Band 2019, Heft 752

We prove weak approximation for isotrivial families of rationally connected varieties defined over the function field of a smooth projective complex curve.

We prove – in arbitrary characteristic – that the genus of abstract modular curves associated to bounded families of continuous geometrically perfect 𝔽ℓ{\mathbb{F}_{\ell}}-linear representations of étale fundamental groups of curves goes to infinity with ℓ{\ell}. This applies to the variation of the Galois image on étale cohomology groups with coefficients in 𝔽ℓ{\mathbb{F}_{\ell}} in 1-dimensional families of smooth proper schemes or, under certain assumptions, to specialization of first Galois cohomology groups.

For a semiring ℛ\mathcal{R}, the relations of shift equivalence over ℛ\mathcal{R} (SE-⁢ℛ\textup{SE-}\mathcal{R}) and strong shift equivalence over ℛ\mathcal{R} (SSE-⁢ℛ\textup{SSE-}\mathcal{R}) are natural equivalence relations on square matrices over ℛ\mathcal{R}, important for symbolic dynamics. When ℛ\mathcal{R} is a ring, we prove that the refinement of SE-⁢ℛ\textup{SE-}\mathcal{R} by SSE-⁢ℛ\textup{SSE-}\mathcal{R}, in the SE-⁢ℛ\textup{SE-}\mathcal{R} class of a matrix A , is classified by the quotient N⁢K1⁢(ℛ)/E⁢(A,ℛ)NK_{1}(\mathcal{R})/E(A,\mathcal{R}) of the algebraic K-theory group N⁢K1⁢(ℛ)NK_{1}(\mathcal{R}). Here, E⁢(A,ℛ)E(A,\mathcal{R}) is a certain stabilizer group, which we prove must vanish if A is nilpotent or invertible. For this, we first show for any square matrix A over ℛ\mathcal{R} that the refinement of its SE-⁢ℛ\textup{SE-}\mathcal{R} class into SSE-⁢ℛ\textup{SSE-}\mathcal{R} classes corresponds precisely to the refinement of the GL⁢(ℛ⁢[t])\mathrm{GL}(\mathcal{R}[t]) equivalence class of I-t⁢AI-tA into El⁢(ℛ⁢[t])\mathrm{El}(\mathcal{R}[t]) equivalence classes. We then show this refinement is in bijective correspondence with N⁢K1⁢(ℛ)/E⁢(A,ℛ)NK_{1}(\mathcal{R})/E(A,\mathcal{R}). For a general ring ℛ\mathcal{R} and A invertible, the proof that E⁢(A,ℛ)E(A,\mathcal{R}) is trivial rests on a theorem of Neeman and Ranicki on the K-theory of noncommutative localizations. For ℛ\mathcal{R} commutative, we show ∪AE⁢(A,ℛ)=N⁢S⁢K1⁢(ℛ)\cup_{A}E(A,\mathcal{R})=NSK_{1}(\mathcal{R}); the proof rests on Nenashev’s presentation of K1K_{1} of an exact category.

We are interested in the algebraic properties of groups of local biholomorphisms and their consequences. A natural question is whether the complexity of solvable groups is bounded by the dimension of the ambient space. In this spirit we show that 2⁢n+1{2n+1} is the sharpest upper bound for the derived length of solvable subgroups of the group Diff⁢(ℂn,0){\mathrm{Diff}({\mathbb{C}}^{n},0)} of local complex analytic diffeomorphisms for n=2,3,4,5{n=2,3,4,5}.

In previous work, we have introduced δ-forms on the Berkovich analytification of an algebraic variety in order to study smooth or formal metrics via their associated Chern δ-forms. In this paper, we investigate positivity properties of δ-forms and δ-currents. This leads to various plurisubharmonicity notions for continuous metrics on line bundles. In the case of a formal metric, we show that many of these positivity notions are equivalent to Zhang’s semipositivity. For piecewise smooth metrics, we prove that plurisubharmonicity can be tested on tropical charts in terms of convex geometry. We apply this to smooth metrics, to canonical metrics on abelian varieties and to toric metrics on toric varieties.

We construct a compatible family of global cohomology classes (an Euler system) for the symmetric square of a modular form, and apply this to bounding Selmer groups of the symmetric square Galois representation and its twists.

A classical theorem by K. Ribet asserts that an abelian variety defined over the maximal cyclotomic extension K of a number field has only finitely many torsion points. We show that this statement can be viewed as a particular case of a much more general one, namely that the absolute Galois group of K acts with finitely many fixed points on the étale cohomology with 𝐐/𝐙{\mathbf{Q}/\mathbf{Z}}-coefficients of a smooth proper K¯{\overline{K}}-variety defined over K . We also present a conjectural generalization of Ribet’s theorem to torsion cycles of higher codimension. We offer supporting evidence for the conjecture in codimension 2, as well as an analogue in positive characteristic.

We give new sufficient conditions for the integrability and unique integrability of continuous tangent subbundles on manifolds of arbitrary dimension, generalizing Frobenius’ classical theorem for C1{C^{1}} subbundles. Using these conditions, we derive new criteria for uniqueness of solutions to ODEs and PDEs and for the integrability of invariant bundles in dynamical systems. In particular, we give a novel proof of the Stable Manifold Theorem and prove some integrability results for dynamically defined dominated splittings.

A special cubic fourfold is a smooth hypersurface of degree 3 and dimension 4 that contains a surface not homologous to a complete intersection. Special cubic fourfolds give rise to a countable family of Noether–Lefschetz divisors 𝒞d{{\mathcal{C}}_{d}} in the moduli space 𝒞{{\mathcal{C}}} of smooth cubic fourfolds. These divisors are irreducible 19-dimensional varieties birational to certain orthogonal modular varieties. We use the “low-weight cusp form trick” of Gritsenko, Hulek, and Sankaran to obtain information about the Kodaira dimension of 𝒞d{{\mathcal{C}}_{d}}. For example, if d=6⁢n+2{d=6n+2}, then we show that 𝒞d{{\mathcal{C}}_{d}} is of general type for n>18{n>18}, n∉{20,21,25}{n\notin\{20,21,25\}}; it has nonnegative Kodaira dimension if n>13{n>13} and n≠15{n\neq 15}. In combination with prior work of Hassett, Lai, and Nuer, our investigation leaves only twenty values of d for which no information on the Kodaira dimension of 𝒞d{{\mathcal{C}}_{d}} is known. We discuss some questions pertaining to the arithmetic of K3 surfaces raised by our results.