18 problems

Kuznetsov's conjecture. For any $k$ and $n$, the Pfaffian variety

Generalized Pfaffian HPD conjecture. For every $0\leq t\leq \lfloor n/2\rfloor$, there exist noncommutative resolutions of singularities of $X$ and $Y$ which are Homologically Proj…

Higher-dimensional Pfaffian HPD conjecture. There exists a noncommutative resolution of singularities $(Y,\CR)$ of $Y$ which is Homologically Projectively Dual to $X$.

Let $\mathbf{G}$ be the Grassmannian in its Plücker embedding, let $\mathbf{Pf}$ be its projective-dual Pfaffian variety, and write $\mathbf{P}^*$ for the dual projective space. A…

Let $B$ be a generalized del Pezzo surface with $b_2(X)=b_2(B)+1$, and let … be a smooth genus-one fibered Calabi–Yau threefold with a 5-section. Assume that $X$ is the degeneracy…

Categorical crepant resolution conjecture. The categorical crepant resolution $\widetilde{D}^b({\mathrm {Pf}}(2k,V^{\vee}))$ has the non-rectangular Lefschetz decomposition

Let $V$ be a vector space of dimension $2n$, let $X_1$ be the source's variety, and let $\pi\colon X_1\to\mathbb{P}(V)$ be the projection. Let…

Let $V$ be a vector space of dimension $2n$, let $X_1$ be the source's variety, and let $p\colon X_1\to\mathbb{P}(\Lambda^2V^\vee)$ be its projection. Let…

Let $V$ be a vector space of dimension $2n$ over an algebraically closed field of characteristic zero. Let $X_1$ be the variety in the source, with projection…

Let $V$ be a vector space of dimension $N$ over an algebraically closed field $k$ of characteristic zero, and consider the Plücker embedding…

Let the large-volume limit at $\tau=1/2$ and the point at $\tau=i\infty$ be points in the relevant moduli space. Homological-projective-duality conjecture. The large-volume limit a…

Homological projective duality conjecture. The homological projective dual variety of the Cayley plane is a non-commutative resolution of the Cartan cubic hypersurface in…

Kuznetsov's duality conjecture. Being generalized partners or generalized duals should imply an equivalence of GM categories:

Let $V$ be a vector space, let $n=\dim V$, and for an integer $0\le r\le n$ define … The varieties $X_r(V)$ parameterize symmetric $2$-tensors of rank at most $r$. Hori–Knapp–Kuzne…

Homological projective duality (HPD) is a framework relating derived categories of projective varieties and their dual data; an abelian gauged linear sigma model (GLSM) is a physic…

Let $W$ be a vector space of dimension $m$, and let $\operatorname{Pf}(2k,W)$ denote the closure of the locus of bivectors in $\Lambda^2W$ of rank $2k$. Let…

Homologically projectively dual varieties are varieties related by homological projective duality. A gauged linear sigma model is a gauged linear sigma model in the sense of the ci…

Let $W$ be a vector space of dimension $2n$, let $V\subset\Lambda^2W^*$ satisfy $\dim V=2n=\dim W$, and let $V^\perp\subset\Lambda^2W$ be its orthogonal. Define the Pfaffian hypers…