講演者:斉藤秀司 氏 (東京工業大学)
講演題目:Existence conjecture for smooth sheaves on varieties over finite fields

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 $$r=1$$.