Pushforward of $K_X+D$ on the non-snc locus

Question

Let $f:Y\rightarrow X$ be a birational morphism of smooth projective varieties, $F$ an effective divisor on $X$, $D=f^{-1}F_{\mathrm{red}}+\mathrm{Ex}(f)$, $B$ a smooth subvariety of $Y$ contained in the non-snc locus of $D$, is $f_*(\mathcal{O}_B(K_Y+D))$ always nonzero on $X$? If the dimension of $B$ and $f(B)$ are equal the question is trivial, but I have little idea about the general case.

Answer 1

This can be zero. For instance, let $X$ be $\mathbb{P}^3$. Let $q$ be a $k$-point. Let $G$ and $H$ be smooth hypersurfaces in $\mathbb{P}^3$ that contain $q$ and such that, as linear subspaces of the Zariski tangent space $T_q(X)$, $T_q(G)$ equals $T_q(H)$. Call this common subspace $S$. Let the effective divisor $\underline{F}$ be $\underline{G}+\underline{H}$.

Let $f:Y\to X$ be the blowing up of $\mathbb{P}^3$ at $q$. Thus the exceptional divisor $E=\text{Ex}(f)$ is $\mathbb{P} T_q(\mathbb{P}^3)$, which is isomorphic to $\mathbb{P}^2$. The strict transforms $f^{-1}G$ and $f^{-1}H$ are smooth hypersurfaces in $Y$ whose intersections with $E$ are both equal to $\mathbb{P} S$.

Let $B$ be $\mathbb{P} S$, so that $B$ is isomorphic to $\mathbb{P}^1$. Then $D$ is not a simple normal crossings divisor at $B$; $D$ is "triple" along $B$. By direct computation, $\mathcal{O}_Y(K_Y+D)|_E$ is isomorphic to $\mathcal{O}_{\mathbb{P}^2}(-3+1+1) = \mathcal{O}_{\mathbb{P}^2}(-1)$. Thus the restriction to the line $B$ is $\mathcal{O}_{\mathbb{P}^1}(-1)$. Therefore the pushforward $f_*(\mathcal{O}_Y(K_Y+D)|_B)$ is zero.