22 problems

Kobayashi's conjecture. A general projective hypersurface $X$ of degree $d$ in $\mathbb{P}^n$ with $d\geq 2n-1$ is hyperbolic.

Hyperbolicity conjecture for general hypersurfaces. A general hypersurface

Hyperbolicity conjecture for general projective hypersurfaces. If the degree of $X$ is sufficiently high, then both $X$ and its complement $\mathbb{P}^{n}\backslash X$ are hyperbol…

Degree-three mirror-transformation conjecture. The mirror transformation for $d=3$ is

Let $P_m$ denote the set of partitions of $m$, let $l(\sigma_m)$ be the length of a partition, and let $C_m^d(\sigma_m)$ be the coefficients associated with partitions in the propo…

The paper considers quantum cohomology of projective hypersurfaces, with virtual structure constants obtained from recursive formulas for Fano hypersurfaces and a generalized mirro…

Hypergeometric-coefficient conjecture. This recursive procedure should recover the coefficients of the hypergeometric series appearing in the mirror calculation, without using the…

Let $A$ be a Gorenstein local algebra of dimension $n+1$ with maximal ideal $\mathfrak{m}$, and let $U$ be a hyperplane in $\mathfrak{m}$ complementary to $\operatorname{Soc} A$. B…

Let $A$ be a Gorenstein local algebra of dimension $n+1$ with maximal ideal $\mathfrak{m}$, and let $U\subseteq\mathfrak{m}$ be a hyperplane complementary to $\operatorname{Soc} A$…

Let $N$ be the ambient projective dimension parameter, let ${\cal O}_{h^a}$ and ${\cal O}_{h^b}$ be the cohomology insertions, and let $t^0,\ldots,t^{N-2}$ be the perturbed Gromov–…

Let $X\subseteq\mathbb{P}^{n+1}$ be a degenerate hypersurface admitting an induced additive action, where an induced additive action is an additive action obtained from the ambient…

Let $k>2$, let $V\subset\mathbb{P}^{n}$ be a projective hypersurface with only isolated singular points, and let $d=\deg V$. Huh's finiteness conjecture. For sufficiently large $n$…

Let $V\subset \mathbb{P}^{n}$ be a projective hypersurface with only isolated singularities, defined by a homogeneous polynomial $f$ of degree $d$, and let the polar degree of $V$…

Let $f$ define a singular projective hypersurface in $\mathbb{P}^n$, and let $E_*(f)$ be the spectral sequence associated with $f$, with second page $E_2(f)$ and limit…

The hypothesis conjecture. Whenever $\alpha_Z>\frac{n}{d}$, condition (W) should imply that the hypothesis of Corollary 3 is satisfied, with the additional restriction $n=3$ if nec…

Higher-dimensional asymptotic conjecture. For each $k$ and every odd $d$ relatively prime to $k$, $\Cal N_d^e$ is nonzero; moreover, for fixed $k$ and such $d$ tending to infinity,

Asymptotic conjecture for real 3-plane counts.

Carlson–Griffiths conjecture. If $V(f)$ does not possess singularities which are large in relation to the degree, then $f$ is determined by $E(f)$, up to a nonzero scalar multiple.

Let $k$ be a positive integer. A projective hypersurface is defined by a homogeneous polynomial in a projective space, and it has only isolated singular points when its singular lo…

Let $V(h)\subseteq\mathbb{P}^n$ be a projective hypersurface defined by a homogeneous polynomial $h$ of degree $d$, with only isolated singular points, and let its polar degree be…

Let $M_k^k$ be the degree-$k$ projective hypersurface, let $\tau_k(x)$ and $F_o^k(x)$ be the functions introduced above, and define the mirror map … Write $x(t)$ for its inverse an…

Let $f\in \mathbb{C}[x_0,\ldots,x_n]$ be a reduced homogeneous polynomial, and let $V(f)$ be its associated projective hypersurface. Write $d=\operatorname{deg}(f)$ and let $d(f)$…