2012/11/30 by Christopher Brav, Vittoria Bussi, Delphine Dupont +2 · 3 citations
Mathematics · #math.AG #math.CV #math.DG #msc:14C30 #msc:32S30 #msc:32S60
paper · pdf · doi:10.5427/jsing.2015.11e
published as Journal of Singularities 11 (2015), 85-151 · 77 pages, LaTeX. (v4) corrections, new Appendix by Joerg Schuermann
arxiv created 2015/06/04 · arxiv updated 2015/06/05
Let U be a smooth \mathbb C-scheme, f:U→\mathbb A1 a regular function, and X=Crit(f) the critical locus, as a \mathbb C-subscheme of U. Then one can define the "perverse sheaf of vanishing cycles" PVU,f, a perverse sheaf on X. This paper proves four main results: (a) Suppose Φ:U→ U is an isomorphism with f∘Φ=f and Φ\vertX=idX. Then Φ induces an isomorphism Φ_*:PVU,f→ PVU,f. We show that Φ_* is multiplication by det(dΦ\vertX)=1 or -1. (b) PVU,f depends up to canonical isomorphism only on X(3),f(3), for X(3) the third-order thickening of X in U, and f(3)=f\vertX(3):X(3)→\mathbb A1. (c) If U,V are smooth \mathbb C-schemes, f:U→\mathbb A1, g:V→\mathbb A1 are regular, X=Crit(f), Y=Crit(g), and Φ:U→ V is an embedding with f=g∘Φ and Φ\vertX:X→ Y an isomorphism, there is a natural isomorphism ΘΦ:PVU,f→Φ\vertX^*(PVV,g)⊗\mathbb Z2PΦ, for PΦ a natural principal \mathbb Z2-bundle on X. (d) If (X,s) is an oriented d-critical locus in the sense of Joyce arXiv:1304.4508, there is a natural perverse sheaf PX,s on X, such that if (X,s) is locally modelled on Crit(f:U→\mathbb A1) then PX,s is locally modelled on PVU,f. We also generalize our results to replace U,X by complex analytic spaces, and PVU,f by \mathcal D-modules, or mixed Hodge modules. We discuss applications of (d) to categorifying Donaldson-Thomas invariants of Calabi-Yau 3-folds, and to defining a 'Fukaya category' of Lagrangians in a complex symplectic manifold using perverse sheaves. This is the third in a series of papers arXiv:1304.4508, arXiv:1305.6302, arXiv:1305.6428, arXiv:1312.0090, arXiv:1403.2403, arXiv:1404.1329, arXiv:1504.00690.