Given a morphism of algebraic stacks $f: X \to Y$ (possibly non-representable) what sufficient conditions will guarantee that the derived push-forward of the structure sheaf on $X$ is isomorphic to the structure sheaf on $Y$? I am looking for a stacky version of a result of Buch-Mihalcea (Thm 3.1 of Quantum K-theory of Grassmannians <https://arxiv.org/abs/0810.0981>, which itself is a variation of a result of Kollar)


Thm 3.1 of BM: Let $f : X \rightarrow Y$ be a surjective map of projective varieties with rational singularities. Assume that the general fiber of $f$ is rational, i.e. $f^{−1}(y)$ is an irreducible rational variety for all closed points in a dense open subset of $Y$. Then $f\_∗[\mathcal{O}\_X]=[\mathcal{O}\_Y]$ in the K-theory of Y. 


I am happy to assume, in any proposed generalization, that the stacks are proper DM stacks (even smooth) and that the general fiber is an irreducible rational variety. 



Derived push-forwards of structure sheaves for morphisms of algebraic stacks
I am amplifying the above comments. The theorem of Buch-Mihalcea also holds for stacks.


Let $k$ be a field of characteristic $0$. Let $Y$ be a Deligne-Mumford stack that is finite type over $k$ and that is normal: there exists a surjective, étale morphism $h:\widetilde{Y}\to Y$ with $\widetilde{Y}$ a normal scheme. Let
$f:X\to Y$ be a proper, surjective morphism of Deligne-Mumford stacks. Assume that $X$ is integral, and assume that $X$ has rational singularities: there exists a surjective, étale morphism $g:\widetilde{X}\to X$ such that $\widetilde{X}$ is a $k$-scheme that has rational singularities. Finally, assume that the geometric generic fiber of $f$ is isomorphic (as a stack) to a smooth proper variety that is rationally connected, or even just $\mathcal{O}$-acyclic.


**Proposition**. The natural map $\mathcal{O}\_Y\to f\_\*\mathcal{O}\_X$ is an isomorphism, and $R^q f\_\*\mathcal{O}\_X$ equals $0$ for every $q>0$.


Because $h$ is flat, $h^\*R^qf\_\*\mathcal{O}\_X$ equals $R^q \widetilde{f} \mathcal{O}\_{X\times\_Y \widetilde{Y}}$, where $\widetilde{f}:X\times\_Y \widetilde{Y} \to \widetilde{Y}$ is the projection. Thus, without loss of generality, assume that $Y$ is a normal scheme. As in the comment, there exists a coarse moduli space $p:X \to |X|$. The morphism $f$ factors as the composition of $p$ and a surjective, proper $k$-morphism, $$|f|:|X|\to Y.$$ Because the characteristic is $0$, $X$ is a tame stack. Thus, $\mathcal{O}\_{|X|} \to p\_\*\mathcal{O}\_X$ is an isomorphism and $R^qp\_\*\mathcal{O}\_X$ equals $0$ for all $q>0$. Therefore, it suffices to prove that $\mathcal{O}\_Y\to |f|\_\*\mathcal{O}\_{|X|}$ is an isomorphism and $R^q|f|\_\*\mathcal{O}\_{|X|}$ equals $0$ for all $q>0$. Via Buch-Mihalcea, it suffices to prove that $|X|$ has rational singularities. This follows from Proposition 5.13, p. 157 of "Birational Geometry of Algebraic Varieties" by Kollár and Mori.


