Fibrations and stable generalized complex structures

Abstract

A generalized complex structure is called stable if its defining anticanonical section vanishes transversally, on a codimension-two submanifold. Alternatively, it is a zero elliptic residue symplectic structure in the elliptic tangent bundle associated to this submanifold. We develop Gompf-Thurston symplectic techniques adapted to Lie algebroids, and use these to construct stable generalized complex structures out of log-symplectic structures. In particular we introduce the notion of a boundary Lefschetz fibration for this purpose and describe how they can be obtained from genus one Lefschetz fibrations over the disc. ContentsLet X be a 2n-dimensional manifold equipped with a closed three-form H ∈ Ω 3 cl (X). Recall that the double tangent bundle TX := T X ⊕ T * X is a Courant algebroid whose anchor is the projection p : TX → T X. It carries a natural pairing V + ξ,This takes (TX, H) to (TX, H + dB), leading to theŠevera class [H] ∈ H 3 (X; R) determining TX up to Courant isomorphism. The Courant automorphisms of TX are generated by the diffeomorphisms and closed B-field transformations.Definition 2.1. A generalized complex structure on (X, H) is a complex structure J on TX that is orthogonal with respect to ·, · , and whose +i-eigenbundle is involutive underThere is an alternative definition of a generalized complex structure using spinors. To state it, recall that sections v = V + ξ ∈ Γ(TX) of the double tangent bundle act on differential forms via Clifford multiplication, given byDefinition 2.2. A generalized complex structure on (X, H) is given by a complex line bundle K J ⊂ ∧ • T * C X pointwise generated by a differential form ρ = e B+iω ∧ Ω with Ω a decomposable complex k-form, satisfying Ω ∧ Ω ∧ ω n−k = 0, and such that dρ + H ∧ ρ = v · ρ for any local section ρ ∈ Γ(K J ) and some v ∈ Γ(TX).Both definitions are related using that K J = Ann(E J ) is the annihilator under the Clifford action of E J , the +i-eigenbundle of J . The bundle K J is called the canonical bundle of J . For later use, we further introduce the analogue of a Calabi-Yau manifold in generalized geometry. Denote by d H = d + H∧ the H-twisted de Rham differential.Example 2.4. The following provide examples of generalized complex structures on (X, 0).• Let ω be a symplectic structure on X. Then K Jω := e iω defines a generalized complex structure J ω . • Let J be a complex structure on X with canonical bundle K J = ∧ n,0 T * X. Then K JJ := K J defines a generalized complex structure J J . • Let P ∈ Γ(∧ 2,0 T X) a holomorphic Poisson structure with respect to a complex structure J. Then K JP,J := e P K J defines a generalized complex structure J P,J .The automorphisms J ω , J J and J P,J are given by, with π = Im(P ):We next introduce the type of a generalized complex structure J , which colloquially provides a measure for how many complex directions there are. The type is an integer-valued upper semicontinuous function on X whose parity is locally constant.Definition 2.5. Let J be a generalized complex structure on X. The type of J is a m...

Classification of Boundary Lefschetz Fibrations over the Disc

“…. In this example we provide X = S 1 × S 3 with the structure of a boundary fibration over the disc, as described in [5,Example 8.3]. The map f : S 1 × S 3 → D 2 is a composition of maps, namely…”

“…A few relevant facts about boundary Lefschetz fibrations were established in [5]. Beyond the local normal form (2.1) for the map f around points in D there is also a semi-global form for f in a neighbourhood of D: Proposition 5.15]).…”

“…In the case when X is connected, Σ is a surface with boundary Z = ∂Σ, and f : X → Σ is surjective, we can lift f to a cover of Σ so that the fibres of the boundary Lefschetz fibration become connected. That is, this particular hypothesis is not really a restriction on the fibration (see [5,Proposition 5.23]). In what follows we will always assume this is the case.…”

“…Then ∂N ± −k is diffeomorphic to ∂Z with the orientation reversed so that we can form a closed manifold X by gluing Z and N ± −k together, and the orientation of Z extends. Moreover, it was shown in [5] that N ± −k admits a boundary fibration over the annulus which can be used to extend the Lefschetz fibration on Z to a boundary Lefschetz fibration on X, again over a larger disc which we recale to D 2 , in such a way that the boundary vanishing cycle along the straight line from θ 0 to zero is a.…”

“…Proof. This is [5,Example 8.4], but in light of our discussion about blow-ups in terms of cycle systems we can determine it directly. Indeed, any cycle system of f has the form (a; b 1 ) such that B 1 = ±A k .…”