Consider $\mathbb A^n$ and let $\Sigma$ be a subdivision of its toric fan $\mathbb R^n_{\geq 0}$. This induces a toric blowup $\pi : Y \to \mathbb A^n$. Let $X \subseteq Y$ be the preimage of the origin $\vec{0} \in \mathbb A^n$ and $\tau : X \to \vec{0}$ the map.

Q: Does the higher cohomology $H^i(X, \mathcal O_X), i \neq 0$ of the structure sheaf vanish on $X$?

We know $R \pi_* \mathcal O_Y = \mathcal O_{\mathbb A^n}$. If only cohomology and base change applied here, we would win. I know $\tau_* \mathcal O_X = \mathcal O_{\vec 0}$ in my example. It suffices to prove the cohomology vanishes after refining the subdivision $\Sigma$ by the Leray spectral sequence, so you can assume the subdivision is an iterated star subdivision or that the cones of $\Sigma$ are all free.

If the irreducible components $X_i \subseteq X$ are reduced, the ``normalization sequence'' may be useful $$\mathcal O_X \to \bigoplus \mathcal O_{X_i} \to \bigoplus \mathcal O_{X_i \cap X_j} \to \cdots$$ By refining my subdivision, I can assume all the components and their intersections are free. For example, if they are all $\mathbb P^n$'s, it is clear that each of their cohomologies vanish. In dimension $n = 2$, I think this would work, but the $\mathbb P^1$'s need not be reduced.

In general, the irreducible components of $X$ may be nonreduced.

Beware that if I were looking at the fiber of $Y \to \mathbb A^n$ over a closed subscheme $Z \subseteq \mathbb A^n$, it would be false. In the examples I know, the pushforward of the structure sheaf is not the structure sheaf, so it is reassuring that that at least holds in the above case.

I have tried to apply some induction on the number of components, but if you ever push forward a structure sheaf and get something other than the structure sheaf (nonreduced), then it won't work. It seems a version of Cech cohomology for a cover by closed subschemes would be nice, but this can't naively exist when the components are nonreduced.

Consider $\mathbb A^n$ and let $\Sigma$ be a subdivision of its toric fan $\mathbb R^n_{\geq 0}$. This induces a toric blowup $\pi : Y \to \mathbb A^n$. Let $X \subseteq Y$ be the preimage of the origin $\vec{0} \in \mathbb A^n$ and $\tau : X \to \vec{0}$ the map.

Q: Does the higher cohomology $H^i(X, \mathcal O_X), i \neq 0$ of the structure sheaf vanish on $X$?

We know $R \pi_* \mathcal O_Y = \mathcal O_{\mathbb A^n}$. If only cohomology and base change applied here, we would win. I know $\tau_* \mathcal O_X = \mathcal O_{\vec 0}$ in my example. I only need to know the cohomology vanishes after refining the subdivision $\Sigma$, so you can assume the subdivision is an iterated star subdivision or that the cones of $\Sigma$ are all free.

If the irreducible components $X_i \subseteq X$ are reduced, the ``normalization sequence'' may be useful $$\mathcal O_X \to \bigoplus \mathcal O_{X_i} \to \bigoplus \mathcal O_{X_i \cap X_j} \to \cdots$$ By refining my subdivision, I can assume all the components and their intersections are free. For example, if they are all $\mathbb P^n$'s, it is clear that each of their cohomologies vanish. In dimension $n = 2$, I think this would work, but the $\mathbb P^1$'s need not be reduced.

In general, the irreducible components of $X$ may be nonreduced.

Beware that if I were looking at the fiber of $Y \to \mathbb A^n$ over a closed subscheme $Z \subseteq \mathbb A^n$, it would be false. In the examples I know, the pushforward of the structure sheaf is not the structure sheaf, so it is reassuring that that at least holds in the above case.

I have tried to apply some induction on the number of components, but if you ever push forward a structure sheaf and get something other than the structure sheaf (nonreduced), then it won't work. It seems a version of Cech cohomology for a cover by closed subschemes would be nice, but this can't naively exist when the components are nonreduced.