arXiv subjects

Alexandru Scorpan

Publications and source records attributed to Alexandru Scorpan.

4 recordsLinked to original sources

arXiv2003

Existence of foliations on 4-manifolds

We present existence results for certain singular 2-dimensional foliations on 4-manifolds. The singularities can be chosen to be simple, e.g. the same as those that appear in Lefschetz pencils. There seems to be a wealth of such creatures on most 4-manifolds. In certain cases, one can prescribe surfaces to be transverse or be leaves of these foliations. The purpose of this paper is to offer objects, hoping for a future theory to be developed on them. For example, foliations that are taut might offer genus bounds for embedded surfaces (Kronheimer's conjecture).

math.GT↗

arXiv2003

A quick survey of foliations on 4-manifolds

We present a few general results on foliations of 4-manifolds by surfaces: existence, tautness, relations to minimal genus of embedded surfaces; as well as some open problems. We hope to stimulate interest in this area.

math.GT↗

arXiv2003

Nowhere-zero harmonic spinors and their associated self-dual 2-forms

Let M be a closed oriented 4-manifold, with Riemannian metric g, and a spin^C structure induced by an almost-complex structure ω. Each connection A on the determinant line bundle induces a unique connection \nabla^A, and Dirac operator \D^A on spinor fields. Let σ: W^+ --> Λ^+ be the natural squaring map, taking self-dual (= positive) spinors to self-dual 2-forms. In this paper, we characterize the self-dual 2-forms that are images of self-dual spinor fields through σ. They are those αfor which (off zeros) c_1(α) = c_1(ω), where c_1(α) is a suitably defined Chern class. We also obtain the formula: || ϕ||^2 D^A ϕ= i (2 d^* σ(ϕ) + < \nabla^A ϕ, i ϕ>)* ϕ. Using these, we establish a bijective correspondence between: {Kahler forms αcompatible with a metric scalar-multiple of g, and with c_1(α) = c_1(ω)} and {gauge classes of pairs (ϕ, A), with \nabla^A ϕ= 0}, as well as a bijective correspondence between: {Symplectic forms αcompatible with a metric conformal to g, and with c_1(α) = c_1(ω)} and {gauge classes of pairs (ϕ, A), with D^ A ϕ= 0, and < \nabla^A ϕ, i ϕ> = 0, and ϕnowhere-zero}.

math.DG↗

arXiv2003

Spinors as automorphisms of the tangent bundle

We show that, on a 4-manifold M endowed with a spin^c structure induced by an almost-complex structure, a self-dual (= positive) spinor field ϕ\in Γ(W^+) is the same as a bundle morphism ϕ: TM \to TM acting on the fiber by self-dual conformal transformations, such that the Clifford multiplication is just the evaluation of ϕon tangent vectors, and that the squaring map σ: W^+ \to Λ^+ acts by pulling-back the fundamental form of the almost-complex structure. We use this to detect Kahler and symplectic structures.

math.DG↗