International Journal of Mathematics

We study the existence of multiple closed Reeb orbits on some contact manifolds by means of S¹-equivariant symplectic homology and the index iteration formula. We prove that a certain class of contact manifolds which admits displaceable exact contact embeddings, a certain class of prequantization bundles, and Brieskorn spheres have multiple closed Reeb orbits.

In the 1990s, Bertram, Feinberg and Mukai examined Brill–Noether loci for vector bundles of rank 2 with fixed canonical determinant, noting that the dimension was always bigger in this case than the naive expectation. We generalize their results to treat a much broader range of fixed-determinant Brill–Noether loci. The main technique is a careful study of symplectic Grassmannians and related concepts.

By gluing two copies of surface S_0,g+2 along g + 1 holes, we get surface S_g,1. The pillar switching is a self-homeomorphism of S_g,1 which switches two pillars of surfaces by 180° horizontal rotation. We analyze the actions of the pillar switchings on π₁S_g,1 and then give concrete expressions of the pillar switchings in terms of standard Dehn twists. The map ψ : B_g → Γ_g,1 sending the generators of B_g to the pillar switchings on S_g,1 is defined by extending the embedding B_g ↪ Γ_0,(g+1),1. We show that this map is injective by analyzing the actions of pillar switchings on π₁S_g,1. We also prove that this map induces a trivial homology homomorphism in the stable range. For the proof we use the categorical delooping method. We construct a suitable monoidal 2-functor from tile category to surface category and show that this functor thus induces a map of double loop spaces.

Via the XJC-correspondence proved in [L. Pirio and F. Russo, Extremal varieties 3-rationally connected by cubics, quadro-quadric Cremona transformations and rank 3 Jordan algebras, submitted] we provide some structure theorems for quadro-quadric Cremona transformations and for extremal varieties 3-covered by twisted cubics by reinterpreting for these objects the algebraic results on the solvability of the radical of Jordan algebras. In this way, we can define the semi-simple part and the radical part of a quadro-quadric Cremona transformation, respectively of an extremal variety 3-covered by twisted cubics, and then describe how general objects are constructed from the semi-simple ones, which are completely classified modulo certain equivalences, via suitable null radical extensions.

We discuss genericity and stability of transversality of holomorphic maps to complex analytic stratifications. We prove that the set of maps between Stein manifolds and Oka manifolds transverse to a countable collection of submanifolds in the target is dense in the space of holomorphic maps with the weak topology. This greatly generalizes earlier results on the genericity of transverse maps by Forstnerič and by Kaliman and Zaidenberg. As an application we show that the Whitney (a)-regularity of a complex analytic stratification is necessary and sufficient for the stability of transverse holomorphic maps between a Stein manifold and an Oka manifold. This gives an analogue of a theorem in the real case due to Trotman.

We give an optimal upper bound for the anti-canonical volume of an ϵ-lc weak log del Pezzo surface. Moreover, we consider the relation between the bound of the volume and the Picard number of the minimal resolution of the surface. Furthermore, we consider blowing up several points on a Hirzebruch surface in general position and give some examples of smooth weak log del Pezzo surfaces.

We establish the following criterion for divisibility in the local ring of those quasianalytic function germs at 0 ∈ ℝⁿ which are definable in a polynomially bounded structure. A sufficient (and necessary) condition for the divisibility of two function germs in is that of their Taylor series at 0 ∈ ℝⁿ in the formal power series ring.