If $X$ is a smooth $G$-variety with trivial canonical bundle, then does $X^G$ also have trivial canonical bundle?

Question

Let $G$ be a reductive group and $X$ a smooth $G$-variety. Then the fixed point subvariety $X^G$ is also smooth (this is theorem 13.1 of Milne's book on algebraic groups). Suppose in addition that the canonical bundle $\omega_X$ is trivial. Then is $\omega_{X^G}$ also trivial?

One idea I had was to use the adjunction formula, which gives that $\omega_{X^G} = \det(I/I^2)^\vee$, where $I$ is the ideal sheaf of $X^G\subset X$. When $X = \text{Spec}(A)$ is affine, we have $I=(g\cdot f - f)_{g\in G, f\in A}$, but in the general case I'm having trouble giving a sufficiently effective description of $I/I^2$ to calculate its determinant.

Answer 1

No. Let $C \subset \mathbb{P}^2$ be a smooth sextic curve, let $X$ be the double covering of $\mathbb{P}^2$ ramified over $C$, and let $G = \mathbb{Z}/2$ acting on $X$ be the involution of the double covering. Then $X$ is a K3 surface, hence its canonical bundle is trivial. But $X^G = C$ is a curve of genus $10$, its canonical class is not trivial.

Answer 2

We can generalize Sasha's counterexample in all dimensions as follows.

Let $f \colon X \to \mathbb{P}^n$ be a double cover branched over a smooth hypersurface $Y=Y_{2n+2}$ of degree $2n+2$. Then $$\omega_X = f^*(\omega_{\mathbb{P^n}} \otimes \mathcal{O}_{\mathbb{P}^n}(n+1)) = \mathcal{O}_X$$ is trivial. On the other hand, the double cover involution gives an action of $G=\mathbb{Z}/2$ on $X$ such that $X^G$ is isomorphic to $Y$ and, by adjunction, we find that $$\omega_Y = \mathcal{O}_Y(n+1)$$ is non-trivial.

Answer 3

For a non-projective example, you could take your favorite smooth variety $Y$ with nontrivial canonical bundle, and let $X$ be the total space of the cotangent bundle of $Y$. Then I believe $X$ will have trivial canonical bundle. You can let $\mathbb{G}_m$ act by scaling on the fibers of $X$. The $\mathbb{G}_m$-fixed points will be the original variety $Y$ (sitting as the zero section inside the cotangent bundle $X$).