5 problems

Let $X$ be a compact Kähler or complex projective manifold that is aspherical or has a nef cotangent bundle, and let $\mathcal{M}$ be a mixed Hodge module on $X$. For an integer…

Beauville's conjecture. If

Catanese's conjecture. The dimension of the paracanonical system satisfies

Schreieder–Kotschick conjecture. The following three statements are equivalent: (1) $X$ admits a global holomorphic 1-form without zeros; (2) $X$ admits a smooth real closed 1-form…

Debarre–Pareschi–Popa conjecture. The following are equivalent: (1) $X$ is reduced, of pure dimension, and has minimal cohomology class