The virtual Todd class formula for almost perfect obstruction theories

Suppose that $f\colon X\to Y$ is a morphism equipped with an almost perfect obstruction theory with virtual tangent class $[T_{X/Y}]-[\mathcal{O}b_X]\in K_0(X)$. Assume that this class lies in the image of the natural map

$$\kappa\colon K^0(X)\longrightarrow K_0(X),$$

so that $[T_{X/Y}]-[\mathcal{O}b_X]=\kappa(F)$ for some $F\in K^0(X)$. Virtual Todd class conjecture. For any $F\in K^0(X)$ satisfying

$$\kappa(F)=[T_{X/Y}]-[\mathcal{O}b_X],$$

we have

$$\mathop{\rm td}\nolimits^{\mathrm{vir}}(X/Y)=\mathop{\rm td}\nolimits(F)\cap[X]^{\mathrm{vir}}.$$

This would extend the usual virtual Riemann–Roch expression from perfect obstruction theories to almost perfect obstruction theories, where the relevant $K$-theory class need not itself be represented by a global complex of vector bundles. The preceding proposition establishes the formula when the almost perfect obstruction theory is induced by a perfect obstruction theory with a global locally free resolution; the general case remains open.