A 'Darboux Theorem' for shifted symplectic structures on derived Artin stacks, with applications

Abstract

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, arXiv:1111.3209. This paper extends the previous three from (derived) schemes to (derived) Artin stacks. We prove four main results: (a) If $(X,\omega )$ is a $k$-shifted symplectic derived Artin stack for $k<0$ in the sense of arXiv:1111.3209, then near each $x\in X$ we can find a 'minimal' smooth atlas $\phi :U\to X$ with $U$ an affine derived scheme, such that $(U,{\phi }^{\ast }(\omega ))$ may be written explicitly in coordinates in a standard 'Darboux form'. (b) If $(X,\omega )$ is a $-1$-shifted symplectic derived Artin stack and $X^{\prime }$ the underlying classical Artin stack, then $X^{\prime }$ extends naturally to a 'd-critical stack' $(X^{\prime },s)$ in the sense of arXiv:1304.4508. (c) If $(X,s)$ is an oriented d-critical stack, we can define a natural perverse sheaf $P_{X,s}^{\bullet }$ on $X$, such that whenever $T$ is a scheme and $t:T\to X$ is smooth of relative dimension $n$, then $T$ is locally modelled on a critical locus Crit$(f:U\to {\mathbb{A}}^1)$ for $U$ smooth, and $t^{\ast }(P_{X,s}^{\bullet })[n]$ is locally modelled on the perverse sheaf of vanishing cycles $P{V}_{U,f}^{\bullet }$ of $f$. (d) If $(X,s)$ is a finite type oriented d-critical stack, we can define a natural motive $M{F}_{X,s}$ in a ring of motives ${\bar{\mathcal{M}}}_X^{st,\hat{\mu }}$ on $X$, such that whenever $T$ is a finite type scheme and $t:T\to X$ is smooth of dimension $n$, then $T$ is locally modelled on a critical locus Crit$(f:U\to {\mathbb{A}}^1)$ for $U$ smooth, and ${\mathbb{L}}^{-n/2}\odot t^{\ast }(M{F}_{X,s})$ is locally modelled on the motivic vanishing cycle $M{F}_{U,f}^{mot,\varphi }$ of $f$ in ${\bar{\mathcal{M}}}_T^{st,\hat{\mu }}$. Our results have applications to categorified and motivic extensions of Donaldson-Thomas theory of Calabi-Yau 3-folds

(v2) 61 pages. Minor corrections, foundational material on perverse sheaves shortened

References in corpus (3)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Higher and derived stacks: a global overview
• Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants

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