A note about connectedness theorems à la Barth
In the paper under review, the main result is a Barth-type connectedness theorem (Theorem 17). Let \(P\) be a smooth projective complex variety of dimension \(n+r\) and \(X\subset P\) be a smooth closed subvariety of dimension \(n\). If the effective cone of cycles of codimension \(n\) in \(P\) is closed and \(X\) is strictly nef, then the pushforward map \(H^1(X,\mathbb{Q})\to H^{2r+1}(P,\mathbb{Q})\) is injective if \(n\geq r+1\). The proof uses Hartshorne's approach to Barth's theorem via the hard Lefschetz theorem. Roughly speaking, the assumptions imply that the cycle \([X]\) on \(P\) restricting on \(X\) is homological big, while the later condition is enough for a Lefschetz type result by the work of \textit{L. Fu} [Math. Res. Lett. 19, No. 5, 1097--1115 (2012; Zbl 1284.14015)]. The author also proves a connectedness result when \(P\) is a spherical variety in Section 4 and a conditional connectedness result for cohomology groups in degree \(>1\) in Section 5.
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