Constructibility and duality for simple holonomic modules on complex symplectic manifolds

Kashiwara, Masaki; Schapira, Pierre

doi:10.1353/ajm.2008.0007

Cited by 17 publications

(15 citation statements)

References 18 publications

(24 reference statements)

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“…If D • , E • are simple holonomic DQ-modules on S supported on smooth Lagrangians L, M , then Kashiwara and Schapira [28] show that RH om(D Observe that the work of Behrend-Fantechi and Kashiwara-Schapira cited above supports the existence of (6.12)-(6.13): there are natural products But since (6.13) is a morphism of complexes, not of perverse sheaves, Theorem 2.7(i) does not apply, so we cannot construct µ L,M,N by naïvely gluing data on an open cover, as we have been doing in §3- §6.…”

Section: )mentioning

confidence: 99%

Symmetries and stabilization for sheaves of vanishing cycles

Brav

Bussi²,

Dupont³

et al. 2015

jsing

106

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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

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Section: )mentioning

confidence: 99%

Symmetries and stabilization for sheaves of vanishing cycles

Brav

Bussi²,

Dupont³

et al. 2015

jsing

106

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“…We will work with Fréchet algebras to make things simpler. The correct generalization beyond Fréchet algebras will be bornological algebras as in [14] and has been used recently by [21] and [18]. The generalization to this context is straightforward.…”

Section: 2mentioning

confidence: 99%

Duality and equivalence of module categories in noncommutative geometry

Block¹

2010

A Celebration of the Mathematical Legacy Of Raoul Bott

154

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Abstract. This is the second in a series of papers intended to set up a framework to study categories of modules in the context of non-commutative geometries. In [3] we introduced the basic DG category P A • , the perfect category of A• , which corresponded to the category of coherent sheaves on a complex manifold. In this paper we enlarge this category to include objects which correspond to quasi-coherent sheaves. We then apply this framework to proving an equivalence of categories between derived categories on the noncommutative complex torus and on a holomorphic gerbe on the dual complex torus.

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“…Brav et al [BBDJS15] and Bussi [Bus14] give a construction of the virtual de Rham cohomology of a pair of Lagrangians in terms of the hypercohomology of a perverse sheaf supported on . As explained in [BBDJS15, Remark 6.15], it follows from the work of Kashiwara and Schapira [KS08] that the perverse sheaf can be recovered as . The same remark speculates on a connection between and .…”

Section: Floer Cohomology and The Fukaya Categorymentioning

confidence: 97%

Locality in the Fukaya category of a hyperkähler manifold

Solomon

Verbitsky²

2019

Compositio Math.

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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 finite energy (xJ + yK)-holomorphic curve with boundary in a collection of I-holomorphic Lagrangians must be constant. By an argument based on the Lojasiewicz inequality, this result holds no matter how the Lagrangians intersect each other. It follows that one can choose perturbations such that the holomorphic polygons of the associated Fukaya category lie in an arbitrarily small neighborhood of the Lagrangians. That is, the Fukaya category is local. We show that holomorphic Lagrangians are tautologically unobstructed in the sense of Fukaya-Oh-Ohta-Ono. Moreover, the Fukaya A ∞ algebra of a holomorphic Lagrangian is formal. Our result also explains why the special Lagrangian condition holds without instanton corrections for holomorphic Lagrangians.

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Constructibility and duality for simple holonomic modules on complex symplectic manifolds

Cited by 17 publications

References 18 publications

Symmetries and stabilization for sheaves of vanishing cycles

Symmetries and stabilization for sheaves of vanishing cycles

Duality and equivalence of module categories in noncommutative geometry

Locality in the Fukaya category of a hyperkähler manifold

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