The logarithmic Hochschild–Kostant–Rosenberg conjecture

Let $X$ be a log scheme for which the constructions in the paper are defined, let $i:X\to B$ be the local complete intersection morphism used to define logarithmic Hochschild cohomology, and let $Td_X^{\log}$ be the log Todd class. Write

$$HKR:\operatorname{Ext}^\star_B(i_*\mathcal{O}_X,i_*\mathcal{O}_X)\xrightarrow{\sim}\bigoplus_{p+q=\star}H^p(X,\wedge^q T_X^{\log})$$

for the composite induced by adjunction and the formality isomorphism, and let $\iota_{\sqrt{Td_X^{\log}}}$ denote contraction with the square root of the log Todd class. The logarithmic Hochschild–Kostant–Rosenberg conjecture. The composite morphism

$$I=HKR^{-1}\circ\iota_{\sqrt{Td_X^{\log}}}$$

provides an isomorphism of dg-algebras

$$\bigoplus_{p+q=\star}H^p(X,\wedge^q T_X^{\log})\to R^\star\Gamma(X,\operatorname{HH}^{\ell X}(\mathcal{O}_X)).$$

Here $HKR^{-1}$ is the inverse of the displayed composite HKR map. The conjecture is a logarithmic analogue of the Hochschild–Kostant–Rosenberg theorem with the Todd-class correction; its status is not resolved in the supplied text.