16
$\begingroup$

Let $X$ be a smooth proper algebraic curve over $\mathbb{C}$. Say a complex local system $\mathbb{V}$ on $X$ is motivic if there exists a dense Zariski-open subset $U\subset X$, and a smooth proper morphism $\pi: Y\to U$, so that $\mathbb{V}|_U$ arises as a subquotient of $R^i\pi_*\mathbb{C}$ for some $i\geq 0$. Fix an integer $r\geq 0$.

Is the set of rank $r$ motivic local systems on $X$, with connected geometric monodromy group, finite?

Here the geometric monodromy group is the Zariski closure of the image of the natural monodromy representation $$\pi_1(X,x)\to GL(\mathbb{V}_x).$$

The question arises from this very interesting recent paper of Joshua Lam, who gives a positive answer when $r=2$ and the genus of $X$ is $2$. Ultimately Lam shows that all such local systems are pulled back from Shimura curves, and then uses Takeuchi's results on finiteness of Shimura curves of bounded genus.

Some remarks:

  1. If instead of $\mathbb{C}$-local systems, one considers motivic $\mathbb{Q}$-local systems, finiteness follows from Deligne's results on finiteness of $\mathbb{Z}$-local systems underlying a polarizable VHS. The same argument shows that there are finitely many motivic $K$-local systems, where $K$ ranges over all number fields of degree at most some constant. One can slightly improve this via arithmetic methods; e.g. if one fixes an $\ell$-adic field $L$, there are finitely many motivic local systems valued in number fields contained in $L$, using the results of this paper of mine.

  2. The analogous statement is false for curves over the algebraic closure of a finite field, which carry a huge collection of motivic local systems.

  3. The statement is false for open curves, as witnessed by e.g. the example of $\mathbb{P}^1\setminus\{0,1,\infty\}$ -- any rank $2$ local system on this curve with quasi-unipotent monodromy at infinity is motivic. Probably to formulate the correct analogue of this question for open curves one needs to fix the conjugacy classes of local monodromy at infinity.

  4. It is necessary to restrict to some condition along the lines of connected geometric monodromy group, to rule out tensor products with finite monodromy local systems, etc.

Improve this question
$\endgroup$
7
  • 2
    $\begingroup$ Given a motivic local system on an open subset of $C \times \mathbb P^1$, one can take a local system on $\mathbb P^1$ minus three points, pull back, tensor, and pushforward to get a local system on an open subset of $C$. If the finiteness is true, then doing this to the hypergeometric local systems you mention in example 3 must always produce ramified local systems (or ones with disconnected monodromy groups, or determinants of different orders, but ramified seems most likely.) Proving this might be a way to get some weak evidence for the original question. $\endgroup$
    – Will Sawin
    Commented Dec 3, 2022 at 12:34
  • 1
    $\begingroup$ Actually maybe this is not so hard - if some kernel sends a hypergeometric system to an unramified local system, then a dual kernel must send the dual of the unramified local system to a system with that hypergeometric as a summand, but the eigenvalues of local monodromy that can arise from a kernel applied to an unramified local system are bounded by vanishing cycles theory. $\endgroup$
    – Will Sawin
    Commented Dec 3, 2022 at 12:36
  • $\begingroup$ @WillSawin: Thanks! Can you say a bit about the vanishing cycle argument you have in mind? $\endgroup$
    – Daniel Litt
    Commented Dec 3, 2022 at 20:03
  • 1
    $\begingroup$ Any eigenvalues of the local monodromy at a point must arise as eigenvalues of the inertia action on vanishing cycles of the tensor product of the kernel with the pullback of the local system somewhere in the fiber over that point. However, the vanishing cycles is an etale-local invariant. Because an unramified local system is etale-locally trivial, the vanishing cycles will not (locally) depend on the choice of local system, so the set of eigenvalues that can appear is independent of the local system. By constructibility of vanishing cycles, this set is finite. $\endgroup$
    – Will Sawin
    Commented Dec 6, 2022 at 17:56
  • $\begingroup$ Since there are finitely many possible local monodromy representations we get by applying the dual kernel to an unramified local system, we can only apply the kernel to get an unramified local system to an original local system with finitely many local monodromy representations. So we would need an example of infinitely many local systems on an open curve with fixed monodromy at infinity to get infinitely many on a closed curve by any kernel. $\endgroup$
    – Will Sawin
    Commented Dec 6, 2022 at 17:59

0

Reset to default

You must log in to answer this question.