Let $G$ be the topological fundamental group of a nonsingular complex projective surface. Troncoso and Urz\'ua proved that the Chern slopes $c_1^2/c_2$ of minimal surfaces of general type $S$ with $\pi_1(S)\simeq G$ are dense in $[1,3]$, and left $[1/3,1)$ open. We prove that they are dense in $[1/2,3]$, an interval that cannot be enlarged without contradicting either a theorem of Mendes Lopes and Pardini or Reid's conjecture. We prove more: the slopes of such surfaces with $K_S$ ample are dense in $[1/2,2]$, the first case in which the conjecture of Troncoso and Urz\'ua on ample canonical classes is established. The tool is an exact ampleness criterion for their product construction, which shows in particular that their own surfaces never have ample canonical class, whatever the defining sections.
We prove that for a very general abelian variety of dimension $\geq 4$, a divisor $D\in {\rm CH}^1(A)$ that satisfies $D^2=0$ in ${\rm CH}^2(A)$ is of torsion. The same result is also established for a very general Jacobian in genus $4$. We use then the second statement in order to prove a conjecture of Pirola, which s...
Let $X$ be a compact log-canonical K\"ahler $n$-fold, $n\ge3,$ with trivial canonical sheaf and with isolated singularities. We prove that the generic fibres of a semi-universal deformation of $X$ have Du Bois invariant $b^{1,n-2}=0$ at the singular points. Under a certain topological hypothesis on $X$ the generic fibr...
A well-known conjecture of Chern, do Carmo, and Kobayashi asserts that, for $n,m \geq 1$, the scalar curvature of a closed, minimally immersed $n$-submanifold of $\mathbb{S}^{n+m}$ with second fundamental form of constant length takes values in a discrete set. This property holds in every codimension when $n \in \{1,2\...
Let $X$ be a smooth, projective, geometrically connected variety over a field $k$ containing a root of unity of order $p$. If $X$ has a Chern number prime to $p$, we show that every action of a finite $p$-group on $X$ factors through a subgroup of $\operatorname{GL}_n(k)$, where $n=\dim X$. This allows one to transfer...
Building on earlier work of Katsura and Rybakov, we complete the classification, in every characteristic, of all groups $G$ acting on an abelian surface $A$ by automorphisms preserving the group law such that the resolution of the quotient $A/G$ is a K3 surface. In order to do so, we study actions of groups with $p\mid...
In this paper, over the field of complex numbers, we prove that a numerical Godeaux surface contains at most six pairwise disjoint $(-2)$-curves, and that this bound is sharp. As an application, we refine the classification of involutions on smooth minimal surfaces of general type with $p_g=0$ and $K^2=7$: the divisori...
Yi-Fan Chen, Y. Shin· 0 citations
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