Equivariant Derived Categories Associated to a Sum of Two Potentials

Abstract

Suppose $f,g$ are homogeneous polynomials of degree $d$ defining smooth hypersurfaces $X_f=V(f)\subset {\mathbb{P}}^{m-1}$ and $X_g=V(g)\subset {\mathbb{P}}^{n-1}$. Then the sum $f(x)+g(y)$ defines a smooth hypersurface $X=V(f(x)+g(y))\subset {\mathbb{P}}^{m+n-1}$ with an action of ${\mu }_d$ scaling the $g$ variables. Motivated by the work of Orlov, we construct a semi-orthogonal decomposition of the derived category of coherent sheaves on $[X/{\mu }_d]$ provided $d\ge \max \{m,n\}$.

arXiv commentSubstantial revisions based on referee comments. To appear in the Journal of Geometry and Physics