Existence of Kähler-Ricci solitons on smoothable \(\mathbb{Q}\)-Fano varieties (Q2232724)

The author studies \(\mathbb Q\)-Fano variety \(M\) which is a normal projective variety with at worst log-terminal singularities and with ample \(\mathbb Q\)-Cartier anticanonical divisor \(K^{-1}_M\). A \(\mathbb Q\)-Fano variety M is called \(\mathbb Q\)-Gorestein smoothable if there is a flat projective family \(\pi :\mathcal M \rightarrow \Delta\) over a disk \(\Delta\) in \(\mathbb C\) such that \(M\cong M_0 :=\pi^{-1}1(0)\), \(M_t := \pi^{-1}(t)\) is smooth for \(t \ne 0\) and \(\mathcal M\) has a relatively ample \(\mathbb Q\)-Cartier anticanonical divisor \(K^{-1}_{M / \Delta}\). The main result the author proves is the following: Theorem. Let \(\pi :\mathcal M \rightarrow \Delta\) be a \(\mathbb Q\)-Gorestein smoothing of a \(\mathbb Q\)-Fano variety \(M_0\) and \(\mathcal V\) be a reductive holomorphic vector field on \(M\), which preserves the fibers. If \((M_0, V_0)\) is K-stable, where \(V_0=\mathcal V_{|M_0}\) then \(M_0\) admits a Kähler-Ricci soliton.