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 ) ⊂ P m−1 and Xg = V (g) ⊂ P n−1 . Then the sum of f and g defines a smooth hypersurface X = V (f ⊕ g) ⊂ P m+n−1 with an action of µ 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/µ d ] provided d ≥ max{m, n}.

Derived categories of resolutions of cyclic quotient singularities

“…If n = 1, the quotient is already smooth and we have Y = X/G -here the semi-orthogonal decomposition categorifies the natural decomposition of the orbifold cohomology; compare [PVdB15]. The n = 1 case is also proven in [Lim16,Thm. 3.3.2].…”