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Dinh and Nguy\^en asked whether the determinant of a Griffiths positive matrix of $(1,1)$-forms belongs to the Hodge--Riemann cone. For every $n\geqslant4$ and $2\leqslant k\leqslant n-2$, we present counterexamples constructed by AI: Nakano positive $k\times k$ matrices of constant $(1,1)$-forms on $\mathbb{C}^n$ whos...
Alp\"oge's construction produces a period family of compact complex threefolds fibred over $\mathbb{P}^1$, with complex two-tori as general fibres. We give a direct proof that every admissible member is Oka, using only the complex-analytic construction. Du--Guo--Kusakabe--Xie independently established the Oka property...
We address Problems~2.2(i), 2.2(ii), and~2.4 in Dinh--Sibony's list. For a saturated holomorphic curve foliation on a compact K\"ahler manifold with finitely many singularities, every positive directed $dd^c$-closed current has finite Poincar\'e mass. Thus linearizability is not needed for the geometric ergodic theorem...
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