Bott vanishing for elliptic surfaces

Abstract: We explore Bott Vanishing for elliptic surfaces over $\mathbb{P}^1$. We show that Bott Vanishing is singnificantly affected by the geometric properties that whether there exists certain type of singular fibers on the elliptic fibration such as cuspidal fibers. For ample line bundle on the surface with large self-intersection, these geometric properties give criteria for Bott Vanishing.