SYMMETRIC THETA DIVISORS OF KLEIN SURFACES

Abstract. This is a slightly expanded version of the talk given by Ch.O. at the conference “Instantons in complex geometry”, at the Steklov Institute in Moscow. The purpose of this talk was to explain the algebraic results of our paper ”Abelian Yang-Mills theory on Real tori and Theta divisors of Klein surfaces ” [10]. In this paper we compute determinant index bundles of certain families of Real Dirac type operators on Klein surfaces as elements in the corresponding Grothendieck group [7] of Real line bundles in the sense of Atiyah. On a Klein surface these determinant index bundles have a natural holomorphic description as theta line bundles. In particular we compute the first Stiefel-Whitney classes of the corresponding fixed point bundles on the real part of the Picard torus. The computation of these classes is important, because they control to a large extent the orientability of certain moduli spaces in Real gauge theory and Real algebraic geometry [11]. Let C be a Riemann surface, and g: = h0(ωC) its genus. The geometric theta divisor of C is the effective divisor of Picg−1(C) defined by Θ: = {[L] ∈ Picg−1(C) | h0(L)> 0}. The set θ of theta characteristics of C is the set of square roots of [ωC], i.e. θ: = {[κ] ∈ Picg−1(C) | κ⊗2 ' ωC} ⊂ Picg−1(C). Note that #θ = 22g. For a theta characteristic [κ] ∈ θ we obtain an associated symmetric theta divisor Θ[κ]: = Θ − [κ] ⊂ Pic0(C) and a holomorphic line bundle L[κ]: = OPic0(C)(Θ[κ]) on the Abelian variety Pic0(C). Definition 1. A Klein surface is a pair (C, ι), where C is a Riemann surface and ι: C → C an anti-holomorphic involution. The fixed point locus Cι decomposes is a finite union ∐n i=1 Ci of n: = #pi0(C ι) circles Ci. The topological type of the Klein surface (C, ι) is the triple (g, n, a), where a:= 0 when C/〈ι 〉 is orientable 1 when C 〈ι 〉 is non-orientable. These invariants are subject to the following conditions [6]: (1) 1 ≤ n ≤ g + 1, g + 1 − n ≡ 0 (mod 2), when a = 0, (2) 0 ≤ n ≤ g, when a = 0. 1 a