Stable generalized complex structures
A stable generalized complex structure is one that is generically symplectic but degenerates along a real codimension two submanifold, where it defines a generalized Calabi-Yau structure. We introduce a Lie algebroid which allows us to view such structures as symplectic forms. This allows us to construct new examples of stable structures, and also to define period maps for their deformations in which the background three-form flux is either fixed or not, proving the unobstructedness of both deformation problems. We then use the same tools to establish local normal forms for the degeneracy locus and for Lagrangian branes. Applying our normal forms to the four-dimensional case, we prove that any compact stable generalized complex 4-manifold has a symplectic completion, in the sense that it can be modified near its degeneracy locus to produce a compact symplectic 4-manifold.
Generalized complex geometry [MR2013140, MR2811595] is a common generalization of complex and symplectic geometry in which the pointwise structure may be described as a symplectic subspace with transverse complex structure. This symplectic distribution is controlled by a real Poisson structure, and so its rank may vary in any given example. Four-dimensional generalized complex manifolds have been thoroughly investigated, the main focus being on structures which are generically symplectic and degenerate along a 2-dimensional submanifold, which then inherits a complex structure rendering it a Riemann surface of genus one. In [MR2312048, MR2574746, Goto:2013vn, MR2958956, MR3177992], many examples of generalized complex four-manifolds were found, the most interesting of which were on manifolds, such as , which admit neither symplectic nor complex structures.
In this paper we develop the main properties of stable generalized complex structures, in which the structure is generically symplectic but degenerates along a real codimension 2 submanifold , a direct generalization of the four-dimensional case described above. We show that inherits a generalized Calabi-Yau structure (of type 1) as well as a holomorphic structure on its normal bundle, and we prove that a tubular neighbourhood of is completely classified by this data, a result which was not available even in dimension four. As an application of this result, we prove that any compact stable generalized complex 4-manifold has a symplectic completion, in the sense that it can be modified near to produce a compact symplectic 4-manifold. We prove a similar normal form theorem for Lagrangian branes, half-dimensional submanifolds analogous to Lagrangians in symplectic geometry. This involves a generalization of the cotangent bundle construction in symplectic geometry, where for example, we associate a natural stable generalized 6-manifold to any co-oriented link , by modifying the cotangent bundle of along in a certain way. We also provide a construction of stable structures on torus fibrations, obtaining, for instance, a stable structure on . We then move to deformation theory and define two period maps controlling deformations of stable generalized complex structures on compact manifolds . The first describes deformations with fixed background 3-form and is a map to , independently discovered by Goto [Goto:2015fk]. The second describes simultaneous deformations of the pair comprised of a stable structure integrable with respect to the 3-form , and is a map to In both cases, we obtain the unobstructedness of the deformation problem. Finally, we describe a number of topological constraints which the pair must satisfy in order to admit a stable generalized complex structure. These exclude, for example, the possibility that is a positive divisor in a compact complex manifold .
The main insight behind the above results is that a stable generalized complex structure is equivalent to a complex log symplectic form, a complex 2-form with a type of logarithmic singularity along the divisor and whose imaginary part defines an elliptic symplectic form, which is a symplectic form but for a Lie algebroid which we introduce called the elliptic tangent bundle. This approach, analogous to that taken in holomorphic log symplectic geometry [MR1953353] as well as in the recent development of real log symplectic geometry [MR3250302, MR3214314, Marcut-Osorno, MR3245143, Cavalcanti13],
justifies the intuition that a stable generalized complex structure is a type of singular symplectic structure, and it allows us to apply symplectic techniques such as Moser interpolation. For this reason, we carefully develop the theory of logarithmic and elliptic forms associated to smooth codimension 2 submanifolds. In particular, we compute the Lie algebroid cohomology of the elliptic tangent bundle and give an explicit description of its cup product.
Organization of the paper:
In Section 1, we introduce the notion of a complex divisor in the smooth category and its associated pair of Lie algebroids, the logarithmic tangent bundle (§LABEL:logderhamcxsect) and the elliptic tangent bundle (§LABEL:elltn), which facilitate working with codimension 2 logarithmic forms. We describe the various residues of an elliptic form (§LABEL:elllogcohmg), allowing an explicit description of the elliptic de Rham cohomology and its cup product. We then compare (§LABEL:compelllog) the logarithmic and elliptic de Rham complexes in the case that the elliptic residue vanishes, a condition which is relevant since stable generalized complex structures satisfy it. We end with an alternative geometric definition of the key Lie algebroids we use, identifying them with certain generalized Atiyah algebroids (§LABEL:atiyahsect) and using this we obtain a key lemma (§LABEL:isodiffdiv) that any family of complex divisors may be rectified in the smooth category.
In Section LABEL:sec2 we introduce the main object of study: stable generalized complex structures. Sections §LABEL:canbun–LABEL:hollinbu establish general results about the geometry of canonical line bundles and of generalized Calabi-Yau manifolds, and in §LABEL:stablestructures we define stable structures and determine in Theorem LABEL:gholstrnd the inherited geometry of the anticanonical divisor, ending with a method for constructing new examples (§LABEL:exst).
In Section LABEL:equivloggc, we establish the equivalence between stable structures and complex log symplectic structures (Theorem LABEL:equivsgcils), or with co-oriented elliptic symplectic structures (§LABEL:elllogsymp) if we consider only gauge equivalence classes of stable structures, and we use this to define two period maps, one for deformations in which is fixed (§LABEL:perfixH) and one where it is not (§LABEL:pernofixH). These then impose certain topological constraints on and on its complement (§LABEL:topconst). In the remainder of this section we establish three main local normal form theorems: Theorem LABEL:darbouximaglog is a Darboux theorem for the neighbourhood of a point in , Theorem LABEL:linaboutd classifies a tubular neighbourhood of , and Theorem LABEL:canonicalsymp is a Lagrangian brane neighbourhood theorem. We prove our symplectic completion result for stable 4-manifolds (Theorem LABEL:theo:symplectization) using the second of these.
The period maps defined in Section LABEL:equivloggc establish the unobstructedness of the deformation problems (with and without fixed 3-form flux), which suggests that the algebras controlling them are formal. In Section LABEL:sec4 we prove that the dgLa controlling the first problem is in fact formal, and we prove that the algebra controlling the second is quasi-isomorphic to a formal dgLa; we conjecture (Conjecture LABEL:conjlinfty) that our quasi-isomorphism extends to a morphism.
We thank R. Goto for the opportunity to visit Kyoto in 2013 and share some of the results of the present work, including Corollary LABEL:independentgoto concerning unobstructedness for deformations with fixed 3-form, which was independently discovered by him and has recently appeared [Goto:2015fk].
G. C. was supported by a VIDI grant from NWO, the Dutch science foundation. M. G. was supported by an NSERC Discovery Grant and acknowledges support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network).
1 Complex divisors on smooth manifolds
Let be a smooth complex line bundle over the smooth –manifold , and let be a section transverse to the zero section. We refer to the pair as a complex divisor.
Our nomenclature is by analogy with the well-known correspondence between holomorphic line bundles with section and divisors on complex manifolds. In our case, we regard the pair as the divisor, though we may abuse notation and use to refer to the smooth real codimension 2 submanifold given by the zero set of . Note that as vanishes transversely along , it has a nonvanishing normal derivative which establishes an isomorphism between the real normal bundle of and the restriction of to .