Windows for cdgas

Abstract

We study a Fourier-Mukai kernel associated to a GIT wall-crossing for arbitrarily singular (not necessarily reduced or irreducible) affine varieties over any field. This kernel is closely related to a derived fiber product diagram for the wall-crossing and simple to understand from the viewpoint of commutative differential graded algebras. However, from the perspective of algebraic varieties, the kernel can be quite complicated, corresponding to a complex with multiple homology sheaves. Under mild assumptions in the Calabi-Yau case, we prove that this kernel provides an equivalence between the category of perfect complexes on the two GIT quotients. More generally, we obtain semi-orthogonal decompositions which show that these categories differ by a certain number of copies of the derived category of the derived fixed locus. The derived equivalence for the Mukai flop is recovered as a very special case.

arXiv comment30 pages; v2: updated to discuss the nature of the wall-crossing kernel and its relationship to a fiber product, v3: accepted for publication in Advances in Mathematics

References in corpus (10)

• Semiorthogonal decomposition for algebraic varieties
• Koszul duality and modular representations of semi-simple Lie algebras
• Equivalences between GIT quotients of Landau-Ginzburg B-models
• On algebraic spaces with an action of G_m
• Derived equivalence for stratified Mukai flop on G(2,4)
• The telescope conjecture for algebraic stacks
• Equivalences and stratified flops
• Kernels for Grassmann Flops
• Kernels from Compactifications
• Iwahori-Matsumoto involution and linear Koszul Duality

Cited by in corpus (1)

• Derived equivalence for the simple flop of type $G_2^{†}$ via tilting bundles