Research Statement. Michael Bailey. November 6, 2017

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1 Research Statement Michael Bailey November 6, Introduction to generalized complex geometry 1 2 My work so far 2 3 Overview of planned projects 4 4 Deformation quantization 5 5 Integration of generalized complex structures 8 6 Holomorphic stacks and derived geometry 9 7 A neighbourhood theorem for submanifolds, and blow-downs 10 My research has centered around generalized complex geometry. This is a common generalization of complex-analytic, symplectic and holomorphic-poisson geometry. Though generalized complex geometry is fascinating in its own right and in the ways it connects diverse mathematical phenomena it has strong motivation from physics, where it has applications in 2-dimensional supersymmetric quantum field theories in particular, topological string theory and mirror symmetry. Among my most notable works to date are a local classification theorem for generalized complex structrures (Theorem 1 here, and [2]) and, more recently, an integration theorem [8], which reveals that a generalized complex structure may be understood as complex structure on a shifted symplectic stack. My research plan involves further investigations in generalized complex geometry and adjacent topics, such as deformation quantization, and holomorphic stacks, along with the application of some of the arguments and tools I have developed to other kinds of geometry, such as higher stacks and derived geometry. In Section 1, I define generalized complex structures, describe some important contexts in which they appear, and give a very brief overview of the now-burgeoning field; In Section 2, I describe my work so far, and how it fits into the field; in Section 3, I outline my current and future research. I hope Sections 1 to 3 are accessible at some level to nonspecialists. In Sections 4 to 7, I go into detail on some of my projects. 1 Introduction to generalized complex geometry Generalized complex structures were defined by Hitchin in 2003 [9], and given a first thorough treatment by Gualtieri [10, 11]. They have been rapidly taken up by physicists and geometers (see, eg., [12, 13, 14]). Definition. A generalized complex structure on a manifold M is given by a complex structure, I : T M T M T M T M, I 2 = Id 1

2 on the sum of the tangent plus cotangent bundles, which is orthogonal for the standard symmetric pairing on T M T M, and which satisfies an integrability condition given by the Courant bracket, an extension of the Lie bracket to T M T M. Or, more generally, one considers such an I on an exact Courant algebroid, of which T M T M is the standard example. I will briefly describe four areas in which generalized complex geometry appears very naturally in the Calabi-Yau geometry of string theory, in Poisson geometry, in mirror symmetry, and in deformation quantization. These are all closely related (from the generalized complex point of view), and each highlights the physical importance of the theory. One place complex analytic geometry meets physics is via string theory and supersymmetry [15] (among other points of contact). The existence of an N = 1 supersymmetry on a compactification of spacetime in string theory entails that the compact cross section, or internal manifold, is Calabi-Yau. This is an especially nice class of complex manifolds, with vanishing Chern class. However, a physically natural weakening of the supersymmetry condition [16] gives instead the structure of a generalized Calabi-Yau manifold [9], the analogue of a Calabi-Yau manifold in the context of generalized complex geometry. It is in the context of explaining these manifolds that generalized complex geometry was given its first explicit formulation. Poisson geometry, given its modern form by Weinstein [17], studies Poisson brackets, which determine the time-evolution and symmetries of Hamiltonian classical mechanics, as geometric structures on manifolds. A special case is the nondegenerate Poisson structures, corresponding to the well-known symplectic structures. As Gualtieri and I have shown [2, 8], generalized complex structures may be viewed as weakly holomorphic Poisson structure Mirror symmetry is a duality originally proposed by physicists between complex geometry and symplectic geometry. In particular, its topological version, homological mirror symmetry (proposed by Kontsevich [18]), relates two invariants described by Witten [19]: the B-model, or derived category of coherent sheaves, of a complex manifold; and the A-model, or Fukaya category, of a symplectic manifold. By putting these two kinds of geometry on equal footing, generalized complex geometry is expected to provide insights into mirror symmetry and, indeed, already has in the special case of T-duality [20, 21, 13]. Kontsevich showed that every Poisson structure on a manifold is the first-order part of a formal deformation of the manifold s algebra of smooth functions to a noncommutative associative algebra [22]. When applied to complex manifolds (or algebraic varieties) [23], this construction naturally yields an extended deformation theory which is equivalent to the deformation theory of these spaces when treated as generalized complex. Therefore, we expect that generalized complex manifolds are the natural geometric models of noncommutative complex manifolds. (More on this in Section 4.) This is related to the A-model, since the A-model is also a kind of quantization, and, at least in the hyperkahler case, the deformation quantization is expected to be the perturbative version of the A-model [24]. 2 My work so far The symmetries of Courant algebroids consist of the diffemorphisms of the underlying manifold, along with certain gauge transformations corresponding to the B-fields of physics [25, 26]. Therefore, one would like to classify generalized complex structures up to diffeomorphism and gauge symmetry. Partial results on local classification were achieved by Gualtieri [10] and then Abouzaid and Boyarchenko. However, the hard part of the problem remained until my thesis and subsequent paper [1, 2]. Complex, symplectic and holomorphic-poisson structures may each be seen 2

3 as generalized complex structures, and I have shown that these special cases give a complete local classification: Theorem 1 (B ). A generalized complex structure is equivalent (up to diffeomorphism and gauge), in a neighbourhood of any point, to a product of a symplectic manifold with a holomorphic Poisson manifold. In fact, given a parity condition, it is locally equivalent just to a holomorphic Poisson structure. This is a generalization of both Darboux s theorem for symplectic structures and the Newlander- Nirenberg theorem for complex structures. This result is surprising, since a generalized complex structure does not, in general, determine a complex structure. The complex structure underlying the holomorphic Poisson structure in the theorem exists locally, but it is non-canonical, i.e., it depends on a choice of gauge. (However, as Gualtieri and I showed [5], any two holomorphic Poisson structures about a point which are equivalent as generalized complex structures are diffeomorphic as holomorphic Poisson so there is a weak uniqueness.) Theorem 1 tells us that generalized complex structures may be studied in the realm of complex analytic geometry, at least locally. In recent work with Gualtieri (See Section 5 and [8]), we explained the global form of this relationship: a generalized complex structure is just a Poisson structure which is holomorphic as a stack, i.e., only up to Morita equivalence. This is reflected in the fact that a generalized complex structure integrates to what we call a weakly holomorphic symplectic groupoid integration is meant in analogy with the integration of Lie algebras to Lie groups or Poisson structures to symplectic groupoids. Remark: As an indication of how my work feeds back into the larger research community, I should mention that the ideas from our integrations paper [8] have already helped to resolve a puzzle coming from physics. A Generalized Kahler manifold is a manifold equipped with a pair of commuting generalized complex structures, satisfying a positivity condition. (Ordinary Kahler structures are the classical example.) These have been studied for some time in the context of string theory. Lindstrom, Rocek, von Unge and Zabzine [27] proposed that a generalized Kahler potential should determine a generalized Kahler structure analogously to how a Kahler potential determines a Kahler structure. However, their proposal had serious limitations: it covered a bunch of special cases and couldn t deal with type change between them; and the coordinate formulas were not covariant, and thus it was not clear on which space the potential function was defined. Bischoff, Gualtieri and Zabzine, in recent work [28], have used ideas from Gualtieri s and my integrations paper to characterize generalized Kahler potentials on a GC manifold, M, as the shadow on M of Lagrangians in a larger space, namely, an integrating holomorphic symplectic bimodule. Other work In addition to my local classification (Theorem 1, [2]), and my recent work on integrations of generalized complex structures, and holomorphic stacks ([8], covered in Section 5), I have completed a number of other projects, which I describe here. A generalized complex structure determines a tranvsersely holomorphic foliation by symplectic leaves. Going the other way, I have characterized the obstruction to a transversely holomorphic symplectic foliation lifting to a generalized complex structure [3]. With Vestislav Apostolov and Georges Dloussky, I have worked on the deformation theory of bi- Hermitian structures, finding new examples of bi-hermitian structures by adapting a deformation argument of Goto from the context of generalized Kahler geometry [4]. 3

4 With Marco Gualtieri, I showed that the local normal form of Theorem 1 is unique up to local diffeomorphism, and furthermore that the complex locus of a generalized complex structure is a complex analytic space [5] in a canonical way. With Gil Cavalcanti and Joey van der Leer Duran, I have demonstrated that certain natural submanifolds of generalized complex manifolds admit a blow-up construction, in analogy with complex geometry [6]. With Gil Cavalcanti and Marco Gualtieri, I have found necessary and sufficient topological conditions for the existence of type-1 generalized Calabi-Yau structures on a manifold [7]. I have shown that generalized complex branes subobjects of generalized complex manifolds which naturally arise in the A-model of physics are locally equivalent just to holomorphic coisotropic submanifolds [29], and globally may be understood as weakly holomorphic coisostropics, in analogy with the weakly holomorphic view on generalized complex structures. 3 Overview of planned projects One reason Theorem 1 and the holomorphic integration result are interesting is that they provide a route to quantization of generalized complex manifolds. As discussed above, the deformation quantizations of a complex manifold appear to be parametrized by deformations of the manifold as a generalized complex structure. Therefore, it should be possible to define and construct deformation quantizations of generalized complex manifolds. Thanks to Theorem 1 and the holomorphic integration result, I believe I now have the tools to make this precise. In Section 4, I discuss an expected route to the deformation quantization of generalized complex structures which makes fundamental use of the Kontsevich quantization for holomorphic Poisson structures [22, 23]. There will be something new to say even for quantizations of ordinary symplectic structures. I have an ongoing project to construct deformation quantizations of symplectic groupoids, i.e., smooth groupoids with a multiplicative symplectic structure. The result of such a quantization should be a Hopf algebroid. Not only is this interesting in its own right, but it is a key step in the project of deformation quantization of generalized complex structures: the holomorphic stack associated to a generalized complex structure has, as an atlas, a holomorphic symplectic groupoid. In some sense, this is a project to quantize stacks. The quantization I propose should in some sense be analogous to (a special case of) Calaque, Pantev, Toen, Vaquie and Vezzosi s deformation quantization of shifted symplectic structures [30, 31], though my approach is considerably more down-to-earth, being inspired by Karabegov s explicit constructions of formal symplectic groupoids [32]. Theorem 1 has nontrivial analytic content, whose implications reach beyond generalized complex geometry. In Section 6, I discuss a project to use the analytic tools I developed in [2] to give a derived Newlander-Nirenberg theorem for differential stacks and, indeed, higher stacks and derived manifolds. In fact, there are a number of connections between generalized complex geometry and stacky/derived geometry which are worth exploring. With Cavalcanti and van der Leer Duran, I am working to find holomorphic classification results like Theorem 1 on more complicated domains, such as in tubular neighbourhoods of submanifolds. I discuss this project in Section 7, and explain how it will make possible new surgeries on generalized complex manifolds, such as blow-downs. 4

5 A note about working with graduate students I hope to be supervising graduate students soon after starting in a faculty position. Each of the proposed projects I describe here can yield sub-projects and related projects that would be suitable for a Ph.D. student (though ideally a student s thesis project is discovered in collaboration between student and advisor). In particular, the quantization problem has many details such as describing the relationship between generalized complex branes and bimodules over the quantization which are important and new but which have close enough analogies with existing theory to be approachable for a student. 4 Deformation quantization One of the reasons that generalized complex structures are of special interest to physicists is that they purport to be semi-classical, geometric models of noncommutative manifolds. I will discuss this in the context of deformation quantization. On a smooth manifold M, a (formal) deformation quantization of C (M) consists of an (in general, noncommutative) associative bilinear star product, : [[ ]]C (M) [[ ]]C (M) [[ ]]C (M) f g = fg + C 1 (f, g) + 2 C 2 (f, g) +... over formal power series in C (M) such that, on functions, f g = fg at 0-th order in, and such that each C k is a bi-differential operator. This is understood as the algebra of observables in quantum mechanics, where corresponds to Planck s constant. The antisymmetric first-order part, {f, g} := C 1 (f, g) C 1 (g, f), determines a Poisson bracket on C (M), which is called the semiclassical limit of. It is natural to ask if a given Poisson structure π has a quantization, i.e., whether it is the first-order commutator of such a quantization. In 1997, Kontsevich showed [22], as a corollary to his Formality Theorem, that any smooth Poisson manifold has a formal deformation quantization. (Whether the series actually converges for finite is in general a difficult question.) In fact, Poisson structures parametrize all first-order quantizations of C (M), up to certain equivalences. In the complex-analytic setting, one would like to quantize the structure sheaf, O(X), of a complex manifold X to a sheaf of algebras whose semiclassical limit is a given holomorphic Poisson structure. However, this is not possible in general. In an affine chart, the formulas for the star-product work equally well for complex coordinates as for real coordinates; but on a manifold M, Kontsevich s construction must choose a rigidification, namely, a connection on T M. In this way, the affine formulas may be transported over the whole space. But it is rare for a complex manifold to admit a global holomorphic connection. The solution to this problem is also due to Kontsevich [23]. One may quantize O(X) in affine charts by choosing local connections, but these local quantizations do not fit together into a sheaf of algebras. Instead, one passes to a stack of algebroids, i.e., a sheaf in which one has, over an open U X, all such quantizations of O(U) (parametrized by holomorphic Poisson structures as well as connections on T U), related to each other by canonical bimodules, with all these elements coherent in a certain way. 5

6 Unlike the C case, the formal deformations of O(X) as a stack of algebroids are parametrized to first order, not only by holomorphic Poisson structures, but by the extended deformation space [33], H 0 (X, 2 T X) H 1 (X, T X) H 2 (X, O(X)), (4.1) (subject to a Maurer-Cartan condition). The first term corresponds to the Poisson structures, the second term corresponds to deformations of X itself, and the third term corresponds to deformations of the gerbe. 4.1 Deformation quantization of generalized complex structures This extended deformation space (4.1) also parametrizes the generalized complex deformations of X, to first order [11]. This suggests that generalized complex manifolds can be given a deformation quantization in this framework especially now, given the relationship I have established between generalized complex and holomorphic Poisson structures. Deformation quantization of generalized complex structures faces difficulties not present in the holomorphic case, and is as yet an unsolved problem. First, the equivalence between generalized complex and holomorphic Poisson manifolds is only local, whereas many generalized complex manifolds are not global deformations of complex manifolds for example, many symplectic manifolds do not admit complex structures. Another view on these problems is that it is not immediately obvious what should be the classical limit of a generalized complex structure whereas an ordinary Poisson structure may be scaled linearly by, with the 0 limit being the underlying commutative manifold (or complex manifold, or algebraic variety). Nonetheless, given a generalized complex manifold (M, I) satisfying the parity constraint, one can choose local gauges in which I is holomorphic Poisson, then construct deformation quantizations locally and hope to glue them together. But the space (4.1) only parametrizes formal deformations of O(X) among stacks of algebroids on X, whereas generalized complex structures are (locally) actual, finite deformations of complex structures. This presents an obstacle to gluing the local quantizations mentioned above, and presses us to move beyond the setting of stacks of algebroids to noncommutative stacks In Section 5, I will describe how, given an integrability condition, a generalized complex manifold determines a holomorphic stack, i.e., a holomorphic Lie groupoid up to Morita equivalence. Quantization of such objects has recently been studied by Calaque, Pantev, Toen, Vaquie and Vezzosi [30, 31]. Then, one should anchor this quantization to the original manifold so that one can recover the geometry in the semi-classical limit. I expect this to give the right deformation quantization of generalized complex manifolds, satisfying a number of criteria. To make this precise, I will take a different approach to quantizing my stacks of interest than CPTVV [31]. Their approach is very abstract (relatively speaking!), using the full apparatus of derived geometry, and is not well adapted to more real world situations like generalized complex geometry (though in some sense our approaches should be equivalent). 4.2 Deformation quantization of Lie groupoids I am working towards the deformation quantization of symplectic Lie groupoids. By this I mean: a deformation quantization of the symplectic arrow space, along with a deformation quantization of 6

7 the Poisson base, such that the structure maps now pullbacks in the opposite direction become algebra homomorphisms (or anti-homomorphisms where the maps were anti-poisson). The difficulty here is that, famously, quantization is not functorial at least, not in general. However, as Karabegov [32] and others have shown, there is a very close connection between a deformation quantization of a Poisson manifold M and its integrating formal symplectic groupoid. (Formal geometry is enough for my purposes.) The formal symplectic groupoid is modeled on a formal neighbourhood of the zero section of T M, and, in fact, the data of a product on M precisely determine the symplectic groupoid structure maps (in the formal geometry sense). The challenge, then, is to find a quantization of T M which incorporates the data of the product on M in just such a way that the structure maps become algebra homomorphisms. A formal symplectic groupoid is naturally expressed as a commutative Hopf algebroid, and its deformation quantization will be a noncommutative Hopf algebroid (see [34] for the general definition). Thus, one fruit of this project should be interesting new examples of Hopf algebroids. Another important application of this project, and my original reason for starting on it, is that deformation quantization of symplectic groupoids will tell us precisely how a Morita equivalence between Poisson structures translates through to their quantizations. For special cases (eg., a pair of real Poisson structures related by a B-transform), it is known that the quantizations will be related by a bimodule in the usual algebraic sense. However, in general there will be a Hopf algebroid bimodule between them, coming from the proposed groupoid quantization. In the deformation quantization of generalized complex structures, these Hopf bimodules are precisely what we need to glue the quantizations of different choices of holomorphic gauge (since the underlying holomorphic Poisson structures will be Morita equivalent). 4.3 Branes and bimodules A good criterion for the correct quantization of a generalized complex manifold is that the quantization should have well-defined categories of modules, or even left-right-bimodules. These should be noncommutative versions of the category of coherent sheaves. Given such a module (after possibly passing to a resolution), we ought to recover a generalized holomorphic vector bundle in the semi-classical limit. Furthermore, given a bimodule, the semi-classical limit should be a generalized complex brane. Generalized complex branes are natural sub-objects of generalized complex manifolds [11], and play a central role in string theory as the boundary conditions for open strings. On complex manifolds, branes are just holomorphic vector bundles with co-higgs fields and, on symplectic manifolds, they include flat vector bundles on Lagrangian submanifolds (they also include the coisotropic A-branes of Kapustin and Orlov [35], which are of special interest to me). In an ongoing project, I show (up to a certain notion of parity) that generalized complex branes are locally equivalent to holomorphic, flat Poisson modules supported on holomorphic coisotropic submanifolds of holomorphic Poisson manifolds. But these are precisely the local models of (resolutions of) bimodules over the deformation quantization given by Kontsevich [23]. I also find a global version of this characterization of branes which fits nicely with the holomorphic stack described in Section 5, and which presumably will verify the good criterion mentioned above once the deformation quantizations are fully understood. I stress that ordinary symplectic structures are included in this approach. If one were to quantize a symplectic manifold along these lines (by taking non-canonical choices of local holomorphic gauge), then one would get, as a category of bimodules, something different from the usual realsymplectic deformation quantization. This will allow us to see coisotropic A-branes as bimodules 7

8 over the weakly holomorphic quantization of a symplectic manifold. 5 Integration of generalized complex structures In this section I will describe my recent work with Gualtieri [8]. This will help us be precise about the holomorphic stacks referenced in previous sections, as well as set the stage for future projects discussed in Section 6. (Indeed, as disscussed previously, the methods of our integration paper have already found an application [28] with the first properly geometric description of generalized complex branes. Two major issues complicate the attempt to understand generalized complex structures through the lens of holomorphic geometry: First, the local complex structure with respect to which a generalized complex structure is holomorphic is not unique. For example, a 4k-real-dimensional symplectic structure is locally the imaginary part of many different 2k-complex-dimensional holomorphic symplectic structures, as we can see from Darboux coordinates. Second, the complex structure is not globally defined. Given the parity condition, one can cover a generalized complex manifold with local holomorphic charts, but the complex structures on two different charts may not agree on their intersection. Then where does this complex structure live, canonically, if not on the generalized complex manifold itself? As I explain below, it lives on the shifted symplectic stack [30] associated to the underlying real Poisson structure, which is a geometric model for the space of symplectic leaves. A related question is: what are the integrating objects of generalized complex manifolds? Many differential-geometric structures can be understood as the infinitesimal version of structures on Lie groupoids. This is in analogy with how Lie algebras are the infinitesimal versions of Lie groups (which are just Lie groupoids with a single object). For example, a Lie groupoid differentiates to a Lie algebroid (a vector bundle over the base of the groupoid, with a certain Lie bracket on its sections); or, a symplectic groupoid (a Lie groupoid with a multiplicative symplectic form) differentiates to a Poisson structure on its base. Crainic [36] and Weinstein [37] did previous work on the integration question for generalized complex structures, but a satisfactory answer was out of reach until my local holomorphicity result (Theorem 1) became available. Both questions about where the complex structure naturally lives, and the integration question, are answered in a recent paper by myself and Gualtieri. We call the integrating object of a generalized complex structure a weakly holomorphic symplectic groupoid, or, alternatively, a 1-shifted holomorphic symplectic stack. Definition. A weakly holomorphic symplectic groupoid, (G, E, Φ), consists of a real symplectic groupoid G, a holomorphic symplectic groupoid Φ, and a real symplectic Morita equivalence G E Φ between G and the imaginary part of Φ. A Lie groupoid, G, acts on its space of objects, M, also called its base. If G is a Lie groupoid with base M, then a Morita equivalence is a weak notion of equivalence between groupoids which, roughly speaking, preserves the geometry of the quotient space M/G. (For example, for the groupoid of paths-up-to-homotopy of a foliation, Morita equivalence preserves the leaf space.) An equivalence between two weakly holomorphic symplectic groupoids, (G, E, Φ) and (G, E, Φ ), consists of a strict equivalence G G of symplectic groupoids, but only a (holomorphic symplectic) Morita equivalence between Φ and Φ (satisfying the obvious commutative diagram). A Lie groupoid up to Morita equivalence is a differential stack. Thus, a weakly holomorphic symplectic groupoid should be understood as a real symplectic groupoid, equipped with a compatible complex structure on the associated stack M/G. 8

9 In analogy with symplectic groupoids, a weakly holomorphic symplectic groupoid differentiates in a certain way to a generalized complex structure on the base, M, of the real groupoid G. The main result of our recent paper [8] says: Theorem 2 (B.-Gualtieri, 2016). A (unique up to topology) weakly holomorphic symplectic groupoid integrating a generalized complex structure exists precisely when the associated real Poisson structure is integrable. Thus, an integrable generalized complex structure may be seen as a real Poisson structure, along with a compatible holomorphic structure on its integrating stack. This holomorphic stack is the object I would like to quantize, as I discussed in Section 4. In the course of proving Theorem 2, we develop a new formalism for Courant reduction (extending those of [38] to groupoid actions), as well as new techniques for integrating multiplicative structures to Lie groupoids with difficult topology. 6 Holomorphic stacks and derived geometry Theorems 1 and 2 can likely be generalized beyond the setting of generalized complex geometry, to that of stacks, higher stacks, and even derived geometry, giving a Newlander-Nirenberg theorem for these kinds of spaces stacks. To explain this generalization, I will describe a different perspective on the work described in Section 5. Pym and Safronov have also recently been investigating the connection between higher stacks and Courant algebroids, and it seems there is much territory to be explored here. As I said in Section 5, given a symplectic groupoid G integrating a Poisson manifold (M, π), specifying a complex structure on the differential stack M/G is equivalent to upgrading the Poisson structure to a generalized complex structure. One way to specify such a stacky complex structure is via a weakly holomorphic symplectic groupoid, that is, by giving a Morita-equivalent holomorphic symplectic groupoid. Alternatively, one could specify a stacky complex structure in a way more analogous with specifying an integrable almost complex structure on a manifold, as follows. Recall that associated to any Lie groupoid G with base manifold M is its infinitesimal counterpart, a Lie algebroid on M that is, a vector bundle L M with a Lie bracket on its space of sections and an anchor L T M, compatible with the brackets in a certain way. Given G and L, we have a tangent complex 0 L T M 0. (6.1) In derived geometry, this realizes the tangent bundle to the stack M/G as a formal quotient, T M/L. Then, to give an almost complex structure on M/G, one could specify a map of complexes I 1 : L L, I 0 : T M T M, along with homotopy from (I 1, I 0 ) to ( Id, Id). This homotopy involves extra data a map C : T M L and some relations. There is also a natural integrability condition. In the spcial case when L = T M and the( anchor ) π : T M T M is a Poisson structure, I0 π these conditions are precisely those such that is a generalized complex structure. C I 1 In general, given such data and conditions, we would like to know whether M/G is holomorphic in the first sense, i.e., if it is Morita equivalent to a holomorphic groupoid in a way compatible with the data. In the Poisson/generalized complex case the answer is yes, as I have explained 9

10 in Section 5. In the more general setting of Lie groupoids/algebroids, or even higher stacks and derived geometry, this should still be true (similar arguments should work), and would constitute something like a Newlander-Nirenberg theorem for derived stacks. There is some recent work by Millès on formal complex structures on manifolds [39] which presages these ideas in some respects, but this result has never been worked out. 7 A neighbourhood theorem for submanifolds, and blow-downs While the holomorphic stack of Section 5 does indeed describe the canonical holomorphic structure associated to a generalized complex structure, it is sometimes useful to have the holomorphic models of the manifold itself, non-canonical though they may be. As a general question, I would like to understand how the holomorphic Poisson structures sit in the moduli space of generalized complex structures. In particular, suppose S is a submanifold of a generalized complex manifold (M, I), and that we have a choice of gauge in a tubular neighbourhood of S for which I is holomorphic Poisson. This would allow us to transfer the full arsenal of surgeries about S from holomorphic Poisson geometry to generalized complex geometry. For example, we would like to have a blow-down construction. Blow-ups of generalized complex manifolds have been studied in [40] and [41] (for blow-ups at a point), and by myself and collaborators in [6] (for more general submanifolds). But, while a blow-up is fundamentally a local construction, a blow-down the reverse construction requires some kind of global understanding of the geometry around the exceptional divisor. Thus, it is useful in the construction of generalized complex manifolds to have semi-local classifications. This is a current project with Cavalcanti and van der Leer Duran. There are a number of expected obstructions to a submanifold S M having a holomorphic neighbourhood some geometric, and some involving Hodge theory on manifolds with boundary. We face many of the same analytic issues which come up for complex analysts when studying the deformation theory of neighbourhoods of complex submanifolds, and we are working to apply some of the same tools [42]. References [1] M. Bailey, On the local and global classification of generalized complex structures. PhD thesis, University of Toronto, [2] M. Bailey, Local classification of generalized complex structures, J. Differential Geom., vol. 95, pp. 1 37, [3] M. Bailey, Symplectic foliations and generalized complex structures, Canad. J. Math., vol. 66, no. 1, pp , [4] V. Apostolov, M. Bailey, and G. Dloussky, From locally conformally Kahler to bi-hermitian structures on non-kahler complex surfaces, Math. Res. Lett., vol. 22, no. 2, pp , [5] M. Bailey and M. Gualtieri, Local analytic geometry of generalized complex structures, Bulletin of the London Mathematical Society, vol. 49, no. 2, pp , [6] M. Bailey, G. R. Cavalcanti, and J. van der Leer Duran, Blow-ups in generalized complex geometry, Trans. Amer. Math. Soc., To appear. 10

11 [7] M. Bailey, G. Cavalcanti, and M. Gualtieri, Type one generalized calabi yaus, Journal of Geometry and Physics, vol. 120, no. Supplement C, pp , [8] M. Bailey and M. Gualtieri, Integration of generalized complex structures, ArXiv e-prints, Nov arxiv: [math.sg]. [9] N. Hitchin, Generalized Calabi-Yau manifolds, Q. J. Math., vol. 54, no. 3, pp , [10] M. Gualtieri, Generalized complex geometry. PhD thesis, University of Oxford, [11] M. Gualtieri, Generalized complex geometry, Ann. Math., vol. 174, no. 1, pp , [12] A. Kapustin and Y. Li, Topological sigma-models with H-flux and twisted generalized complex manifolds, Adv. Theor. Math. Phys., vol. 11, no. 2, pp , [13] O. Ben-Bassat, Mirror symmetry and generalized complex manifolds. I. The transform on vector bundles, spinors, and branes, J. Geom. Phys., vol. 56, no. 4, pp , [14] A. S. Cattaneo, J. Qiu, and M. Zabzine, 2D and 3D topological field theories for generalized complex geometry, Adv. Theor. Math. Phys., vol. 14, no. 2, pp , [15] P. Candelas, G. T. Horowitz, A. Strominger, and E. Witten, Vacuum configurations for superstrings, Nuclear Physics B, vol. 258, pp , [16] M. Graa, R. Minasian, M. Petrini, and A. Tomasiello, Generalized structures of n = 1 vacua, Journal of High Energy Physics, vol. 2005, no. 11, p. 020, [17] A. Weinstein, The local structure of poisson manifolds, J. Differential Geom., vol. 18, no. 3, pp , [18] M. Kontsevich, Homological algebra of mirror symmetry, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zurich, 1994), pp , Birkhauser, Basel, [19] E. Witten, Topological sigma models, Comm. Math. Phys., vol. 118, no. 3, pp , [20] J. Persson, T-duality and generalized complex geometry, J. High Energy Phys., no. 3, pp. 025, 18 pp. (electronic), [21] G. R. Cavalcanti and M. Gualtieri, Generalized complex geometry and T-duality, in A celebration of the mathematical legacy of Raoul Bott, vol. 50 of CRM Proc. Lecture Notes, pp , Amer. Math. Soc., Providence, RI, [22] M. Kontsevich, Deformation quantization of Poisson manifolds, I, in eprint arxiv:qalg/ , Sept [23] M. Kontsevich, Deformation quantization of algebraic varieties, Letters in Mathematical Physics, vol. 56, no. 3, pp , [24] A. KAPUSTIN, Topological strings on noncommutative manifolds, International Journal of Geometric Methods in Modern Physics, vol. 01, no. 01n02, pp ,

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