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 ${\Omega }_X^{ch}(\log D)$, whose conformal weight $0$ subspace is the algebra ${\Omega }_X(\log D)$ of forms with log poles along $D$. We prove various basic structural results about ${\Omega }_X^{ch}(\log D)$. In particular, if $X^{\ast }=X\setminus D$ has a volume form then we show that ${\Omega }_X^{ch}(\log D)$ 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 ${\Theta }^{\otimes d}$ on the elliptic curve $E={\mathbf{C}}^{\ast }/q^{\mathbf{Z}}$. We further show how ${\Omega }_X^{ch}(\log D)$ can be understood in terms of a simple birational modification of the space of jets into $X$.

Accepted version of article in Ann. Henri. Poincaré