# Symmetries and stabilization for sheaves of vanishing cycles

@article{Brav2012SymmetriesAS,
title={Symmetries and stabilization for sheaves of vanishing cycles},
author={Christopher Brav and Vittoria Bussi and Delphine Dupont and Dominic Joyce and Bal{\'a}zs Szendrői},
journal={arXiv: Algebraic Geometry},
year={2012}
}
• Published 14 November 2012
• Mathematics
• arXiv: Algebraic Geometry
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$(d\Phi… A classical model for derived critical loci Let$f:U\to{\mathbb A}^1$be a regular function on a smooth scheme$U$over a field$\mathbb K$. Pantev, Toen, Vaquie and Vezzosi (arXiv:1111.3209, arXiv:1109.5213) define the "derived critical Virtual fundamental classes for moduli spaces of sheaves on Calabi-Yau four-folds • Mathematics • 2015 Let$({\bf X},\omega_{\bf X}^*)$be a separated,$-2$-shifted symplectic derived$\mathbb C$-scheme, in the sense of Pantev, Toen, Vezzosi and Vaquie arXiv:1111.3209, of complex virtual dimension A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes • Mathematics Annales de la Faculté des sciences de Toulouse : Mathématiques • 2019 Pantev, Toen, Vaqui\'e and Vezzosi arXiv:1111.3209 defined$k$-shifted symplectic derived schemes and stacks${\bf X}$for$k\in\mathbb Z$, and Lagrangians${\bf f}:{\bf L}\to{\bf X}$in them. They Generalized Donaldson-Thomas theory over fields K$\neq$C Generalized Donaldson-Thomas invariants defined by Joyce and Song arXiv:0810.5645 are rational numbers which `count' both$\tau$-stable and$\tau$-semistable coherent sheaves with Chern character A ‘Darboux theorem’ for shifted symplectic structures on derived Artin stacks, with applications • Mathematics • 2015 This is the fifth in a series arXiv:1304.4508, arXiv:1305,6302, arXiv:1211.3259, arXiv:1305.6428 on the '$k$-shifted symplectic derived algebraic geometry' of Pantev, Toen, Vaquie and Vezzosi, Perversely categorified Lagrangian correspondences • Mathematics • 2016 In this article, we construct a$2$-category of Lagrangians in a fixed shifted symplectic derived stack S. The objects and morphisms are all given by Lagrangians living on various fiber products. A Formality of differential graded algebras and complex Lagrangian submanifolds Let$i: \mathrm{L} \hookrightarrow \mathrm{X}$be a compact Kahler Lagrangian in a holomorphic symplectic variety$\mathrm{X}/\mathbf{C}$. We use deformation quantisation to show that the Locality in the Fukaya category of a hyperkähler manifold • Mathematics Compositio Mathematica • 2019 Let$(M,I,J,K,g)$be a hyperkähler manifold. Then the complex manifold$(M,I)$is holomorphic symplectic. We prove that for all real$x,y$, with$x^{2}+y^{2}=1$except countably many, any ## References SHOWING 1-10 OF 58 REFERENCES A theory of generalized Donaldson–Thomas invariants • Mathematics • 2008 This book studies generalized Donaldson-Thomas invariants$\bar{DT}{}^\alpha(\tau)$. They are rational numbers which 'count' both$\tau$-stable and$\tau$-semistable coherent sheaves with Chern A classical model for derived critical loci Let$f:U\to{\mathbb A}^1$be a regular function on a smooth scheme$U$over a field$\mathbb K$. Pantev, Toen, Vaquie and Vezzosi (arXiv:1111.3209, arXiv:1109.5213) define the "derived critical Constructible sheaves and the Fukaya category • Mathematics • 2006 Let$X$be a compact real analytic manifold, and let$T^*X$be its cotangent bundle. Let$Sh(X)$be the triangulated dg category of bounded, constructible complexes of sheaves on$X$. In this paper, Motivic degree zero Donaldson–Thomas invariants • Mathematics • 2009 Given a smooth complex threefold X, we define the virtual motive$[\operatorname{Hilb}^{n}(X)]_{\operatorname {vir}}$of the Hilbert scheme of n points on X. In the case when X is Calabi–Yau, The intrinsic normal cone • Mathematics • 1997 Abstract.Let$X$be an algebraic stack in the sense of Deligne-Mumford. We construct a purely$0$-dimensional algebraic stack over$X$(in the sense of Artin), the intrinsic normal cone${\frak
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Let M and N be Lagrangian submanifolds of a complex symplectic manifold S. We construct a Gerstenhaber algebra structure on $$\mathcal{T}or_\ast^{\mathcal{O}_S}(\mathcal{O}_M,\mathcal{O}_N)$$ and a
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Let X be a compact real analytic manifold, and let T* X be its cotangent bundle. In a recent paper with Zaslow (J Am Math Soc 22:233–286, 2009), we showed that the dg category Shc(X) of constructible
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We briefly review the formal picture in which a Calabi-Yau n-fold is the complex analogue of an oriented real n-manifold, and a Fano with a fixed smooth anticanonical divisor is the analogue of a
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Let Y be a complex manifold, and S an open disc. Put X=YxS9 and identify Y with Fx{0}. Let (M, F) be a holonomic filtered .S^-Module, i.e. M is holonomic and GrM is coherent over Gr3)x, where ^5Z is
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