Varieties fibred over abelian varieties with fibres of log general type

Let (X,B)$(X,B)$ be a complex projective klt pair, and let f:X→Z$f:X\to Z$ be a surjective morphism onto a normal projective variety with maximal albanese dimension such that _KX+B$K_X+B$ is relatively big over Z. We show that such pairs have good log minimal models.     

Hitchin–Mochizuki morphism,opers and Frobenius-destabilized vector bundles over curves

Let X   be a smooth projective curve of genus g≥2$g\ge 2$ defined over an algebraically closed field k   of characteristic p>0$p>0$. For p>r(r−1)(r−2)(g−1)$p>r(r-1)(r-2)(g-1)$ we construct an atlas for the locus of all Frobenius-destabilized bundles of rank r (i.e. we construct all Frobenius-destabilized bundles of rank r and degree zero up to isomorphism). This is done by exhibiting a surjective morphism from a certain Quot-scheme onto the locus of stable Frobenius-destabilized bundles. Further we show that there is a bijective correspondence between the set of stable vector bundles E over X   such that the pull-back F^?(E)$F^{?}(E)$ under the Frobenius morphism of X has maximal Harder–Narasimhan polygon and the set of opers having zero p-curvature. We also show that, after fixing the determinant, these sets are finite, which enables us to derive the dimension of certain Quot-schemes and certain loci of stable Frobenius-destabilized vector bundles over X. The finiteness is proved by studying the properties of the Hitchin–Mochizuki morphism; an alternative approach to finiteness has been realized in [3]. In particular we prove a generalization of a result of Mochizuki to higher ranks.     

Pull-back morphisms for reflexive differential forms

Let f:X→Y$f:X\to Y$ be a morphism between normal complex varieties, where Y$Y$ is Kawamata log terminal. Given any differential form σ$\sigma$, defined on the smooth locus of Y$Y$, we construct a “pull-back form” on X$X$. The pull-back map obtained by this construction is _?Y${?}_Y$-linear, uniquely determined by natural universal properties and exists even in cases where the image of f$f$ is entirely contained in the singular locus of Y$Y$.     

Lê–Greuel type formula for the Euler obstruction and applications

The Euler obstruction of a function f   can be viewed as a generalization of the Milnor number for functions defined on singular spaces. In this work, using the Euler obstruction of a function, we establish several Lê–Greuel type formulas for germs f:(X,0)→(C,0)$f:(X,0)\to (\mathbb{C},0)$ and g:(X,0)→(C,0)$g:(X,0)\to (\mathbb{C},0)$. We give applications when g is a generic linear form and when f and g have isolated singularities.     

Algebra of bounded polynomials on a set Zariski closed at infinity cannot be finitely generated

We prove that if a set S⊂Rⁿ$S\subset {\mathbb{R}}^n$ is Zariski closed at infinity, then the algebra of polynomials bounded on S cannot be finitely generated. It is a new proof of a fact already known to Plaumann and Scheiderer (2012) [1]. On the way we show that if the ring R[_ζ1,…,_ζk]⊂R[X]$\mathbb{R}[{\zeta }_1,\dots ,{\zeta }_k]\subset \mathbb{R}[X]$ contains the ideal (_ζ1,…,_ζk)R[X]$({\zeta }_1,\dots ,{\zeta }_k)\mathbb{R}[X]$, then the mapping (_ζ1,…,_ζk):Rⁿ→R^k$({\zeta }_1,\dots ,{\zeta }_k):{\mathbb{R}}^n\to {\mathbb{R}}^k$ is finite.     

Locally torsion-free quasi-coherent sheaves

Let X   be an arbitrary scheme. It is known that the category Qcoh(X)$\mathfrak{Qcoh}(X)$ of quasi-coherent sheaves admits arbitrary products. However its structure seems to be rather mysterious. In the present paper we will describe the structure of the product object of a family of locally torsion-free objects in Qcoh(X)$\mathfrak{Qcoh}(X)$, for X an integral scheme. Several applications are provided. For instance it is shown that the class of flat quasi-coherent sheaves on a Dedekind scheme X   is closed under arbitrary direct products, and that the class of all locally torsion-free quasi-coherent sheaves induces a hereditary torsion theory on Qcoh(X)$\mathfrak{Qcoh}(X)$. Finally torsion-free covers are shown to exist in Qcoh(X)$\mathfrak{Qcoh}(X)$.     

Common properties of bounded linear operators AC and BA: Local spectral theory

Let X, Y   be Banach spaces, A:X→Y$A:X\to Y$ and B,C:Y→X$B,C:Y\to X$ be bounded linear operators satisfying operator equation ABA=ACA$ABA=ACA$. In this paper, we show that AC and BA share the local spectral properties such as Bishop's property (β), subscalarity and Dunford's property (C). Also, the quasi-nilpotent part and the analytic core for AC and BA are studied.     

Continuous weak selections for products

A weak selection on an infinite set X   is a function σ:[X]²→X$\sigma :{[X]}^2\to X$ such that σ({x,y})∈{x,y}$\sigma (\{x,y\})\in \{x,y\}$ for each {x,y}∈[X]²$\{x,y\}\in {[X]}^2$. A weak selection on a space is said to be continuous if it is a continuous function with respect to the Vietoris topology on [X]²${[X]}^2$ and the topology on X  . We study some topological consequences from the existence of a continuous weak selection on the product X×Y$X\times Y$ for the following particular cases:

(i)

Both X and Y are spaces with one non-isolated point.     

Decomposition of small diagonals and Chow rings of hypersurfaces and Calabi–Yau complete intersections

On the one hand, for a general Calabi–Yau complete intersection X$X$, we establish a decomposition, up to rational equivalence, of the small diagonal in X×X×X$X\times X\times X$, from which we deduce that any decomposable 0-cycle of degree 0 is in fact rationally equivalent to 0, up to torsion. On the other hand, we find a similar decomposition of the smallest diagonal in a higher power of a hypersurface, which provides us an analogous result on the multiplicative structure of its Chow ring.     

Rigidity of symmetric products

Given a metric continuum X, we consider the following hyperspaces of X  : 2^X$2^X$, _Cn(X)$C_n(X)$ and _Fn(X)$F_n(X)$ (n∈N$n\in \mathbb{N}$). Let _F1(X)={{x}:x∈X}$F_1(X)=\{\{x\}:x\in X\}$. A hyperspace K(X)$K(X)$ of X   is said to be rigid provided that for every homeomorphism h:K(X)→K(X)$h:K(X)\to K(X)$ we have that h(_F1(X))=_F1(X)$h(F_1(X))=F_1(X)$. In this paper we study under which conditions a continuum X   has a rigid hyperspace _Fn(X)$F_n(X)$.