Logarithmic jets and the chiral de Rham complex of a pair
Emile Bouaziz
TL;DR
This work introduces the logarithmic chiral de Rham complex $Ω^{ch}_{X}(\operatorname{log}D)$ for a smooth variety $X$ with simple normal crossings divisor $D$, connecting vertex algebraic structures to logarithmic geometry. It establishes a topological (and, when $D$ is anticanonical, extended topological) symmetry on $Ω^{ch}_{X}(\operatorname{log}D)$, and defines an elliptic genus $\operatorname{Ell}_{X,D}(q,y)$ that lives as a section of $Θ^{\otimes d}$ on the elliptic curve $E_q$ in the log CY case. The paper develops a birational perspective via the log jet space $J_{\operatorname{log}D}(X)$, showing a Lagrangian (logarithmic) subalgebra corresponding to a log $cγ$-system and providing an explicit isomorphism with a log jet differential sheaf. It also computes the cohomological character in the toric (e.g., $\mathbb{P}^d$ with toric boundary) case and outlines the general ellipticity and dependence on compactification, thereby extending the classic MSV theory to open varieties through the log geometry framework.
Abstract
To a smooth variety $X$ with simple normal crossings divisor $D$, we associate a sheaf of vertex algebras on $X$, denoted $Ω^{ch}_{X}(\operatorname{log}D)$, whose conformal weight $0$ subspace is the algebra $Ω_{X}(\operatorname{log}D)$ of forms with log poles along $D$. We prove various basic structural results about $Ω^{ch}_{X}(\operatorname{log}D)$. In particular, if $X^{*}=X\setminus D$ has a volume form then we show that $Ω^{ch}_{X}(\operatorname{log}D)$ 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 $Θ^{\otimes d}$ on the elliptic curve $E=\mathbf{C}^{*}/q^{\mathbf{Z}}$. We further show how $Ω^{ch}_{X}(\operatorname{log}D)$ can be understood in terms of a simple birational modification of the space of jets into $X$.