An introduction to perverse sheaves

Rietsch, Konstanze

doi:10.1090/fic/040/19

Cited by 9 publications

(6 citation statements)

References 21 publications

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“…Three other books are Kashiwara and Schapira [27], Schürmann [50], and Hotta, Tanisaki and Takeuchi [21]. Massey [36] and Rietsch [43] are surveys on perverse sheaves, and Beilinson, Bernstein and Deligne [5] is an important primary source, who cover both Q-perverse sheaves on C-schemes as in (a), and Q l -perverse sheaves on K-schemes as in (g) below.…”

Section: Background On Perverse Sheavesmentioning

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: Background On Perverse Sheavesmentioning

confidence: 99%

Symmetries and stabilization for sheaves of vanishing cycles

Brav

Bussi²,

Dupont³

et al. 2015

jsing

106

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“…20.5. The proof of Theorem 20.3 is tedious but does not require sophisticated methods; it is completely worked out in online notes [79] (following an outline of T. A. Springer [86], as communicated to him by R. MacPherson) and [46] . Here is the outline.…”

Section: 4mentioning

confidence: 99%

Lecture notes on sheaves and perverse sheaves

Goresky¹

2021

Preprint

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“…Proof. The simples perverse sheaves on a variety V are shifted intersection cohomology complexes, namely IC(L)[− codim V W ], for L an irreducible local system on an open subset of a closed W ⊆ V , by [7]: for another reference see [31,Theorem 6.9]. The first part follows by setting V = ∆, and W = ∆ or 0 respectively.…”

Section: Proposition 32 [22 Section 9]mentioning

confidence: 99%

Perverse Schobers and Wall Crossing

Donovan

2017

International Mathematics Research Notices

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For a balanced wall crossing in geometric invariant theory, there exist derived equivalences between the corresponding GIT quotients if certain numerical conditions are satisfied. Given such a wall crossing, I construct a perverse sheaf of categories on a disk, singular at a point, with half-monodromies recovering these equivalences, and with behaviour at the singular point controlled by a GIT quotient stack associated to the wall. Taking complexified Grothendieck groups gives a perverse sheaf of vector spaces: I characterise when this is an intersection cohomology complex of a local system on the punctured disk.

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An introduction to perverse sheaves

Cited by 9 publications

References 21 publications

Symmetries and stabilization for sheaves of vanishing cycles

Symmetries and stabilization for sheaves of vanishing cycles

Lecture notes on sheaves and perverse sheaves

Perverse Schobers and Wall Crossing

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