Let $X$ be a Polish space and let $\mathcal K(X)$ be its Vietoris hyperspace. A family $\mathcal I\subseteq\mathcal K(X)$ is hereditary if it is downward closed under inclusion. Matheron and Zelen\'y asked whether every comeager hereditary family in $\mathcal K(X)$ contains a dense hereditary $G_\delta$ subfamily. We give an affirmative answer in ZFC.
Let $G$ be a finite group and $x \in G$. In 2013, Hai-Reuven proved that the cardinality of the centralizer of $x$ divides the cardinality of the solubilizer of $x$. Settling a conjecture of Mousavi, Poozesh and Zamani, we show that the cardinality of the normalizer of $\langle x \rangle$ is also a divisor. We actually...
Let $f:\mathbf P^n_{\mathbb C}\to\mathbf P^n_{\mathbb C}$ be an endomorphism of degree greater than one. We prove that every irreducible Zariski closed subset $V$ with $f^{-1}(V)=V$ is a linear subspace.
Let $\Gamma$ be a property (T) group admitting an injective, non-surjective endomorphism and let $G$ be the associated ascending HNN-extension. Let $W=\left(\bigoplus_{G/\Gamma} \mathbb{Z}/2\mathbb{Z}\right) \rtimes G.$ We show that $W$ is not an MF group and that $C^*(G)$ is not a finite C*-algebra. The ideas and proo...
In this article, we show that flips and divisorial contractions preserve the K\"ahler condition (for strongly $\mathbb{Q}$-factorial compact K\"ahler generalized klt pairs with $B+\beta_X$ big), and we give a criterion for varieties in Fujiki's class $\mathcal C$ to be K\"ahler. We also prove the existence of small $\m...
Christopher D. Hacon, Yi Li, Lingyao Xie· 0 citations
Let $\mathcal F=\{2K_2,C_4,C_5\}$, so that $\operatorname{Forb}(\mathcal F)$ is the class of split graphs. Resolving a conjecture of Wang and Tang, we prove that the class of split graphs is generalized spectrally closed and that it admits no walk-realizable $\mathcal F$-supporter of any finite order.
Let $p:X\rightarrow\Delta$ be a proper holomorphic submersion from a K\"ahler manifold $X$. We prove that, if $K_{X_t}$ is nef for every $t\in\Delta$, then $\dim_{\mathbb C}H^0(X_t,mK_{X_t})$ is independent of $t$ for every integer $m\geq1$. This gives a partial answer to Siu's conjecture on the invariance of plurigene...
F. Deng, Jia-Fu Ning, Zhi-Wei Wang et al.· 2 citations· ⚡1
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