Asymptotic nonvanishing of syzygies of algebraic varieties

We establish precise nonvanishing results for asymptotic syzygies of smooth projective varieties. This refines Ein-Lazarsfeld’s asymptotic nonvanishing theorem. Combining with the author’s previous asymptotic vanishing result, we completely determine the asymptotic shapes of the minimal free resolutions of the graded section modules of a line bundle on a smooth projective variety as the positivity of the embedding line bundle grows.