90 problems

Let $M$ be a manifold homotopy equivalent to $\mathbb{C}P^n$ and admitting a nontrivial circle action. Petrie's conjecture. Any homotopy equivalence between $M$ and $\mathbb{C}P^n$…

Aluffi–Harris conjecture. The same formula should admit a natural generalization to arbitrary projective varieties, including singular ones, by replacing the Euler characteristic w…

Cappell–Shaneson conjecture.

Deligne–Grothendieck conjecture. There is a unique natural transformation

Let $X$ be a complex algebraic variety, let $IT_{y*}(X)$ denote its intersection Hirzebruch class transformation, and let $L_*(X)$ denote the homology $L$-class of Goresky–MacPhers…

Rationality conjecture. If the polynomials $\varphi_i$ have rational coefficients, then

Epsilon-class conjecture. If $\nabla$ is an integrable, admissible, absolute connection, then

Let a vector bundle be associated with a matroid bundle. MacPherson's conjecture. All characteristic classes of the vector bundle should be carried by the associated matroid bundle…

Let $M$ be a smooth manifold with a Poisson structure $\pi_0$, and let $\widehat{A}(\pi_0)\in H^{\mathrm{ev}}(M,\mathbb{C}[[t]])$ be the class obtained from the periodic cyclic for…

Esnault's conjecture. The Chern classes of Deligne canonical extensions of motivic flat bundles vanish in the rational Chow groups.

Mumford's conjecture. This homomorphism should be an isomorphism in the stable range of Harer's stability theorem; in the formulation stated earlier, it is an isomorphism in degree…

Let $N$ be a $1$-connected $6$-manifold, and let $\bar W_3(N)$ denote its normal Stiefel–Whitney class. Six-manifold embedding conjecture. There exists such an $N$ with … that does…

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 $M$ be a Weil–Petersson variety, and let $X$ be a Weil–Petersson subvariety of dimension $q$ in $M$. Let $c_k(\omega_{WP})$ denote the $k$-th elementary polynomial of the curva…

Let $\phi\colon\widehat V\to V$ be a blowing down between nonsingular complex projective varieties, with $\widehat V$ obtained by blowing up $V$ along a nonsingular irreducible cen…

Let $V$ be a nonsingular complex projective variety of complex dimension $n$, viewed as a complex manifold, with fundamental class $[V]\in H_{2n}(V,\mathbb Z)$ and total Todd class…

Let $C_{2r+1}$ be an odd cycle, let $K_n$ be the complete graph on $n$ vertices, and let $\varpi_1^{n-2}(\text{\tt Hom}(C_{2r+1},K_n))$ denote the indicated Stiefel-Whitney charact…

Linear-combination conjecture. The class $v_1$ is a linear combination of $\alpha$ and $e_1$:

Let $B$ be the base of a surface bundle, let $p$ be a prime, and let $\kappa_i$ denote its Morita–Mumford classes. Vanishing conjecture. For $s\geq 1$ and $v\geq 0$, one has … The…

Let $M$ be a smooth closed oriented $2n$-dimensional manifold, let $E$ and $F$ be smooth complex vector bundles over $M$, and let $D_+\colon\Gamma(E)\to\Gamma(F)$ be an elliptic di…

Let $T_g$ be the Torelli group of a closed oriented surface $\Sigma_g$ of genus $g$. The higher FR-torsion classes are cohomology classes associated to the universal surface bundle…

Characteristic-class conjecture. The characteristic classes of codimension-$p$ foliations on $X$ coincide with the elements of

Conjecture on twisted K-theory characteristic classes. The classes $c_{r,r}$ are the only integral characteristic classes for twisted $K$-theories.

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…

Beilinson–Cheeger–Chern class conjecture. For all $p \geq 1$ and $n \in {\mathbb N} \cup\{\infty\}$,