Volume 21, Issue 4

In 1895 Wiman introduced the Riemann surface 𝒲 of genus 6 over the complex field ℂ defined by the equation X 6 + Y 6 + ℨ 6 +( X 2 + Y 2 + ℨ 2 )( X 4 + Y 4 + ℨ 4 )−12 X 2 Y 2 ℨ 2 = 0, and showed that its full automorphism group is isomorphic to the symmetric group S 5 . We show that this holds also over every algebraically closed field 𝕂 of characteristic p ≥ 7. For p = 2, 3 the above polynomial is reducible over 𝕂, and for p = 5 the curve 𝒲 is rational and Aut(𝒲) ≅ PGL(2,𝕂). We also show that Wiman’s 𝔽 19 2 -maximal sextic 𝒲 is not Galois covered by the Hermitian curve H 19 over the finite field 𝔽 19 2 .

We develop a variational method to find pseudo-algebraic Ricci solitons on connected Lie groups.As applications, we prove that every Einstein nilradical admits a non-Riemannian algebraic Ricci soliton, and that any algebraic Ricci soliton on a semi-simple Lie group is Einstein. Furthermore, we construct several Lorentz algebraic Ricci solitons on the nilpotent Lie groups which have a codimension one abelian ideal.

Let X be a smooth projective complex curve of genus g ≥ 2 and let M X (2, Λ ) be the moduli space of semi-stable rank-2 vector bundles over X with fixed determinant Λ . We show that the wobbly locus, i.e. the locus of semi-stable vector bundles admitting a non-zero nilpotent Higgs field, is a union of divisors 𝓦 k ⊂ M X (2, Λ ). We show that on one wobbly divisor the set of maximal subbundles is degenerate. We also compute the class of the divisors 𝓦 k in the Picard group of M X (2, Λ ).

We consider the family ℰ ( s , r , d ) of all singular complex analytic vector fields X(z)=Q(z)P(z)eE(z)∂∂z$X(z)=\frac{Q(z)}{P(z)}{{e}^{E(z)}}\frac{\partial }{\partial z}$on the Riemann sphere where Q , P , ℰ are polynomials with deg Q = s , deg P = r and deg ℰ = d ≥ 1. Using the pullback action of the affine group Aut(ℂ) and the divisors for X , we calculate the isotropy groups Aut(ℂ) X of discrete symmetries for X ∈ ℰ ( s , r , d ). The subfamily ℰ ( s , r , d ) id of those X with trivial isotropy group in Aut(ℂ) is endowed with a holomorphic trivial principal Aut(ℂ)-bundle structure. A necessary and sufficient arithmetic condition on s , r , d ensuring the equality ℰ ( s , r , d ) = ℰ ( s , r , d ) id is presented. Moreover, those X ∈ ℰ ( s , r , d ) \ ℰ ( s , r , d ) id with non-trivial isotropy are realized. This yields explicit global normal forms for all X ∈ ℰ ( s , r , d ). A natural dictionary between analytic tensors, vector fields, 1-forms, orientable quadratic differentials and functions on Riemann surfaces M is extended as follows. In the presence of nontrivial discrete symmetries Γ < Aut( M ), the dictionary describes the correspondence between Γ -invariant tensors on M and tensors on M / Γ .

The 2-Fano varieties, defined by De Jong and Starr, satisfy some higher-dimensional analogous properties of Fano varieties. We consider (weak) k -Fano varieties and conjecture the polyhedrality of the cone of pseudoeffective k -cycles for those varieties, in analogy with the case k = 1. Then we calculate some Betti numbers of a large class of k -Fano varieties to prove some special case of the conjecture. In particular, the conjecture is true for all 2-Fano varieties of index at least n − 2, and we complete the classification of weak 2-Fano varieties answering Questions 39 and 41 in [2].

In this paper we compute the generating rank of k -polar Grassmannians defined over commutative division rings. Among the new results, we compute the generating rank of k -Grassmannians arising from Hermitian forms of Witt index n defined over vector spaces of dimension N > 2 n . We also study generating sets for the 2-Grassmannians arising from quadratic forms of Witt index n defined over V ( N , 𝔽 q ) for q = 4, 8, 9 and 2 n ≤ N ≤ 2 n + 2. We prove that for N > 6 and anisotropic defect (polar corank) d ≠ 2 they can be generated over the prime subfield, thus determining their generating rank.

We give a short, elementary and non-computational proof for the classical Beckman–Quarles theorem asserting that a map of a Euclidean space into itself that preserves distance 1 must be an isometry.

Let X ⊂ ℙ n be an unramified real curve with X (ℝ) ≠ 0. If n ≥ 3 is odd, Huisman [9] conjectured that X is an M -curve and that every branch of X (ℝ) is a pseudo-line. If n ≥ 4 is even, he conjectures that X is a rational normal curve or a twisted form of such a curve. Recently, a family of unramified M -curves in ℙ 3 providing counterexamples to the first conjecture was constructed in [11]. In this note we construct another family of counterexamples that are not even M -curves. We remark that the second conjecture follows for generic curves of odd degree from the de Jonquières formula.

We study the interaction between the g.o. property and certain flag curvature conditions. A Finsler manifold is called g.o. if each constant speed geodesic is the orbit of a one-parameter subgroup. Besides the non-negatively curved condition, we also consider the condition (FP) for the flag curvature, i.e. in any flag we find a flag pole such that the flag curvature is positive. By our main theorem, if a g.o. Finsler space ( M , F ) has non-negative flag curvature and satisfies (FP), then M is compact. If M = G / H where G has a compact Lie algebra, then the rank inequality rk 𝔤 ≤ rk 𝔥+1 holds. As an application,we prove that any even-dimensional g.o. Finsler space which has non-negative flag curvature and satisfies (FP) is a smooth coset space admitting a positively curved homogeneous Riemannian or Finsler metric.

We extend the concept of the Segre invariant to vector bundles on a surface X . For X = ℙ 2 we determine what numbers can appear as the Segre invariant of a rank 2 vector bundle with given Chern classes. The irreducibility of strata with fixed Segre invariant is proved and their dimensions are computed. Finally, we present applications to the Brill–Noether Theory for rank 2 vector bundles on ℙ 2 .

Coarse geometry, and in particular coarse homotopy theory, has proven to be a powerful tool for approaching problems in geometric group theory and higher index theory. In this paper,we continue to develop theory in this area by proving a Coarse Lifting Lemma with respect to a certain class of bornologous surjective maps. This class is wide enough to include quotients by coarsely discontinuous group actions, which allows us to obtain results concerning the coarse fundamental group of quotients which are analogous to classical topological results for the fundamental group. As an application, we compute the fundamental group of metric cones over negatively curved compact Riemannian manifolds.

We show that any compact metric f - K -contact, respectively S -manifold is obtained from a compact K -contact, respectively Sasakian manifold by an iteration of constructions of mapping tori, rotations, and type II deformations.

In this paper, we investigate an extremum problem for the power moment of a convex polygon contained in a disc. Our result is a generalization of a classical theorem: among all convex n -gons contained in a given disc, the regular n -gon inscribed in the circle (up to rotation) uniquely maximizes the area functional. It also implies that, among all convex n -gons contained in a given disc and containing the center in those interiors, the regular n -gon inscribed in the circle (up to rotation) uniquely maximizes the mean of the length of the chords passing through the center of the disc.

Here is a simplified proof that every sharply transitive subset of PGL 2 ( K ) is a coset of a subgroup, for every finite field K .