Let $K$ be a field of characteristic different from $2$, $\overline{K}$ its algebraic closure. Let $n \geq 3$ be an odd prime such that $2$ is a primitive root modulo $n$. Let $f(x)$ and $h(x)$ be degree $n$ polynomials with coefficients in $K$ and without repeated roots. Let us consider genus $(n-1)/2$ hyperelliptic curves $C_f : y^2 = f(x)$ and $C_h : y^2 = h(x)$, and their jacobians $J(C_f)$ and $J(C_h)$, which are $(n-1)/2$-dimensional abelian varieties defined over $K$.
Suppose that one of the polynomials is irreducible and the other reducible. We prove that if $J(C_f)$ and $J(C_h)$ are isogenous over $\overline{K}$ then both jacobians are abelian varieties of CM type with multiplication by the field of $n$th roots of $1$.
We also discuss the case when both polynomials are irreducible while their splitting fields are linearly disjoint. In particular, we prove that if $\operatorname{char}(K)=0$, the Galois group of one of the polynomials is doubly transitive and the Galois group of the other is a cyclic group of order $n$, then $J(C_f)$ and $J(C_h)$) are not isogenous over $\overline{K}$.
We prove Qingyuan Jiang’s conjecture on semiorthogonal decompositions of derived categories of Quot schemes of locally free quotients. The author’s result on categorified Hall products for Grassmannian flips is applied to prove the conjecture.
We prove a version of the Arezzo–Pacard–Singer blow-up theorem in the setting of Poincaré type metrics. We apply this to give new examples of extremal Poincaré type metrics. A key feature is an additional obstruction which has no analogue in the compact case. This condition is conjecturally related to ensuring the metrics remain of Poincaré type.
We give an example of a homogeneous reflexive sheaf over $\mathbb{C}^3$ which admits a non-conical Hermitian Yang–Mills connection. This is expected to model bubbling phenomenon along complex codimension $2$ submanifolds when the Fueter section takes zero value.
Let $\mathbf{k}$ be a perfect field and let $X \subset \mathbb{P}^N$ be a hypersurface of degree $d$ defined over $\mathbf{k}$ and containing a linear subspace $L$ defined over $\overline{\mathbf{k}}$ with $\operatorname{codim}_{\mathbb{P}^N} L = r$. We show that $X$ contains a linear subspace $L_0$ defined over $\mathbf{k}$ with $\operatorname{codim}_{\mathbb{P}^N} L \leq dr$. We conjecture that the intersection of all linear subspaces (over $\mathbf{k}$) of minimal codimension $r$ contained in $X$, has codimension bounded above only in terms of $r$ and $d$. We prove this when either $d \leq 3$ or $r \leq 2$.
We give a simple argument to prove Nagai’s conjecture for type II degenerations of compact hyperkähler manifolds and cohomology classes of middle degree. Under an additional assumption, the techniques yield the conjecture in arbitrary degree. This would complete the proof of Nagai’s conjecture in general, as it was proved already for type I degenerations by Kollár, Laza, Saccà, and Voisin [10] and independently by Soldatenkov [18], while it is immediate for type III degenerations. Our arguments are close in spirit to a recent paper by Harder [8] proving similar results for the restrictive class of good degenerations.
The Surface Group Conjectures are statements about recognising surface groups among one-relator groups, using either the structure of their finite-index subgroups, or all subgroups. We resolve these conjectures in the two generator case. More generally, we prove that every two-generator one-relator group with every infinite-index subgroup free is itself either free or a surface group.
We compute the global dimension function $\operatorname{gldim}$ on the principal component $\operatorname{Stab}^\dagger (\mathbb{P}^2)$ of the space of Bridgeland stability conditions on $\mathbb{P}^2$. It admits $2$ as the minimum value and the preimage $\operatorname{gldim}^{-1} (2)$ is contained in the closure $\overline{\operatorname{Stab}^{\operatorname{Geo}} \mathbb{P}^2}$ of the subspace consisting of geometric stability conditions. We show that $\operatorname{gldim}^{-1} [2, x)$ contracts to $\operatorname{gldim}^{-1} (2)$ for any real number $x \geq 2$ and that $\operatorname{gldim}^{-1} (2)$ is contractible.
In this short note, we compute higher extension groups for all irreducible representations and deduce the multiplicity formula for finite length representations in triple product case.