Logarithmic jets and the chiral de Rham complex of a pair

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