14 problems

Seiberg–Witten invariant conjecture. The constants are the Seiberg–Witten invariants of the link, namely

Let $X$ be a closed, oriented, smooth four-manifold with $b_2^+(X)>0$. Let ${\mathfrak{s}}={\mathfrak{s}}_0\otimes L_1$ be a $\operatorname{Spin}^c$ structure defined by…

Let $V$ be a surface with $p_g(V)>0$, let $K$ be an effective canonical divisor, and fix $m\in H^2(V,\mathbb{Z})$. Let … be the zero-cycle on…

Let $(V,g)$ be a surface with a Kähler metric $g$, let $m\in H^2(V,\mathbb{Z})$, and let $\beta$ be a real closed $2$-form of type $(1,1)$. Let $\tau$ be a $Spin^c(4)$-structure re…

Let $(M^{4},g)$ be a compact anti-self-dual Einstein manifold with negative scalar curvature. Anti-self-dual Einstein vanishing conjecture. All the Seiberg–Witten invariants vanish…

Let $M^{4}=\overline{{\mathbb C}{\mathcal H}_{2}/\Gamma}$ be a reverse-oriented compact complex-hyperbolic $4$-manifold. Reverse-oriented complex-hyperbolic vanishing conjecture. A…

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 $S$ be the surface and $M$ the moduli space defined above, let $s\in\mathbb Z$, and let $\alpha\in K(S)$ have $\operatorname{rk}(\alpha)=s$. The universal functions…

Let $X$ be a simply connected, closed, oriented, smooth $4$-manifold with $b_2^+(X)>1$ and $b_2^-(X)>1$. Orientation-vanishing conjecture. The Seiberg–Witten invariant vanishes for…

Let $X$ be a closed, connected, oriented, smooth $4$-manifold with $b_2^+(X)>1$. Simple type conjecture. $X$ has Seiberg–Witten simple type. This conjecture asserts a fundamental c…

Let $X$ be a closed oriented homology-oriented smooth 4-manifold satisfying … Let $\lambda_{\rm FO}(X)$ and $\lambda_{\rm SW}(X)$ denote the Furuta–Ohta and Mrowka–Ruberman–Savelie…

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 $X_i$ for $i=1,2,3$ be either a 4-torus $T^4$, or a closed oriented almost complex 4-manifold with … … and $SW_{X_i}(\Gamma_{X_i})\equiv 1\pmod 2$, where $\Gamma_{X_i}$ is a sp…