Derived Categories of Coherent Sheaves and Triangulated Categories of Singularities
Dmitri Orlov
TL;DR
The paper builds a rigorous bridge between graded D-branes in Landau-Ginzburg models with homogeneous superpotentials and triangulated categories of singularities, via a chain of equivalences and fully faithful functors. It establishes that the graded D-brane category $DGrB(W)$ is equivalent to the graded singularity category $D_{Sg}^{gr}(A)$ of the fiber $W=0$, and that $D_{Sg}^{gr}(A)$ embeds into (and sometimes is equivalent to) ${D^b(qgr A)}$ and hence to ${D^b(coh(Y))}$ for the projective geometry defined by $W$. The results yield concrete semiorthogonal decompositions and equivalences for hypersurfaces, Calabi-Yau, Fano, and general-type cases, connecting B-branes, matrix factorizations, and coherent-sheaf theory in a unified homological framework. This provides a concrete realization of homological mirror-symmetry phenomena in the graded setting and clarifies how singularity and projective-geometry data encode D-brane categories in Landau-Ginzburg models.
Abstract
In this paper we establish an equivalence between the category of graded D-branes of type B in Landau-Ginzburg models with homogeneous superpotential W and the triangulated category of singularities of the fiber of W over zero. The main result is a theorem that shows that the graded triangulated category of singularities of the cone over a projective variety is connected via a fully faithful functor to the bounded derived category of coherent sheaves on the base of the cone. This implies that the category of graded D-branes of type B in Landau-Ginzburg models with homogeneous superpotential W is connected via a fully faithful functor to the derived category of coherent sheaves on the projective variety defined by the equation W=0.