10 problems

Let the base field be an algebraically closed field of characteristic zero. Let $f:X\to Y$ be a morphism from a smooth projective variety $X$ to a smooth projective variety $Y$ of…

Semipositivity conjecture for relative pluricanonical direct images. The sheaf $f'_*\omega^{\otimes m}_{X'/Y'}$ is nef and locally free for every positive integer $m$.

Let $f:Y\to X$ be a surjective morphism from a smooth projective variety to a smooth projective variety of dimension $n$, and suppose that $f$ is smooth over $X_0=X\setminus B$ for…

Let $f:X\to Y$ be a smooth fibration of smooth projective varieties, and let $L$ be a line bundle on $X$. The direct image in question is $f_{\ast}(K_{X/Y}\otimes L)$, where…

Direct-image conjecture. The direct image

Let $f\colon X \to Y$ be a fibration between smooth projective varieties with general fiber $F$ and $\kappa(Y)\ge0$. For a sufficiently divisible positive integer $m$ such that…

Let $f:X\to B$ be a projective morphism of smooth projective varieties $X$ and $B$ with connected fibers, and let $D$ be the discriminant locus of $f$. Consider the determinant…

Let $f\colon X\to Y$ be a projective surjective morphism between complex manifolds. Let $(L,h)$ be a holomorphic line bundle over $X$ equipped with a singular Hermitian metric of s…

Kawamata's conjecture. The locally free sheaf

Local freeness conjecture for weakly semistable morphisms. The sheaf