SYZ for index 1 hypersurfaces in projective space

Abstract

We study homological mirror symmetry of the singular hypersurface $X_0=V(t^{n+1}-x_0\cdots x_n)\subseteq P^{n+1}$. Following an SYZ type approach, we produce an LG-model whose Fukaya-Seidel category recovers line bundles on $X_0$. As a byproduct of our approach, we answer a conjecture of N. Sheridan about generating the small component of the Fukaya category of the smooth index 1 Fano hypersurface in $P^{n+1}$ without bounding co-chains.