Quantum Lefschetz theorem revisited

Abstract

Let $X$ be any smooth Deligne-Mumford stack with projective coarse moduli, and $Y$ be a smooth complete intersection in $X$ associated with a direct sum of semi-positive line bundles. We will introduce a useful and broad class known as admissible series for discussing quantum Lefschetz theorem. For any admissible series on the Givental's Lagrangian cone of $X$, we will show that a hypergeometric modification of the series lies on the Lagrangian cone of $Y$. This confirms a prediction from Coates-Corti-Iritani-Tseng about the genus zero quantum Lefschetz theorem beyond convexity. In our quantum Lefschetz theorem, we use extended variables to formulate the hypergeometric modification, which may be of self-independent interest.