1
$\begingroup$

Apologies in advance for the long setup and question.

Let $L \to X$ be a line bundle. We may take its frame bundle $p \colon Fr(L) \to X$, a $\mathbb{G}_m$-torsor. We have

$$ p_*\mathcal{O}_{Fr(L)} = \bigoplus_{k \in \mathbb{Z}} {L}^{\otimes (-k)}. $$

To see this, we may take a trivializing neighborhood $U$ for $L$, and note that the fiber $L|_U$ looks like $U \times K$, where $K$ is a 1-dimensional vector space. So the functions $\mathcal{O}_{U \times K} \cong \mathcal{O}_U[x]$, where $x \in K^\vee$ is a nonzero functional. Choosing these $x$'s in families amounts to choosing a section of $L^\vee =: L^{\otimes(-1)}$. Requiring that the frames be nowhere vanishing sections of $L$ is like inverting $x$, since we don't care if they're defined where $x=0$; in families it's like adjoining $(L^\vee)^\vee = L$. So we get the above.

This $p_*\mathcal{O}_{Fr(L)}$ is graded by the characters of $\mathbb{G}_m$, with the $L^{\otimes(-k)}$ piece corresponding to the character $\lambda \mapsto \lambda^{-k}$ (or maybe its inverse).

Now let $F$ be a degree $d$ (totally real) number field, and $L_F \to X$ be a rank $d$ vector bundle with real multiplication by $\mathcal{O}_F$; the fibers look like 1-dimensional $F$-vector spaces. Take its $\mathcal{O}_F$-frame bundle $p \colon Fr(L_F) \to X$, which is a Res$_{\mathcal{O}_F/\mathbb{Z}}\mathbb{G}_m$-torsor. Points of $Fr(L_F)$ are nonzero elements of the fiber. Taking a trivializing neighborhood as above gives us $p_*\mathcal{O}_{L_F}(U) \cong U[x_1,\dots,x_d]$ locally, and globally $p_*\mathcal{O}_{L_F} = \bigoplus_{k \geq 0} L_F^{\otimes_\mathbb{Z} (-k)}$. Requiring that sections of the frame bundle don't vanish is making sure not all of the coordinates of the section are $0$. So I want functions that aren't necessarily defined at $(0,\dots,0)$ -- but by the Koecher principle, any function defined on the $d$-dimensional fiber minus the origin extends to the origin!

This leads me to believe that we invert nothing (in contrast to the above situation where we invert $x$, and it gives us the positive tensor powers of $L$), giving $$ p_*\mathcal{O}_{Fr(L_F)} = p_*\mathcal{O}_{L_F} = \bigoplus_{k \geq 0} L_F^{\otimes_\mathbb{Z} (-k)}. $$

Is this true? It seems to disagree with my intuition that this should be graded by characters of the group Res$_{\mathcal{O}_F/\mathbb{Z}}\mathbb{G}_m$, unless some of the graded pieces (i.e., the ones where one of the $k_\sigma$'s is negative) can be trivial. There is also the fact that we take the tensor powers of $L_F$ by tensoring over $\mathbb{Z}$, rather than $\mathcal{O}_F$, which is not obvious to me.

The above is my real question, but I'm also interested in the case when $V \to X$ is just a rank $g$ vector bundle, and $p \colon Fr(V) \to X$ is its usual frame bundle, a GL$_g$-torsor. For this, is there a good description of $p_*\mathcal{O}_{Fr(V)}$ in terms of $V$? I have separate motivation for this question, but it may also provide intuition for my current main one.

$\endgroup$
0
$\begingroup$

I think I've figured out an answer that makes sense and deals with my problems mentioned in the question.

Since $Fr(L_F)$ is a $Res_{\mathcal{O}_F/\mathbb{Z}}\mathbb{G}_m$-torsor, its fibers should look (non-canonically) like $Res_{\mathcal{O}_F/\mathbb{Z}}\mathbb{G}_m$, whose structure sheaf is $\mathcal{O}_{Res_{\mathcal{O}_F/\mathbb{Z}}\mathbb{G}_m} \cong \mathcal{O}_F[t,t^{-1}]$. So over a trivializing neighborhood $U$ we should have a (non-canonical) isomorphism with $\mathcal{O}_U \otimes_\mathbb{Z} \mathcal{O}_F[t,t^{-1}]$. The only way I can see to make this happen is to have $$ p_*\mathcal{O}_{Fr(L_F)} = \bigoplus_{k \in \mathbb{Z}} L_F^{\otimes_{\mathcal{O}_F} (-k)}. $$ Here each piece has the right size, since $t^{-k}\mathcal{O}_F \leftrightarrow L_F^{\otimes_{\mathcal{O}_F} (-k)}$ and the fibers of both look like $\mathcal{O}_F$ tensored with the residue field. When we base change to $\mathbb{R}$ we can see the grading by non-parallel weight characters, since $\mathbb{R} \otimes_\mathbb{Z} \mathcal{O}_F \cong \mathbb{R}^d$ ($F$ was totally real of degree $[F:\mathbb{Q}] = d$). Thus after base changing to $\mathbb{R}$, the structure sheaf looks (again, non-canonically) like $\prod_{\sigma \colon F \to \mathbb{R}} \mathbb{R}[t_\sigma,t_\sigma^{-1}]$. The character $\lambda \mapsto \prod_{\sigma \colon F \to \mathbb{R}} \lambda^{k_\sigma}$ corresponds to the piece $t^{-k_{\sigma_1}}\mathbb{R} \times \dots \times t^{-k_{\sigma_d}}\mathbb{R}$.

The reasons I am convinced by this answer are because the fibers looks like $Res_{\mathcal{O}_F/\mathbb{Z}}\mathbb{G}_m$ and because we get the grading by characters. I'd love to see if anyone has a reference that confirms this answer (or an issue that would point to it being false)!

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.