116 problems

Let $X_{cSigma}$ be a smooth toric variety satisfying conditions $({\rm homogenous}.1)$ and $({\rm homogenous}.2)$, and let $D=(d_1,\ldots,d_r)$ be a multidegree. Define … where…

Let $X$ be an algebraic variety over $\mathbb{C}$, and let its tangent sheaf be locally free. Zariski–Lipman conjecture. Then $X$ is smooth. The conjecture has been proved under va…

Let $X$ be a smooth proper toric variety. For each positive integer $\ell$, the $\ell$th toric Frobenius morphism $F_{\ell}:X\to X$ is defined by extending…

Let $R$ be a commutative regular ring, let $M$ be a commutative, cancellative, torsion-free monoid without nontrivial units, and let $i$ be a nonnegative integer. For each natural…

For a normal projective toric variety $X$, consider the local Euler obstruction at a point, denoted by $\operatorname{Eu}_X$. Matsui–Takeuchi's conjecture. If the calculations in d…

Battaglia–Zaffran's conjectures. The basic cohomology ring $H^{*}_{\mathcal{F}_{\mathfrak h}}(\mathcal{Z}_{\mathcal{K}})$ should have a description similar to the cohomology ring o…

McKinnon's conjecture. There exists a rational curve $C$ on $X$ passing through $Q$ achieving the best approximation constant at $Q$ with respect to $L$.

Modularity conjecture. We conjecture that $\widehat{Ell}({\bf P},q)$ is a modular form. The preceding theorem establishes the stronger specific claim that, for a Gorenstein toric v…

Multiplicity-poset conjecture. The multiplicity $\operatorname{mult}_{P_\tau}X(\mathcal{L})$ is determined by the poset $H(\tau)$: whenever $\tau$ and $\tau'$ have isomorphic assoc…

Multiplicative-formula conjecture. There exists a multiplicative formula in $X_{d,n}$ similar to the known formula for a face that is a single suitable block, for a face that is a…

Bihan et al.'s maximality conjecture. Every toric variety is an M-variety. The paper presents a six-dimensional projective toric counterexample whose total mod $2$ Betti number ove…

Let $X_k=\mathcal O(k)\oplus\mathcal O(-2-k)\rightarrow\mathbb P^1$ be equipped with a diagonal $T^2$ action on the bundle with weights $(\lambda,\lambda)$, and let $I_k^T(q)$ deno…

Let $\mathcal{C}(E,D,X)$ be the toric code associated to the $1$-cycle $E$, the $T$-invariant Cartier divisor $D$, and the toric variety $X$. Assume that $X$ is a nonsingular toric…

Let $I$ be a toric ideal defining a projectively normal $r$-dimensional toric variety. The normality Gröbner-basis conjecture. The ideal $I$ has a Gröbner basis consisting of binom…

The $G^1$-degeneration conjecture. The spectral sequence $G^k_{p,q}$ for $H_*(X(\mathbb{R}))$ degenerates at the $G^1$ level.

Totaro's degeneration conjecture. The spectral sequence degenerates at the $E^2$ level for every toric variety, as it does with rational coefficients.

A real algebraic variety is a complex variety $X$ defined over the real numbers, with real part $X(\mathbb{R})$ equal to the fixed-point set of complex conjugation. It is an M-vari…

Torsion conjecture. The theorems above are true without assumptions on $q$.

Let $X$ be a target with equivariant quantum cohomology, and suppose its non-equivariant counterpart is semisimple. An asymptotic solution is a formal solution of the quantum diffe…

Let $\Delta$ be a full triangulation of a configuration $\mathcal{A}$ of maximal rank $d$, and let $C\subseteq\mathcal{A}$ be called $r$-partitionable if it is the disjoint union o…

Let $X$ be a toric variety and let $\mathcal{E}$ be a reflexive equivariant sheaf over $X$, described by filtrations $E^\rho(i)$ and an underlying vector space arrangement. Conject…

Let $n$ be a positive integer. Define the commutative ring $\mathbf{M}^{\text{def}}(n)$ generated by variables $\mathsf{S}_i$ for $i\in[n]$ and $\mathsf{S}_\alpha$ for…

Manin's conjecture. As $t\to\infty$, the dominant term of $N_\infty(U(\bold A),t)$ should be asymptotically equivalent to

Canonical product conjecture. This canonical product always induces the correct product on $H^*(X)$, making the above identification an isomorphism of algebras.

Mixed realization conjecture. The functor $\mathop{\rm real}\nolimits_\Delta$ should factor through a mixed realization functor sending ${\mathbf D}(\Delta)$ to mixed Hodge modules…