23 problems

Vistoli's conjecture. If $\alpha\in\ker\tau_X$, then $\xi\alpha=0$ for some class $\xi$ of a perfect complex with everywhere non-zero rank.

Let $X$ be a complex manifold, let $\mathscr{M}$ be a coherent bounded complex of $\mathscr{D}_X$-modules, and let $\Lambda$ contain $\operatorname{char}(\mathscr{M})$. Choose a go…

Köck's conjecture. There is a push-forward of completions

Let $X$ be a $G$-space, let $\Psi$ be a semisimple conjugacy class in $G$, choose $h\in\Psi$, and let $Z={\mathcal Z}_G(h)$. Theorem (c) states that the map…

Let $M$ be a smooth projective variety, let $\pi$ be a group, and let $f:M\to B\pi$ be a continuous map. For $\alpha\in H^*(B\pi;\mathbb{Q})$, let $\operatorname{Td}(M)$ denote the…

Let $X$ be a complex manifold of dimension $d$, let ${\cal M}^\bullet$ be an object of $\operatorname{D}^b_{good}({\cal D}^{op}_X)$, and let $\Lambda$ be a conic subvariety of…

Debarre–Huybrechts–Macrì–Voisin conjecture. Then $c_X=(2n-1)!!$ and the Huybrechts–Riemann–Roch polynomial of $X$ is

Relative tropical Riemann–Roch conjecture.

Moderate-position tropical Riemann–Roch conjecture. If $D$ is in moderate position on $X$, then

Anti-effective tropical Riemann–Roch conjecture.

López de Medrano–Rincón–Shaw's conjecture.

Let $S$ be a compact tropical manifold. Let $H^{\bullet,\bullet}(S;\mathbb{Z})$ be its total tropical cohomology group, let $s$ be a permissible $C^\infty$-divisor, let…

Suppose that $f\colon X\to Y$ is a morphism equipped with an almost perfect obstruction theory with virtual tangent class $[T_{X/Y}]-[\mathcal{O}b_X]\in K_0(X)$. Assume that this c…

Numerical Riemann–Roch conjecture. The Huybrechts–Riemann–Roch polynomial of $X$ satisfies

Riemann--Roch root conjecture. 1. The polynomial $\mathrm{RR}_X(q)$ has $n$ distinct negative real roots forming an arithmetic sequence. 2. If $X$ is smooth, then…

Let $X$ be a hyperkähler manifold, and let … be its Riemann–Roch polynomial. Root and log-concavity conjectures. The following properties are conjectured: 1. The sequence of coeffi…

Cao–Jiang positivity conjecture. All coefficients of $\operatorname{RR}_X(q)$ are non-negative.

Let $\Delta$ be a tropical surface, let $D$ be a Cartier divisor on $\Delta$, and let $K_\Delta$ be the divisor obtained by summing $(\operatorname{deg} r-2)[r]$ over the ridges…

Let $X$ be a complex manifold, let $\mathcal{HD}(\mathscr{O}_X)=\bigoplus_{i=0}^{d_X}\Omega_X^i[i]$, and consider Kashiwara's commutative diagram whose lower horizontal arrow is an…

Let $A_n$ be the root lattice, and let $L$ be a full-rank sub-lattice of $A_n$ satisfying the uniformity and reflection-invariance properties: its min-genus and max-genus are equal…

Let $X$ be a complex manifold, with tangent bundle $TX$, and let $\operatorname{eu}({\mathcal{O}}_X)$ denote the Euler class of its structure sheaf in the relevant cohomology. Kash…

Let $\pi\colon E\to B$ be an oriented surface bundle with closed fibers of genus $g$, equipped with a fiberwise metric. Let $\mathcal{H}(E)$ be the associated Hodge bundle, with fi…

Naive Riemann–Roch conjecture. One might conjecture that