Melrose's b-calculus provides a powerful framework for analyzing elliptic operators on manifolds with boundary. In the context of log geometry, log smooth manifolds offer a natural generalization of manifolds with boundary, suitable for studying phenomena like degenerations of pseudo-holomorphic curves.
I'm interested in whether a "derived" enhancement of the b-calculus exists for log smooth manifolds. Specifically, can we construct a functorial derived enhancement $\mathbb{T}\mathcal{M}$ of the logarithmic tangent sheaf $T\mathcal{M}$ within the framework of derived differential geometry?
Ideally, such a $\mathbb{T}_\mathcal{M} \in \textbf{Perf}^{\geq 0}(\mathcal{M})$ should satisfy the following:
Local Model: On an affine toric variety $X_P$, $\mathbb{T}_{X_P}$ should be locally modeled by the complex $\mathbb{R}[P] \otimes P^{\text{gp}} \to \mathbb{R}[P] \otimes (P^{\text{gp}})^*$.
Underlying: A natural morphism $\mathbb{T}\mathcal{M} \to T\mathcal{M}$ in $\textbf{QCoh}(\mathcal{M})$ should induce the identity on $H^0$.
Compatibility with de Rham: The derived b-de Rham complex $\Omega^\bullet_b (\mathcal{M}) := \text{Sym}_{\mathcal{O}_\mathcal{M}} (\mathbb{T}_\mathcal{M}[-1])$ should have a filtration compatible with the logarithmic de Rham complex.
Has this been explored before? What are the main challenges in constructing such a derived b-calculus? Could it lead to new insights into elliptic operator theory or moduli spaces of maps on log smooth manifolds?
