# A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes

Joyce, D
Safronov, P

23 April 2020

## Journal:

Annales de la Faculte des Sciences de Toulouse

## Last Updated:

2021-06-12T03:42:05.167+01:00

28

## DOI:

10.5802/afst.1616

831-908

## abstract:

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 have important
applications to Calabi-Yau geometry and quantization. Bussi, Brav and Joyce
arXiv:1305.6302 proved a 'Darboux Theorem' giving explicit Zariski or \'etale
local models for $k$-shifted symplectic derived schemes ${\bf X}$ for $k<0$
presenting them as twisted shifted cotangent bundles.
We prove a 'Lagrangian Neighbourhood Theorem' giving explicit Zariski or
etale local models for Lagrangians ${\bf f}:{\bf L}\to{\bf X}$ in $k$-shifted
symplectic derived schemes ${\bf X}$ for $k<0$, relative to the
Bussi-Brav-Joyce 'Darboux form' local models for ${\bf X}$. That is, locally
such Lagrangians can be presented as twisted shifted conormal bundles. We also
give a partial result when $k=0$.
We expect our results will have future applications to $k$-shifted Poisson
geometry (see arXiv:1506.03699), to defining 'Fukaya categories' of complex or
algebraic symplectic manifolds, and to categorifying Donaldson-Thomas theory of
Calabi-Yau 3-folds and 'Cohomological Hall algebras'.

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