Symmetries and stabilization for sheaves of vanishing cycles
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:...
| Päätekijät: | , , , , |
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| Aineistotyyppi: | Journal article |
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Worldwide Center of Mathematics
2015
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| _version_ | 1826302655995052032 |
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| author | Brav, C Bussi, V Dupont, D Joyce, D Szendroi, B |
| author_facet | Brav, C Bussi, V Dupont, D Joyce, D Szendroi, B |
| author_sort | Brav, C |
| collection | OXFORD |
| description | 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 $\mathbb K$-schemes over other fields $\mathbb K$, or 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. |
| first_indexed | 2024-03-07T05:50:54Z |
| format | Journal article |
| id | oxford-uuid:e8d6c20e-b6ee-4996-8962-0cf76429a316 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T05:50:54Z |
| publishDate | 2015 |
| publisher | Worldwide Center of Mathematics |
| record_format | dspace |
| spelling | oxford-uuid:e8d6c20e-b6ee-4996-8962-0cf76429a3162022-03-27T10:49:47ZSymmetries and stabilization for sheaves of vanishing cyclesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:e8d6c20e-b6ee-4996-8962-0cf76429a316Symplectic Elements at OxfordWorldwide Center of Mathematics2015Brav, CBussi, VDupont, DJoyce, DSzendroi, BLet $U$ 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 $\mathbb K$-schemes over other fields $\mathbb K$, or 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. |
| spellingShingle | Brav, C Bussi, V Dupont, D Joyce, D Szendroi, B Symmetries and stabilization for sheaves of vanishing cycles |
| title | Symmetries and stabilization for sheaves of vanishing cycles |
| title_full | Symmetries and stabilization for sheaves of vanishing cycles |
| title_fullStr | Symmetries and stabilization for sheaves of vanishing cycles |
| title_full_unstemmed | Symmetries and stabilization for sheaves of vanishing cycles |
| title_short | Symmetries and stabilization for sheaves of vanishing cycles |
| title_sort | symmetries and stabilization for sheaves of vanishing cycles |
| work_keys_str_mv | AT bravc symmetriesandstabilizationforsheavesofvanishingcycles AT bussiv symmetriesandstabilizationforsheavesofvanishingcycles AT dupontd symmetriesandstabilizationforsheavesofvanishingcycles AT joyced symmetriesandstabilizationforsheavesofvanishingcycles AT szendroib symmetriesandstabilizationforsheavesofvanishingcycles |