20 problems

Let $\mathcal{X}$ be an orbifold with coarse moduli space $X$, and let $\phi\colon Z\to X$ be a crepant resolution. Iritani's conjecture. There exists an isomorphism … induced by a…

Let $G=\operatorname{GL}_r$. Let $\mathcal{B}\subset(\mathrm{B}_{\operatorname{GL}_r}^{\mathrm{red}})^{\mathrm{cl}}$ be an open subset, and set … Let…

Period-preserving involution conjecture. The space $F_{1,1}$ admits a period-preserving birational involution $S\leftrightarrow S'$ such that

A Fourier–Mukai partner of a smooth projective variety is a smooth projective variety with an equivalent derived category. Finiteness conjecture for Fourier–Mukai partners. Every s…

The symplectic biextension conjecture. The derived categories of coherent sheaves on $A$ and $A'$ are equivalent if and only if there is such an isomorphism $f$.

Let $C:\operatorname{Spec} R\to \mathcal{M}_{cub}$ be a cubic curve, and let $\widehat{C}$ denote its formal completion, a formal group over $R$. Write…

Let $C$ be a smooth projective curve of genus $g$, let $L$ be a line bundle on $C$ with $l=\deg L>2g-2$, and let $B^L$ be the Hitchin base. For rank $r$, numerical invariant $hi$,…

Let $C$ be a nonsingular curve and $G$ a reductive group. Let ${\mathcal M}_{\mathrm{Higgs},G}$ and ${\mathcal M}_{\mathrm{Higgs},{}^LG}$ be the Higgs moduli spaces for $G$ and its…

Let $\pi_M:(M,\sigma)\to B$ be an integrable system, with partial Fourier–Mukai transform $\phi_{\mathrm{FM}}$, sheaf of Kähler differentials $\Omega_M^k$, and decomposition-theore…

Let a flop be presented by a wall-crossing with a derived $Q$-construction producing a kernel for the associated window functor. A Fourier–Mukai equivalence is an equivalence induc…

Let $A$ be an abelian variety of dimension $g$, let $K=K_0(A)\otimes\mathbf Q$, and write $K^{(p)}$ and $\widehat K^{(r)}$ for the eigenspace components associated with the relevan…

Let $X$ be the underlying curve, let $\operatorname{M}_n$ be the moduli space of rank-$n$ Higgs bundles, and let $\mathcal{L}\to\operatorname{Jac}^0(X)$ be a topologically trivial…

Let $X$ and $X'$ be smooth cubic fourfolds, with K3 categories … An equivalence between these categories is called a Fourier–Mukai equivalence if the induced composition through…

Let $\Gamma$ be a finite subgroup of $SL_n(\mathbb{C})$, let $Y=\mathbb{C}^n/\Gamma$, and let … be two crepant resolutions. Two varieties are Fourier–Mukai partners if their bounde…

Let $\mathscr{L}^{\mathcal{X}}\subset\mathscr{V}^{\mathcal{X}}$ and $\mathscr{L}^{\mathcal{Y}}\subset\mathscr{V}^{\mathcal{Y}}$ be the equivariant genus-zero Gromov–Witten Lagrangi…

Let $\mathcal A$ be the Hitchin base, let $H:M\to\mathcal A$ be the Hitchin fibration, and let $\overline P$ be the universal Poincaré sheaf from the global conjecture. For each…

Let $C$ be the underlying smooth projective curve, let $r$ and $d$ be the rank and degree, let $M=\mathcal M(d,r,\omega_C)$ be the moduli space of $S$-equivalence classes of Higgs…

Derived finiteness conjecture. There exist only finitely many smooth projective varieties $Y$, up to isomorphism, whose derived categories are equivalent to that of $X$:

Let $W$ be the subscheme under consideration, let $S_2(W)$ denote its second-symmetric locus, and suppose that $R^1_2(W)$ is torsion-free. Scheme-theoretic identification conjectur…

Let $B$ be an abelian variety of dimension $n$, and let $\widehat B$ be its dual. The Fourier–Mukai isomorphism … identifies numerical cycle classes on $B$ with those on…