Can Chern class/character be categorified? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T12:34:56Z http://mathoverflow.net/feeds/question/60403 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/60403/can-chern-class-character-be-categorified Can Chern class/character be categorified? 36min 2011-04-03T03:00:26Z 2011-05-03T19:49:43Z <p>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).</p> <p>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?)</p> http://mathoverflow.net/questions/60403/can-chern-class-character-be-categorified/60405#60405 Answer by David Ben-Zvi for Can Chern class/character be categorified? David Ben-Zvi 2011-04-03T03:41:58Z 2011-04-03T03:41:58Z <p>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.</p> <p>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).</p> <p>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).</p> <p>(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")</p> http://mathoverflow.net/questions/60403/can-chern-class-character-be-categorified/63750#63750 Answer by DamienC for Can Chern class/character be categorified? DamienC 2011-05-02T23:04:28Z 2011-05-03T19:49:43Z <p>The paper <a href="http://arxiv.org/abs/0804.1274" rel="nofollow">http://arxiv.org/abs/0804.1274</a> of Toën-Vezzosi is about categorifying the Chern character. Let me try to summarize their strategy. </p> <p>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). </p> <p>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. </p> <p>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 $F$. </p> <h3>Why does this categorify the Chern character ?</h3> <p>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 $\pi_0(\mathcal O_{LX})=HH_0(X)$). </p> <p>One can show that this constructs the ususal Chern character. The main difficulty is the (conjectural) existence of the $S^1$-invariant trace. </p> <h3>Follow-up</h3> <p>A complete treatment of this approach (together with a proof of the conjecture) has been done by the above mentioned authors in <a href="http://arxiv.org/abs/0903.3292" rel="nofollow">a long paper in french</a>. </p>