underlying variety is smooth (sending a bounded complex to the alternating "sum" of the Chern characters of its cohomology sheaf).

My question is, if I want to think $D^b(X)$ as a certain categorification of $K_0(X)$, is it possible to categorify the chern character map? What will be a good candidate of the target category? (Or is there a heuristic showing this is not likely to be true?)

Answer by David Ben-Zvi for Can Chern class/character be categorified?

2011-04-03T03:41:58Z

There are categorified analogs of the Chern character, but I don't think of them in the way you're proposing. More precisely, you can take an object in the derived category and assign to it a class in cohomology, and this map factors through K-theory, so the two constructions you're discussing seem to me to be the same.

One way to think of the Chern character is the following. Given any associative, dg or $A_\infty$ algebra, you can define its Hochschild homology. This is the recipient for a universal trace map from the algebra, and more generally for any "finite" module (perfect complex) you get a class (its character) in Hochschild homology. Given more generally a (dg or $A_\infty$) category you can similarly define its Hochschild homology and a character map for "finite" objects (which factors through the K-theory of the category), which agrees with the above when your category is modules over an algebra (which it usually is, noncanonically).

To "categorify" you can replace an algebra by an associative algebra object in any symmetric monoidal $\infty$-category. Its Hochschild homology is defined as an object of said category and again there's a Chern character map for "finite" modules. Why is this a categorification? for example you can take your associative algebra to be some derived category of sheaves with a monoidal structure (eg coherent sheaves or $\D$-modules or.. with tensor product or some convolution product), and then its Hochschild homology is itself a category. Thus module categories will have Chern characters which are objects of this homology category. This is (one way to think of) the notion of a "character sheaf" in representation theory (where our associative algebra is sheaves on a group with convolution, and module categories are categories with a nice action of the group, and their Chern character are adjoint-equivariant sheaves on the group, ie categorified class functions).

(This story is by the way a special case of the Cobordism Hypothesis with Singularities of Jacob Lurie -- in fact just of its one-dimensional case.. our algebra objects are assigned to a point, their Hochschild homology is assigned to the circle, modules are allowable "singularities" in the theory and their Chern character is attached to a circle with a marked "singular point")

Answer by DamienC for Can Chern class/character be categorified?

2011-05-02T23:04:28Z

The paper http://arxiv.org/abs/0804.1274 of Toën-Vezzosi is about categorifying the Chern character. Let me try to summarize their strategy.

First of all they introduce a triangulated $2$-category $Dg(X)$ of derived categorical sheaves on a (derived) scheme $X$. It is based on a the idea that a categorification of the theory of modules on a commutative ring $k$ is given by $k$-linear categories: they argue that dg-categories can be used in order to categorify homological algebra in a similar but better way (better in the sens that the non-dg setting seems to be too rigid to allow push-forwards).

The second step is to use, for a given (derived) scheme $X$, the pull-back along the projection $LX\to X$. For a categorical sheaf $F$ on $X$ on consider its pull-back $p^*F$, which naturally come equipped with a self-equivalence $u$. The rough idea to see this is to consider the pull-back (a-k-a >transgression) along the evaluation map $S^1\times LX\to X$, and then to observe that categorical sheaves on $S^1\times LX$ are categorical sheaves on $LX$ together with a $\mathbb{Z}$-action.

Finally, they conjecture the existence of an $S^1$-equivariant trace $Tr^{S^1}(u)\in D^{S^1}(LX)$. Its $K_0$ would be a candidate for the (categorified) Chern character of $F$.

Why does this categorify the Chern character ?

If we do the same construction starting with a sheaf of $X$, then we get in the end an element in $\pi_0(\mathcal O_{LX}^{S^1})=HC_0^{-}(X)$ (while the non-$S^1$-invariant trace takes values in $\pi_0(\mathcal O_{LX})=HH_0(X)$).

One can show that this constructs the ususal Chern character. The main difficulty is the (conjectural) existence of the $S^1$-invariant trace.

Follow-up

A complete treatment of this approach (together with a proof of the conjecture) has been done by the above mentioned authors in a long paper in french.