Let $X$ be a compact, smooth $n$-dimensional Calabi-Yau variety and let $M \subseteq X$ be an $m$-dimensional subvariety. Here, both $n$ and $m$ are complex dimensions. I want to take a rank $N$ holomorphic vector bundle $E$ on $M$ and push it forward $i_{*}E$ to $X$ via the natural inclusion. Finally, what I'm interested in doing is taking the Chern character.

I believe that $\text{ch}_{k}(i_{*}E)=0$ for all $k <n-m$. But what I really want to show is the following:

$$\text{PD}\big(\text{ch}_{n-m}(i_{*}E) \big) = N[M] \in H_{2m}(X, \mathbb{Z})$$

where "$\text{PD}$" of course denotes Poincare duality. What I mean by the righthand side is $N$ times the fundamental class of $M$. Is this true? If so, how can I go about proving it?

Context: In the string theory literature, we see tons of statements like "take a stack of $N$ D$2m$-branes wrapping some $2m$-dimensional cycle $M$." I think this corresponds to simply taking a rank $N$ holomorphic vector bundle on $M$. This indeed would be the correct interpretation if my above claim is true.

By the way, I know my claim is true if $M$ is a curve, but I'm hoping it's more general! I believe it should rely on $X$ being Calabi-Yau.

  • $\begingroup$ What definition of chern character are you using? $\endgroup$ – Saal Hardali May 22 '17 at 19:18

This is a long comment since I don't know if this will solve your whole question.

Under certain assumptions you can use Grothendieck-Riemann-Roch to tackle this sort of questions. For example in Huybrechts' book the statement reads: Consider a smooth projective morphism $f:X\to Y$ of smooth projective varieties. Let $\mathcal{F}$ be a coherent sheaf on $X$. Then, $$ \operatorname{ch}\left(\sum(-1)^iR^if_*\mathcal{F}\right)\operatorname{td}(Y)=f_*(\operatorname{ch}(\mathcal{F})\operatorname{td}(X)). $$ In your particular case the inclusion map is a closed immersion, so the functor $i_*$ is exact and the higher direct images vanish. Then GRR says that $$ \operatorname{ch}\left(i_*E\right)\operatorname{td}(X)=i_*(\operatorname{ch}(E)\operatorname{td}(M))\Rightarrow \operatorname{ch}\left(i_*E\right)=\frac{i_*(\operatorname{ch}(E)\operatorname{td}(M))}{\operatorname{td}(X)}, $$ where to compute the fraction you use the Taylor series of $\frac{1}{1-x}$. As you mention the fact that $X$ is CY comes into play with the Todd class.

This post in MO and this tag of the stacks project may be helpful when calculating the RHS of GRR.


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