4 problems

Let $L$ be a 5-dimensional rational homology sphere with $H_1(L,\mathbb Z)=0$ admitting a fixed-point-free differentiable circle action. For an orbit $O$, define its exceptional mu…

Let $L$ be a 5-dimensional rational homology sphere with $H_1(L,\mathbb Z)=0$ admitting a pseudo-free differentiable circle action. Let $O_1,\dots,O_k$ be the nonfree orbits with s…

Let $M^4$ be a smooth, compact $4$-manifold with spherical boundary components … Assume that $H_1(M,\mathbb Z)=0$ and $H_2(M,\mathbb Z)\cong\mathbb Z$. Smooth topological Bogomolov…

Let $X$ be a semistable regular arithmetic surface of genus $g>1$ over $O_K$. Let $\overline\omega_X$ be its relative dualizing sheaf with its Arakelov metric, let $\delta_X$ denot…