Symmetries of complex analytic vector fields with an essential singularity on the Riemann sphere

We consider the family ℰ (s, r, d) of all singular complex analytic vector fields X ( z ) = Q ( z ) P ( z ) e E ( 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.

Homology of SL2 over function fields I: Parabolic subcomplexes

The present paper studies the group homology of the groups {\operatorname{SL}_{2}(k[C])} and {\operatorname{PGL}_{2}(k[C])} , where {C=\overline{C}\setminus\{P_{1},\dots,P_{s}\}} is a smooth affine curve over an algebraically closed field k. It is well known that these groups act on a product of trees and the quotients can be described in terms of certain equivalence classes of rank two vector bundles on the curve {\overline{C}} . There is a natural subcomplex consisting of cells with suitably non-trivial isotropy group. The paper provides explicit formulas for the equivariant homology of this “parabolic subcomplex”. These formulas also describe group homology of {\operatorname{SL}_{2}(k[C])} above degree s, generalizing a result of Suslin in the case {s=1} .

Reversible homogeneous Finsler metrics with positive flag curvature

In this work, we continue with the classification for positively curve homogeneous Finsler spaces {(G/H,F)}. With the assumption that the homogeneous space {G/H} is odd dimensional and the positively curved metric F is reversible, we only need to consider the most difficult case left, i.e. when the isotropy group H is regular in G. Applying the fixed point set technique and the homogeneous flag curvature formulas, we show that the classification of odd dimensional positively curved reversible homogeneous Finsler spaces coincides with that of L. Bérard Bergery in Riemannian geometry except for five additional possible candidates, i.e. {\mathrm{SU}(4)/\mathrm{SU}(2)_{(1,2)}\mathrm{S}^{1}_{(1,1,1,-3)}}, {\mathrm{Sp}(2)/\mathrm{S}^{1}_{(1,1)}}, {\mathrm{Sp}(2)/\mathrm{S}^{1}_{(1,3)}}, {\mathrm{Sp}(3)/\mathrm{Sp}(1)_{(3)}\mathrm{S}^{1}_{(1,1,0)}}, and {G_{2}/\mathrm{SU}(2)} with {\mathrm{SU}(2)} the normal subgroup of {\mathrm{SO}(4)} corresponding to the long root. Applying this classification to homogeneous positively curved reversible {(\alpha,\beta)} metrics, the number of exceptional candidates can be reduced to only two, i.e. {\mathrm{Sp}(2)/\mathrm{S}^{1}_{(1,1)}} and {\mathrm{Sp}(3)/\mathrm{Sp}(1)_{(3)}\mathrm{S}^{1}_{(1,1,0)}}.

Homogeneous locally conformally Kähler and Sasaki manifolds

We prove various classification results for homogeneous locally conformally symplectic manifolds. In particular, we show that a homogeneous locally conformally Kähler manifold of a reductive group is of Vaisman type if the normalizer of the isotropy group is compact. We also show that such a result does not hold in the case of non-compact normalizer and determine all left-invariant lcK structures on reductive Lie groups.