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" $PV_{U,f}$, 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
$\Phi\vert_X=$id$_X$. Then $\Phi$ induces an isomorphism $\Phi_*:PV_{U,f}\to
PV_{U,f}$. We show that $\Phi_*$ is multiplication by det$(d\Phi\vert_X)=1$ or
$-1$.
(b) $PV_{U,f}$ depends up to canonical isomorphism only on $X^{(3)},f^{(3)}$,
for $X^{(3)}$ the third-order thickening of $X$ in $U$, and
$f^{(3)}=f\vert_{X^{(3)}}:X^{(3)}\to\mathbb A^1$.
(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 $\Phi\vert_X:X\to Y$ an isomorphism,
there is a natural isomorphism
$\Theta_\Phi:PV_{U,f}\to\Phi\vert_X^*(PV_{V,g})\otimes_{\mathbb Z_2}P_\Phi$,
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 $PV_{U,f}$.
We also generalize our results to replace $U,X$ by complex analytic spaces,
and $PV_{U,f}$ 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.Comment: 77 pages, LaTeX. (v4) corrections, new Appendix by Joerg Schuerman
