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" $PV_{U,f}$, 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

Local BPS Invariants: Enumerative Aspects and Wall-Crossing

“…The isomorphism constructed above commutes with the Chow morphisms. Therefore the higher genus BPS invariants as well as their sl 2 × sl 2 refinements as defined in [20,3,25] remain unchanged as well.…”

Relative Donaldson-Thomas theory for Calabi-Yau 4-folds

“…As a categorification of Donaldson-Thomas invariants, Brav, Bussi, Dupont, Joyce and Szendroi [8] and Kiem and Li [32] recently defined a cohomology theory for Calabi-Yau 3-folds whose Euler characteristic is the DT 3 invariant. The point is that moduli spaces of simple sheaves on Calabi-Yau 3-folds are critical points of holomorphic functions locally [9], [31], and we could consider perverse sheaves of vanishing cycles of these functions.…”

“…In general, the virtual dimension of [M • MY is the perverse sheaf constructed by Brav-Bussi-Dupont-Joyce-Szendroi [8] and Kiem-Li [32]. In Cases I-III, if M Y is smooth, the perverse sheaf P…”

On orientations for gauge-theoretic moduli spaces

“…Kontsevich and Soibelman [27] needed orientation data for their motivic Donaldson-Thomas invariants of Calabi-Yau 3-folds (see also [11]). Later, orientation data was found to be necessary in other generalizations of Donaldson-Thomas theory for 3-Calabi-Yau categories, including Kontsevich and Soibelman's Cohomological Hall Algebras [28], and categorification of Donaldson-Thomas theory using perverse sheaves by Ben-Bassat, Brav, Bussi, Dupont, Joyce, and Szendrői [9,10,18].…”