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 states that any rational section of the Kummer fibration $K=J/\pm {\rm Id}\rightarrow \mathcal{M}_4$, where $J\rightarrow \mathcal{M}_4 $ is the Jacobian fibration, must be a multiple of the Griffiths-Pirola section given by the difference of the two trigonal divisors.
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 fibres have also link invariant $l^{1,n-2}=0.$ If $X$ is a projective log-canonical $n$-fold, $n\ge3,$ with ample anti-canonical sheaf and with isolated singularities then the generic fibres have $b^{1,n-2}=l^{1,n-2}=0$ (without the topological hypothesis). These are generalizations of recent results of Tenie `Global smoothing of singular Fano and Calabi--Yau varieties'and an older result of Namikawa `Deformation theory of Calabi--Yau threefolds and certain invariants of singularities.'
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 $n \in \{1,2\}$. We disprove this conjecture for all $n \geq 3$ with $m \geq 4$, and for even $n \geq 4$ with $m \geq 3$.
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 properties of representations of finite $p$-groups to their actions on $X$. We deduce a fixed-point theorem which, unlike previously known results of this kind, is sensitive to the arithmetic of the base field. We also obtain a bound on the orders of cyclic $p$-subgroups of the Cremona groups: for instance $\operatorname{Cr}_n(\mathbb{Q})$ contains no element of order $p^2$ when $p \ge n+2$. The method is based on the following observation, of independent interest. For an affine algebraic group $G$ over a field of characteristic zero, $\operatorname{ed}_p(G) + \dim G$ is the least dimension of a smooth projective variety $Y$ with a generically free $G$-action such that the degree map $\operatorname{CH}_G(Y) \to \mathbb{F}_p$ is nonzero. A key input for our result is Karpenko and Merkurjev's computation of the essential $p$-dimension of $p$-groups.
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 $p\mid|G|$ in characteristics $p=2,3$ and $5$. Using the theory of rational double points in positive characteristic, we show that the possible ADE singularity types of $A/G$ are constrained by the requirement that their local fundamental group contains $G$ as a subgroup, and we determine the singular locus of $A/G$ via the action of $G$ on the $\ell$-adic Tate module of $A$. As a key step in the classification, we prove that if $A$ is a supersingular abelian surface and $p\mid|G|$, then $A/G$ can never be a generalised Kummer surface. Finally, we construct explicit examples of such generalised Kummer surfaces as quotients of products of two elliptic curves.
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 divisorial fixed part $R$ satisfies $R^2=-1$, the involution acts trivially on $H^*(S,\mathbb{Q})$, and, if the minimal resolution of the quotient is of general type, it is a numerical Campedelli surface containing five pairwise disjoint $(-2)$-curves. Another application concerns commuting involutions on smooth minimal surfaces of general type with $p_g=0$ and $K^2=8$.
