Symmetries and stabilization for sheaves of vanishing cycles

Abstract

Let $U$ be a smooth $\mathbb{C}$-scheme, $f:U\to {\mathbb{A}}^1$ 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" $P{V}_{U,f}$, a perverse sheaf on $X$. This paper proves four main results: (a) Suppose $\Phi :U\to U$ is an isomorphism with $f\circ \Phi =f$ and $\Phi {|}_X=$id${}_X$. Then $\Phi$ induces an isomorphism ${\Phi }_{\ast }:P{V}_{U,f}\to P{V}_{U,f}$. We show that ${\Phi }_{\ast }$ is multiplication by det$(d\Phi {|}_X)=1$ or $-1$. (b) $P{V}_{U,f}$ depends up to canonical isomorphism only on $X^{(3)},f^{(3)}$, for $X^{(3)}$ the third-order thickening of $X$ in $U$, and $f^{(3)}=f{|}_{X^{(3)}}:X^{(3)}\to {\mathbb{A}}^1$. (c) If $U,V$ are smooth $\mathbb{C}$-schemes, $f:U\to {\mathbb{A}}^1$, $g:V\to {\mathbb{A}}^1$ are regular, $X=$Crit$(f)$, $Y=$Crit$(g)$, and $\Phi :U\to V$ is an embedding with $f=g\circ \Phi$ and $\Phi {|}_X:X\to Y$ an isomorphism, there is a natural isomorphism ${\Theta }_{\Phi }:P{V}_{U,f}\to \Phi {|}_X^{\ast }(P{V}_{V,g}){\otimes }_{{\mathbb{Z}}_2}{P}_{\Phi }$, for $P_{\Phi }$ a natural principal ${\mathbb{Z}}_2$-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 $P_{X,s}$ on $X$, such that if $(X,s)$ is locally modelled on Crit$(f:U\to {\mathbb{A}}^1)$ then $P_{X,s}$ is locally modelled on $P{V}_{U,f}$. We also generalize our results to replace $U,X$ by complex analytic spaces, and $P{V}_{U,f}$ 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.

77 pages, LaTeX. (v4) corrections, new Appendix by Joerg Schuermann

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