236 problems

Let $X_4\subset\mathbb{P}^3_{\mathbb{C}}$ be the Fermat quartic surface $X_4=\{[x_0:x_1:x_2:x_3]\mid x_0^4+x_1^4+x_2^4+x_3^4=0\}$. Find an explicit finite collection of automorphis…

Let $X$ be a K3 quartic surface defined over a field of characteristic $3$. The 58-line bound conjecture. The surface $X$ contains $112$ lines or at most $58$ lines. The theorem pr…

Let $(F,g)$ be a polarized irreducible holomorphic symplectic fourfold deformation equivalent to the Hilbert scheme of length-two subschemes of a K3 surface. Let…

Let $z(q)$ be the mirror map for a one-parameter deformation of an algebraic K3 surface arising from an orbifold construction whose Picard–Fuchs equation is third order. A Lian–Yau…

Let $S$ be a surface given in ${\bf{TK1}}$, ${\bf{TK2}}$, or ${\bf{TK3}}$. Let $F$ and $C$ denote the divisor classes used in these trigonal K3 surface constructions, and let $l$ b…

Let $S\subset {\mathbb P}^n$ be a primitive K3 surface, and let $d>0$. Consider the family of rational curves in the linear system $|{\mathcal O}_S(d)|$. Transitive monodromy conje…

Joyce's conjecture. For every $\sigma\in\operatorname{Stab}^{\ast}(X)$ and $alphain N(X)$, there is an invariant $J^{\alpha}(\sigma)inLambda$ such that $J^{\alpha}(\sigma)$ does not de…

Let $X$ be a K3 surface. Let … be the homomorphism induced by the action of Fourier--Mukai autoequivalences on the Mukai lattice, and let…

Artin–Shioda conjecture. Every supersingular $K3$ surface is unirational.

Let $(X,\alpha)$ be a projective twisted K3 surface, let ${\rm D}^{\rm b}(X,\alpha)$ be its bounded derived category of twisted coherent sheaves, let ${\mathcal P}_0(X,\alpha)$ be…

Let $M$ be a fixed differentiable manifold underlying a K3 surface, and let … and $'$. Write $[]$ and $[']$ for their periods in ${\mathbb P}(H^2(M,{\mathbb C}))$, and let…

Shioda's conjecture. If $\mathfrak{p}$ is a supersingular prime, then the two lattices $T_Z$ and $T_{\mathfrak{p}}$ are similar.

Polynomial-cycle injectivity conjecture. If the cohomology class of $P$ vanishes, then $P$ vanishes in the Chow group:

Let $X$ be a numerical $(K3)^{[2]}$ and suppose that Items (1) through (6) of Proposition (hodgeprop) hold. O'Grady's conditional theorem. Item (1) of Theorem (mainthm1) holds. The…

Let $X$ be a K3 surface with $H^2=n\in2{\mathbb{Z}}$, and let ${\mathcal{L}}={\mathcal{L}}^n(d,m_1,\dots,m_h)$ be the system of curves in $|dH|$ through general points…

Segre's conjecture. If $\mathcal L$ is non-empty and reduced, then it is non-special.

Let $X$ be a tamed $K3$-fibered Calabi–Yau threefold, and let $n_C$ denote the virtual number of rational curves in the fiberwise class $C\in H_2(X,\mathbf Z)_f$. Suppose that the…

Let $(X^{an},\Omega)$ be the $K$-analytic K3 surface with volume form constructed from a chosen set ${\cal L}$ of lines. Independence conjecture. The isomorphism class of the pair…

Let $X^{an}/K$ be an analytic K3 surface with the lattice $\Lambda_X$, period homomorphism $\rho_X$, valuation vector $v_X:=val_K\circ\rho_X$, and affine piecewise-linear action on…

Let $K={{\bf C}}((t))$ and let $X^{an}/K$ be an analytic K3 surface admitting an analytic torus fibration $X^{an}\to S^2$ with standard singularities. Intrinsic period-lattice conj…

Let ${\cal F}_v$ be a leaf indexed by a positive vector $v$ in the K3 period description, let $\Gamma_v$ be the corresponding stabilizer group, and let $Aut_{{\bf Z}PL,v}(S^2)$ be…

Let $X^{2n}$ be an irreducible holomorphic symplectic manifold equipped with an abelian fibration whose fibres are isomorphic to Jacobians of curves. A multisection is a subvariety…

Let $X$ be an irreducible symplectic manifold deformation equivalent to $(K3)^{[n]}$, let $(\,\ )$ be its Beauville form, and let $h\in H^{1,1}_{\mathbb Z}(X)$ satisfy $(h,h)=2$. F…

Let $n\geq 2$, and let $(X,H)$ be a degree-$2$ polarized irreducible symplectic variety, meaning that $X$ is an irreducible symplectic manifold deformation equivalent to…

Let $S$ and $\mathcal L$ be as above: $S$ is a generic $K3$ surface with ${\rm Pic}(S)=\mathbb{Z}H$, $H^2=2g-2\geq 2$, and $\mathcal L=\mathcal L^n(d,m_1,\ldots,m_r)$ is the system…