The G-noncommutative minimal model program conjecture

Let $G$ be an algebraic group, let $X$ be a smooth projective variety with a $G$-action, and let $\pi:X\to Y$ be a $G$-contraction. Let $\mathrm{Stab}(X_G)$ denote the space of stability conditions and let $\{\sigma_t^{\pi,\psi}\}$ be a family of paths in $\mathrm{Stab}(X_G)/\mathbb C$. For a $G$-equivariant morphism $\pi:X\to X'$ satisfying $R\pi_{\ast}(\mathcal O_X)=\mathcal O_{X'}$, use the induced functor $\pi^{\ast}:D^b_G(X')\to D^b_G(X)$. The $G$-noncommutative minimal model program conjecture. (A) One can associate to $\pi$ a canonical family of quasi-convergent paths $\{\sigma_t^{\pi,\psi}\}$ in $\mathrm{Stab}(X_G)/\mathbb C$. Generic values of $\psi$ yield a $G$-linear semiorthogonal decomposition of $D^b_G(X)$, and different generic values give mutation-equivalent decompositions. (B) For a generic $\psi$, the semiorthogonal factors of $D^b_G(X)$ are closed under tensor product with complexes of the form $\pi^{\ast}(E)$ for $E\in\mathrm{Perf}_G(Y)$. (C) Given a further $G$-contraction $Y\to Y'$, the semiorthogonal decomposition of $D^b_G(X)$ associated with $X\to Y'$ refines the decomposition associated with $X\to Y$. (D) If $\pi:X\to X'$ is a $G$-equivariant morphism of smooth varieties with $R\pi_{\ast}(\mathcal O_X)=\mathcal O_{X'}$, then for suitable parameters, the semiorthogonal decomposition of $D^b_G(X)$ associated with $X\to Y$ refines the decomposition obtained by combining the semiorthogonal decomposition of

$$\pi^{\ast}(D^b_G(X'))\cong D^b_G(X')$$

associated to $X'\to Y$ with

$$D^b_G(X)=\langle\operatorname{ker}(\pi_{\ast}),\pi^{\ast}(D^b_G(X'))\rangle.$$

The conjecture proposes a categorical version of the minimal model program in which $G$-contractions determine canonical, compatible semiorthogonal decompositions of equivariant derived categories. The provided context establishes the relevant equivariant stability-condition and derived-category framework, but gives no evidence that any of (A)--(D) is known or resolved.