A note on the PDE approach to the \(L^\infty\) estimates for complex Hessian equations on transverse Kähler manifolds (Q6647822)

It is known that obtaining the \(L^\infty\) estimate is the hardest part in solving the complex Monge-Ampère equation. In the case of Yau's celebrated work, it was given by using the Moser iteration argument, which works if the right-hand side is \(L^p\) with \(p>n\), where \(n\) is the complex dimension of the manifold. For a long time, a purely PDE approach to the sharp \(L^\infty\) estimates was lacking. By \textit{B. Guo} et al. [Ann. Math. (2) 198, No. 1, 393--418 (2023; Zbl 1525.35121)] and by \textit{B. Guo} et al. [Anal. PDE 17, No. 2, 749--756 (2024; Zbl 1544.32068)], the proof using PDE techniques was established. The achievement of this paper is to extend these results in [\textit{B. Guo} et al., Ann. Math. (2) 198, No. 1, 393--418 (2023; Zbl 1525.35121); Anal. PDE 17, No. 2, 749--756 (2024; Zbl 1544.32068)] to degenerate transverse complex Monge-Ampère equations on homologically orientable transverse Kähler manifolds and as an application, the author gives a purely PDE-based proof of the regularity of conical Ricci flat metrics on an affine \(\mathbb{Q}\)-Gorenstein \(\mathbb{T}\)-varieties.\N\NTo obtain a \(L^\infty\) estimate, the author follows the approach in [\textit{B. Guo} et al., Ann. Math. (2) 198, No. 1, 393--418 (2023; Zbl 1525.35121); Anal. PDE 17, No. 2, 749--756 (2024; Zbl 1544.32068)], and the author uses a specific auxiliary transverse Monge-Ampère equation to prove the necessary estimate. The method of the proof is almost the same as the one in [\textit{B. Guo} et al., Ann. Math. (2) 198, No. 1, 393--418 (2023; Zbl 1525.35121); Anal. PDE 17, No. 2, 749--756 (2024; Zbl 1544.32068)], but some modifications are necessary for the transverse Kähler manifold setting. By following the similar arguments as in the main result (Theorem 1.1), which is slightly modified for the general complex Hessian equations, the author obtains the sharp \(L^\infty\) estimate for the transverse complex Hessian equations. The author also obtains the higher order estimate by applying the \(L^\infty\) estimate. And then, the author shows that any weak Ricci flat cone metric is smooth on \(Y_{reg}\), where \(Y\) is a complex affine normal variety that is supposed to be \(\mathbb{Q}\)-Gorenstein. This smoothness result is rather standard, but the author obtains the \(L^\infty\) estimate for the transverse nef class, whose proof only requires an estimate for the nonnegative transverse metric.