# About the proof that deformation of canonical singularity is still canonical singularity

I was reading Professor Kawamata's paper Deformation of Canonical Singularity (which also appears in the book Algebraic Varieties: Minimal Models and Finite Generation Corollary 2.12.11) the main theorem is stated as follows:

Corollary 2.12.11. Let $$f: \mathcal{X} \rightarrow B$$ be a flat morphism from a germ of an algebraic variety to a germ of a smooth curve. Assume that the central fiber $$X_0=f^{-1}(P)$$ has only canonical singularities. Then so has the total space $$\mathcal{X}$$ as well as any fiber $$X_t$$ of $$f$$. Moreover, if $$\mu: V \rightarrow \mathcal{X}$$ is a birational morphism from a normal variety with the strict transform $$X$$ of $$X_0$$, then $$K_V+X \geq \mu^*\left(K_{\mathcal{X}}+X_0\right)$$.

The key technical lemma that will be used is the following extension theorem

Theorem A. Let $$V$$ be a smooth variety with $$X$$ a smooth divisor on $$V$$, with $$\mu:V\to \mathcal{X}$$. Assume that $$K_V+X$$ is $$\mu$$-big for the pair $$(V, X)$$. Then the natural homomorphism $$\mu_* \mathcal{O}_V\left(m\left(K_V+X\right)\right) \rightarrow \mu_* \mathcal{O}_X\left(m K_X\right)$$ is surjective for any positive integer $$m$$.

The proof of the Corollary 2.12.11 goes as follows

(1) since we require the smoothness assumption in the extension theorem we first take some resolution $$\mu:V\to \mathcal{X}$$ with the strict transform of the central fiber $$X_0$$ being $$X$$, and assume $$X_0$$ is also smooth.

(2) Since both $$\mu$$ and $$\mu|_X$$ is birational, thus $$K_V+X$$ is $$\mu$$-big. Thus we can apply the extension theorem (Theorem A)

(3) Assume $$m$$ be the integer that $$mK_{X_0}$$ is Cartier (always exist some $$m$$ by canonical singularity assumption)

and we get the surjection $$H^0(V,\mathcal{O}_V(m(K_V+X))\to H^0(\mathcal{X},m(K_{\mathcal{X}}+ X_0))\to H^0(X,mK_X)= H^0(X_0,mK_{X_0})\tag{*}$$ (where the last equality is due to $$X_0$$ has canonical singularity)

(4) Therefore the nowhere vanishing section of $$mK_{X_0}$$ lift to nowhere vanishing section on $$m(K_{\mathcal{X}}+X_0)$$ and lift to section on $$m(K_V+X)$$

(5) Therefore $$K_{\mathcal{X}}+X_0$$ is $$\mathbb{Q}$$-Cartier and $$K_V+X\ge \mu^*(K_{\mathcal{X}}+X_0)$$.

The question is why the surjectivity of (*) implies $$K_{\mathcal{X}}+X_0$$ is $$\mathbb{Q}$$-Cartier and $$K_V+X\ge \mu^*(K_{\mathcal{X}}+X_0)$$?

$$K_{X_0}$$ is $$\mathbb Q$$-Cartier, so $$mK_{X_0}$$ is Cartier for some $$m>0$$ and in particular it is (locally) generated by 1 section $$s$$. As discussed above $$s$$ lifts to a section $$S$$ of $$m(K_{\mathcal X}+X_0)$$. Now $$m(K_{\mathcal X}+X_0)$$ is a reflexive sheaf which (by Nakayama's Lemma) is (locally near $$X_0$$) generated by the section $$S$$. Thus $$m(K_{\mathcal X}+X_0)$$ is Cartier (near $$X_0$$). Since $$S$$ lifts to $$\mu _*(m(K_V+X))$$, we have $$\mu _*(m(K_X+V))=m(K_{\mathcal X}+X_0)$$. Write $$\mu ^*(m(K_{\mathcal X}+X_0))+E=m(K_V+X)$$, then $$E\geq 0$$ as otherwise $$\mu _*(m(K_X+V))$$ is strictly contained in $$m(K_{\mathcal X}+X_0)$$.