Transversality for Holomorphic Supercurves

Abstract

We study holomorphic supercurves, which are motivated by supergeometry as a natural generalisation of holomorphic curves. We prove that, upon perturbing the defining equations by making them depend on a connection, the corresponding linearised operator is generically surjective. By this transversality result, we show that the resulting moduli spaces are oriented finite dimensional smooth manifolds. Finally, we examine how they depend on the choice of generic data.2010 Mathematics Subject Classification. 53D35, 58C15.

Compactness for holomorphic supercurves

“…By our first result, Prp. 4.3, which follows from the first mean value inequality in the last section and a compactness result for holomorphic supercurves with bounded L p -norms for p > 2 from [11], the emergence of bubbling points is solely determined by the underlying sequence of holomorphic curves. We may thus follow the classical treatment of bubbling, with one major exception: For the proof of Gromov compactness in the next section, we need the super energy, not just the classical energy, concentrated in a bubbling point to coincide with the super energy of the corresponding bubble.…”

“…3.1 is entirely local. Recall from [11] that every (A, J)-holomorphic supercurve is a local holomorphic supercurve upon choosing conformal coordinates on Σ and trivialising the bundle L by a local holomorphic section. In the following, we denote by z = s + it the standard coordinates on C ∼ = R 2 .…”

“…[9] and [8]). In the case of holomorphic supercurves, the transversality problem was solved in [11] by making the defining equations depend on a connection A such that the corresponding linearised operator is generically surjective. This perturbed definition is referred to as an (A, J)-holomorphic supercurve.…”

“…Under this identification, the standard metric on S 2 is a constant multiple of the Fubini-Study metric h (s,t) := 1 (1+s 2 +t 2 ) 2 (ds 2 + dt 2 ) Based on Def. 1.1, this article solves the compactness problem in important cases and may be read independent of [10] and [11]. It is organised as follows.…”

J-holomorphic curves and Dirac-harmonic maps

“…Remark 6.12. A Dirac-harmonic map (f, ψ), where f is a J-holomorphic curve and ψ ∈ ker∂ ∇ f , is a (∇ g , J)-holomorphic supercurve as studied in [16], cf. also [14].…”