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# 数学談話会（12月26日）

This is a joint work with Moritz Kerz. Let $X$ be a smooth variety over a finite field $\mathbb{F}_q$. For an integer $r>0$, let ${\cal S}_r(X)$ be the set of lisse $\overline{\mathbb{Q}_\ell}$-sheaves on $X$ of rank $r$ up to isomorphism and up to semi-simplification. Let $Cu(X)$ be the set of normalizations of integral curves on $X$. Let ${\cal S}k_r(X)$ be the set of systems $(V_Z)_{Z\in Cu(X)}$ with $V_Z\in {\cal S}_r(Z)$ such that
$(V_Z)_{|Z\times _X Z'}=(V_{Z'})_{|Z\times _X Z'}$ for $Z,Z'\in Cu(X)$.
The question is how to determine the image of the restriction map
$\tau:{\cal S}_r(X)\to {\cal S}k_r(X)$,
i.e. when a system $(V_Z)_{Z\in Cu(X)}$ glues to a lisse $\overline{\mathbb{Q}_\ell}$-sheaf on $X$. We explain a conjecture of Deligne on the problem which describes the image in terms of a ramification condition at infinity and prove the conjecture in case $r=1$.