Logarithmic jets and the chiral de Rham complex of a pair

Abstract

To a smooth variety $X$ with simple normal crossings divisor $D$, we associate a sheaf of vertex algebras on $X$, denoted $\Omega^{ch}_{X}(\operatorname{log}D)$, whose conformal weight $0$ subspace is the algebra $\Omega_{X}(\operatorname{log}D)$ of forms with log poles along $D$. We prove various basic structural results about $\Omega^{ch}_{X}(\operatorname{log}D)$. In particular, if $X^{*}=X\setminus D$ has a volume form then we show that $\Omega^{ch}_{X}(\operatorname{log}D)$ admits a topological structure of rank $d=\operatorname{dim}(X)$, which is enhanced to an extended topological structure if $D\sim -K_{X}$ is in fact anticanonical. In this latter case we also show that the resulting $(q,y)$ character $\operatorname{Ell}(X,D)(q,y)$ is a section of the line bundle $\Theta^{\otimes d}$ on the elliptic curve $E=\mathbf{C}^{*}/q^{\mathbf{Z}}$. We further show how $\Omega^{ch}_{X}(\operatorname{log}D)$ can be understood in terms of a simple birational modification of the space of jets into $X$.