17 problems

Let $S$ be a rational algebraic surface, let $C$ be the first Chern class of a differentiable complex vector bundle on $S$, and let $e$ be the corresponding Donaldson-invariant gra…

Let $\ell$, $F$, $L_1$, $\mathfrak{s}_0$, $SW(\mathfrak{s}_0\otimes L_1)$, and $Q_X$ denote the level, bundle and line-bundle data, Seiberg–Witten invariant, and intersection form…

Let $X$ be a four-manifold, let $E$ be a rank-two bundle, and let $z\in\mathbb A(X)$. Let $n_a$, $d_a$, $\overline{\operatorname{Red}}(W,E)$, $\overline V(z)$,…

Let $(X1, \Sigma1)$ and $(X2,\Sigma2)$ be permissible 4-manifolds with $b1(Xi)=0$, and let $tilde Xi=Xi{Sigmai}Bg$. Write … Choose an identification $\phi$, set $X=X_1\#_{\Sigma}X_2$…

Let $S$ be a rational surface, let $N,d,r$ be the parameters in the wall-crossing formula, let $K^2$ denote $K_S^2$, and let $Q_i(N,d,r,K^2)$ and $P_i(N,d,r,K^2)$ be the polynomial…

Let $h$ and $p$ be the generators used in the paper for the polynomial expressions $B_{4n}$. For every integer $n$, there is a nonnegative integer $l(n)$ such that…

Let $X$ be a $4$-manifold with $b_+(X)=1$ and $b_1(X)=0$, let $\xi\in H^2(X,\mathbb Z)$ satisfy $\xi^2<0$, and let $\lambda_{[X]}(q)$ be the power-series factor associated with the…

Let $X$ be a smooth simply connected $4$-manifold. For a class $xi\in H^2(X,\mathbb Z)$ defining a wall, an integer $N$, and the associated wall-crossing term $\delta^X_{\xi,N}$, l…

Let $X$ be a compact oriented $4$-manifold with $b_+^2(X)=1$, without assuming that $X$ is simply connected, and consider its Donaldson-theoretic moduli spaces and geometric data a…

For integers $(\beta_1,\beta_2,\beta_3,\beta_4)$ and $v\geq0$, let $F_v(\beta_1,\beta_2,\beta_3,\beta_4,y)$ and $G_v(\beta_1,\beta_2,\beta_3,\beta_4,y)$ be the universal functions…

Let $X$ be a four-manifold of ${\rm U}(3)$-simple type, let $\{K_i\}$ be its Kronheimer–Mrowka basic classes, and let $\mathfrak s_i$ be chosen so that the basic class associated t…

Witten's conjecture. Then $X$ has simple type; the basic classes of $X$ are precisely the Seiberg–Witten basic classes of $X$; and there is a nonzero constant $c(X)$ depending only…

Let $\pi:\mathrm{M}(d,1)\to |\mathcal{O}_{\mathbb{P}^{2}}(d)|$ be the Fitting map, and let … Here $g$ denotes the dimension of a general fiber of $\pi$. Donaldson-number formula co…

Consider the complex projective plane $\mathbb{C}\mathrm{P}^2$ with gauge group $\mathrm{SU}(2)$. Its regularized $u$-plane integral has a contribution from the cusp at…

Mochizuki's conjecture. Mochizuki's result holds for $X$, up to sign, when $\widetilde{\mathcal A}(\xi_1,y;a)$ is replaced by the coefficients of $\mathcal B(\xi_1,\xi;a)$ in the p…

Let $c=\operatorname{H}$, let $N_f=0,2,3$, and let $m,n,k$ satisfy … Write $\pmb{\mathrm{D}}^{N_f}_{m,2n}$ for the Donaldson invariant and $\pmb{\Phi}_{k,m,2n}$ and…

For $N_f=2,3$, let $\pmb{\mathrm{D}}^{N_f}_{m,2n}$ denote the generating function for the generalized $\operatorname{SO}(3)$-Donaldson invariants of $\mathbb{C}\mathrm{P}^2$ with…