8 problems

Let $(M,\domega)$ be a minimal symplectic $4$-manifold with symplectic Kodaira dimension $\delta0kappa^s(M)=2$. Bogomolov–Miyaoka–Yau conjecture. The symplectic canonical class sat…

An irreducible four-manifold is a smooth four-manifold that is not the connected sum of two nontrivial four-manifolds. Write $c_1^2$ for its first Chern number squared. The $3/2$ c…

Let $X$ be a closed oriented aspherical 4-manifold. Write $\chi(X)$ for its Euler characteristic and $\sigma(X)$ for its signature. Geography conjecture. One should have … This is…

Symplectic Bogomolov–Miyaoka–Yau conjecture. One has

Let $X^{4}$ be a closed, oriented, aspherical $4$-manifold. Write $\chi(X^{4})$ for its Euler characteristic and $\sigma(X^{4})$ for its signature. Strengthened Euler characteristi…

Let $X^{4}$ be a closed, oriented, aspherical $4$-manifold. Write $\chi(X^{4})$ for its Euler characteristic and $\sigma(X^{4})$ for its signature. Euler characteristic-signature i…

Let $g=2,3,\ldots$. A smooth surface $X$ of general type has canonical self-intersection $K_X^2$ and holomorphic Euler characteristic $\chi(X,\mathcal{O}_X)$. A pencil of curves of…

Bogomolov Watershed Conjecture. Every simply connected surface of general type has negative index, that is,