11 problems

Kotschick's conjecture. The following conditions are equivalent:

Let $\overline{\mathcal{M}}_{3,15}$ and $\overline{\mathcal{M}}_{3,16}$ denote the moduli spaces of stable genus-$3$ curves with, respectively, $15$ and $16$ marked points. Vanishi…

Let $X$ be a smooth minimal model, and let $g$ be the maximal number of pointwise linearly independent (PLI) holomorphic one-forms that $X$ carries. Smooth fibration conjecture. Th…

Smoothness criterion. The morphism $f$ is smooth if and only if there is a holomorphic $1$-form $\omega\in H^0(A,\Omega_A^1)$ such that $f^*\omega$ has no zero.

The p-completeness conjecture. For sufficiently large $g$, the variety $\operatorname{\cM_g^{o}}$ is $p$-complete, or else

Let $X$ be a smooth projective variety. A degree $l$ polynomial decomposition of the diagonal means a decomposition of the diagonal whose non-supported part lies in the subalgebra…

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…

Let $X$ be a simply-connected smooth projective 3-fold, let $\gamma\in H^{p,0}(X)$ with $p=2$ or $p=3$, and let $D\in\mathbb D_{\textup{PT}}^{X}$. For $k\geq 0$, the stable-pairs d…

Let $X$ be a smooth projective variety admitting a holomorphic one-form without zeros. Birational abelian-fibration conjecture. Then $X$ is birational to a smooth projective variet…

Luo–Zhang conjecture. The dimension of $W$ can be at most

Hacon–Kovács–Luo–Zhang conjecture. The zero locus $Z(\omega)$ is nonempty for every global holomorphic one-form $\omega\in H^0(X,\Omega_X^1)$.