Canonical bundle formula for a point blowup ($K_{\tilde{M}}\cong {\varphi }^{\ast }{K}_M\otimes [E{]}^{\otimes (n-1)}$)

For the blowup of a complex manifold at a point in complex dimension $n$, the differential gives a section of $K_{\tilde{M}}\otimes ({\varphi }^{\ast }{K}_M{)}^{-1}$ whose divisor on a complex manifold is $(n-1)E$. Indeed, a local chart has $x_i=u$ and $x_j=u{t}_j$ for $j\ne i$, so its Jacobian determinant is $\pm u^{n-1}$. The holomorphic line bundle associated to a divisor then gives the formula for the canonical bundle.

Let $Y$ be a smooth complex analytic hypersurface through the blowup centre with $[-Y]\cong K_X$. Its multiplicity at that point is one, so ${\sigma }^{\ast }Y=\tilde{Y}+E$. Combining this with the canonical bundle formula for a point blowup gives $[-\tilde{Y}]\cong K_{\tilde{X}}\otimes [E{]}^{\otimes (2-n)}$. The nontrivial powers of the exceptional line bundle of a point blowup prove the equivalence for $n\ge 2$. On a compact complex curve the premise cannot hold for an irreducible hypersurface: it is one point, so its negative divisor has degree $-1$, whereas the canonical bundle has even degree $2g-2$.