Here are the exact definitions of the terms: Let $G$ be a topological group. Then $G$ has the small index property if every subgroup of countable (including finite) index is open in $G$. Furthermore, $G$ has automatic continuity if every group homomorphism from $G$ to any separable topological ...

A topological group is said to have automatic continuity if every homomorphism from it to a second countable topological group is continuous. Various topological groups are known to have this property including $\operatorname{Sym}(\mathbb{N}), \operatorname{Aut}(\mathbb{Q}, <)$ and $\operatornam...

I would like to know if someone has an explicit example for the rank of the Neron-Severi group of a normal crossing scheme (proper over a field) being different from the rank of the kernel of $\operatorname{NS}(X^{(0)})\to \operatorname{NS}(X^{(1)})$, the restriction map from the Neron-Severi gro...

$\DeclareMathOperator\NS{NS}\DeclareMathOperator\Pic{Pic}$I have two questions on a comment from Daniel Hyubrechts's Complex Geometry on pages 133/134. Let $X$ be a compact Kähler manifold. Consider the exponential sequence on cohomology $$ \dotsb \to H^1(X,\mathcal{O}_X^*)=\Pic(X) \to H^2(X, ...

Let $X$ be a smooth projective unirational variety over an algebraically closed field of characteristic $p>0$, and $\ell\neq p$ a prime. My question: can the Neron-Severi group of $X$ contain (non-zero) $\ell$-torsion? This appears to be closely related to the presense of torsion in the etale coh...