A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes

Journal article

Pantev, Toen, Vezzosi and Vaquie arXiv:1111.3209 defined $k$-shifted symplectic derived schemes and stacks $({\bf X} ,\omega_{\bf X} )$ 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 $({\bf X} ,\omega_{\bf X} )$ for $k<0$, where ${\bf X}\simeq{\bf Spec}\,A^\bullet$ for $A^\bullet$ a quasi-free cdga with coordinates $x^i_j,y^{k-i}_j$ and $\omega_{\bf X}\simeq(\omega^0,0,0,\ldots)$ with $\omega^0=\sum_{i,j}{\rm d}_{\rm dR}x^i_j{\rm d}_{\rm dR}y^{k-i}_j$. 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 $({\bf X} ,\omega_{\bf X} )$ for $k<0$, relative to the Bussi-Brav-Joyce 'Darboux form' local models $A^\bullet,x^i_j,y^{k-i}_j,\omega^0$ for $({\bf X} ,\omega_{\bf X} )$. They show ${\bf L}\simeq{\bf Spec}\,B^\bullet$ and ${\bf f}\simeq{\bf Spec}\,\alpha$, for $B^\bullet$ a quasi-free cdga with coordinates $\tilde x^i_j,u^i_j,v^{k-1-i}_j$ and $\alpha:A^\bullet\to B^\bullet$ a cdga morphism with $\tilde x^i_j=\alpha(x^i_j)$, and the Lagrangian structure is $h_{\bf L}\simeq(h^0,0,0,\ldots)$ with $h^0=\sum_{i,j}{\rm d}_{\rm dR}u^i_j{\rm d}_{\rm dR}v^{k-1-i}_j$. 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'.

Authors: Joyce, D; Safronov, P

• Publication status: Not published
• Peer review status: Not peer reviewed
• Publication date: 2015-01-01
• Keywords: math.AG