7 problems

Rational Lagrangian fibration conjecture. One has

A Type B $4$-fold is an irreducible symplectic $4$-fold numerically equivalent to $(K3)^{[2]}$ that is birational to a hypersurface of degree at most $12$. O'Grady's nonexistence c…

Lagrangian-fibration base conjecture. The base $B$ is isomorphic to a quotient of

Let $M$ be a nonsingular irreducible symplectic variety of dimension $2n$, and let $\pi:M\to B$ be a Lagrangian fibration. The base $B$ is a projective variety associated with the…

Voisin's injectivity conjecture. The restriction

Let $Y$ be a bielliptic surface, let $f\colon X\to Y$ be its canonical covering of degree $n=2$, and let $w$ be a sheaf class on $Y$. Write $K_{X;f^*H,f^*A}(f^*w)$ and…

Let $X$ be an irreducible symplectic fourfold and let $H$ be a polarization on $X$ satisfying the properties listed above; write $h=c_1(H)$. Since $\dim |H|=5$, the complete linear…