Existence of the equivariant minimal model program for compact Kähler threefolds with the action of an abelian group of maximal rank

Let X be a $\mathbb{Q}$-factorial compact Kähler klt threefold admitting an action of a free abelian group G, which is of positive entropy and of maximal rank. After running the G-equivariant log minimal model program, we show that such X is either rationally connected or bimeromorphic to a Q-complex torus. In particular, we fix an issue in the proof of our previous paper [23, Theorem 1.3].     

On the Galois covers of degenerations of surfaces of minimal degree

We investigate the topological structures of Galois covers of surfaces of minimal degree (i.e., degree n) in ${\mathbb{CP}}^{n+1}$. We prove that for $n\ge 5$, the Galois covers of any surfaces of minimal degree are simply-connected surfaces of general type.     

Hausdorff operators on compact abelian groups

Necessary and sufficient conditions are given for the boundedness of Hausdorff operators on the generalized Hardy spaces $H_E^p(G)$, real Hardy space $H_{\mathbb{R}}^1(G)$, $\text{BMO}(G)$, and $\text{BMOA}(G)$ for compact Abelian group G. Surprisingly, these conditions turned out to be the same for all groups and spaces under consideration. Applications to Dirichlet series are given. The case of the space of continuous functions on G and examples are also considered.     

Higher dimensional Shimura varieties in the Prym loci of ramified double covers

In this paper, we construct Shimura subvarieties of dimension bigger than one of the moduli space ${\mathsf{A}}_p^{\delta }$ of δ-polarized abelian varieties of dimension p, which are generically contained in the Prym loci of (ramified) double covers. The idea is to adapt the techniques already used to construct Shimura curves in the Prym loci to the higher dimensional case, namely, to use families of Galois covers of ${\mathbb{P}}^1$. The case of abelian covers is treated in detail, since in this case, it is possible to make explicit computations that allow to verify a sufficient condition for such a family to yield a Shimura subvariety of ${\mathsf{A}}_p^{\delta }$.     

Foliations by curves on threefolds

We study the conormal sheaves and singular schemes of one-dimensional foliations on smooth projective varieties X of dimension 3 and Picard rank 1. We prove that if the singular scheme has dimension 0, then the conormal sheaf is μ-stable whenever the tangent bundle $TX$ is stable, and apply this fact to the characterization of certain irreducible components of the moduli space of rank 2 reflexive sheaves on ${\mathbb{P}}^3$ and on a smooth quadric hypersurface $Q_3\subset {\mathbb{P}}^4$. Finally, we give a classification of local complete intersection foliations, that is, foliations with locally free conormal sheaves, of degree 0 and 1 on Q₃.     

Odd-dimensional counterparts of abelian complex and hypercomplex structures

We introduce the notion of abelian almost contact structures on an odd-dimensional real Lie algebra $\mathfrak{g}$. We investigate correspondences with even-dimensional Lie algebras endowed with an abelian complex structure, and with Kähler Lie algebras when $\mathfrak{g}$ carries a compatible inner product. The classification of 5-dimensional Sasakian Lie algebras with abelian structure is obtained. Later, we introduce abelian almost 3-contact structures on real Lie algebras of dimension $4n+3$, obtaining the classification of these Lie algebras in dimension 7. Finally, we deal with the geometry of a Lie group G endowed with a left invariant abelian almost 3-contact metric structure. We determine conditions for G to admit a canonical metric connection with skew torsion, which plays the role of the Bismut connection for hyperKähler with torsion (HKT) structures arising from abelian hypercomplex structures. We provide examples and discuss the parallelism of the torsion of the canonical connection.     

On the average degree of some irreducible characters of a finite group

Let G be a finite group and N be a non-trivial normal subgroup of G, such that the average degree of irreducible characters in $\mathrm{Irr}(G|N)$ is less than or equal to 16/5. Then, we prove that N is solvable. Also, we prove the solvability of G, by assuming that the average degree of irreducible characters in $\mathrm{Irr}(G|N)$ is strictly less than 16/5. We show that the bounds are sharp.     

Geodesics as products of one-parameter subgroups in compact lie groups and homogeneous spaces

We study the geodesic equation for compact Lie groups G and homogeneous spaces $G/H$, and we prove that the geodesics are orbits of products $\exp (t{X}_1)\cdots \exp (t{X}_N)$ of one-parameter subgroups of G, provided that a simple algebraic condition for the Riemannian metric is satisfied. For the group $SO(3)$, we relate this type of geodesics to the free motion of a symmetric top. Moreover, by using series of Lie subgroups of G, we construct a wealth of metrics having the aforementioned type of geodesics.     

Singular rational curves on elliptic K3 surfaces

We show that on every elliptic K3 surface there are rational curves ${(R_i)}_{i\in \mathbb{N}}$ such that $R_i^2\to \infty$, that is, of unbounded arithmetic genus. Moreover, we show that the union of the lifts of these curves to $\mathbb{P}({\mathrm{\Omega }}_X)$ is dense in the Zariski topology. As an application, we give a simple proof of a theorem of Kobayashi in the elliptic case, that is, there are no globally defined symmetric differential forms.     

Vector bundles on flag varieties

We study vector bundles on flag varieties over an algebraically closed field k. In the first part, we suppose $G=G_k(d,n)$ $(2\le d\le n-d)$ to be the Grassmannian parameterizing linear subspaces of dimension d in $k^n$, where k is an algebraically closed field of characteristic $p>0$. Let E be a uniform vector bundle over G of rank $r\le d$. We show that E is either a direct sum of line bundles or a twist of the pullback of the universal subbundle $H_d$ or its dual $H_d^{\vee }$ by a series of absolute Frobenius maps. In the second part, splitting properties of vector bundles on general flag varieties $F(d_1,\text{…},d_s)$ in characteristic zero are considered. We prove a structure theorem for bundles over flag varieties which are uniform with respect to the ith component of the manifold of lines in $F(d_1,\text{…},d_s)$. Furthermore, we generalize the Grauert–M$\ddot{\mathrm{u}}$lich–Barth theorem to flag varieties. As a corollary, we show that any strongly uniform i-semistable $(1\le i\le n-1)$ bundle over the complete flag variety splits as a direct sum of special line bundles.     

Isomorphisms of Galois groups of number fields with restricted ramification

Let K be a number field and S a set of primes of K. We write $K_S/K$ for the maximal extension of K unramified outside S and $G_{K,S}$ for its Galois group. In this paper, we answer the following question under some assumptions: “For $i=1,2$, let $K_i$ be a number field, $S_i$ a (sufficiently large) set of primes of $K_i$ and $\sigma :G_{K_1,S_1}\stackrel{\sim }{\to }{G}_{K_2,S_2}$ an isomorphism. Is σ induced by a unique isomorphism between $K_{1,S_1}/K_1$ and $K_{2,S_2}/K_2$?” Here, the main assumption is about the Dirichlet density of $S_i$.     

On the generic behavior of the metric entropy, and related quantities,of uniformly continuous maps over Polish metric spaces

In this work, we show that if f is a uniformly continuous map defined over a Polish metric space, then the set of f-invariant measures with zero metric entropy is a $G_{\delta }$ set (in the weak topology). In particular, this set is generic if the set of f-periodic measures is dense in the set of f-invariant measures. This settles a conjecture posed by Sigmund (Trans. Amer. Math. Soc. 190 (1974), 285–299), which states that the metric entropy of an invariant measure of a topological dynamical system that satisfies the periodic specification property is typically zero. We also show that if X is compact and if f is an expansive or a Lipschitz map with a dense set of periodic measures, typically the lower correlation entropy for $q\in (0,1)$ is equal to zero. Moreover, we show that if X is a compact metric space and if f is an expanding map with a dense set of periodic measures, then the set of invariant measures with packing dimension, upper rate of recurrence and upper quantitative waiting time indicator equal to zero is residual.     

On topological and combinatorial structures of pointed stable curves over algebraically closed fields of positive characteristic

In this paper, we study anabelian geometry of curves over algebraically closed fields of positive characteristic. Let $X^{•}=(X,D_X)$ be a pointed stable curve over an algebraically closed field of characteristic $p>0$ and ${\mathrm{\Pi }}_{X^{•}}$ the admissible fundamental group of $X^{•}$. We prove that there exists a group-theoretical algorithm, whose input datum is the admissible fundamental group ${\mathrm{\Pi }}_{X^{•}}$, and whose output data are the topological and the combinatorial structures associated with $X^{•}$. This result can be regarded as a mono-anabelian version of the combinatorial Grothendieck conjecture in the positive characteristic. Moreover, by applying this result, we construct clutching maps for moduli spaces of admissible fundamental groups.     

Essential finite generation of extensions of valuation rings

Given a generically finite local extension of valuation rings $V\subset W$, the question of whether W is the localization of a finitely generated V-algebra is significant for approaches to the problem of local uniformization of valuations using ramification theory. Hagen Knaf proposed a characterization of when W is essentially of finite type over V in terms of classical invariants of the extension of associated valuations. Knaf's conjecture has been verified in important special cases by Cutkosky and Novacoski using local uniformization of Abhyankar valuations and resolution of singularities of excellent surfaces in arbitrary characteristic, and by Cutkosky for valuation rings of function fields of characteristic 0 using embedded resolution of singularities. In this paper, we prove Knaf's conjecture in full generality.