2.1 Divisors

Inspired by divisors in algebraic geometry, we will make use of the following notion to deal with singularities of geometric structures:

We will encounter three types of divisors in this article:

In general, the vanishing locus is only (the image of) an immersed submanifold with transverse intersections, and is naturally stratified by the multiplicity of its points. It is convenient to introduce some notation to refer to the different strata of the vanishing locus, so if we let, say, D, denote the vanishing locus of any of the divisors above, the set of points with multiplicity at least k will be denoted by $D(k)$ and the set of points with multiplicity precisely k will be denoted by $D[k]$ . Each of the $D(k)$ is itself an immersed submanifold, and hence we may speak of:

Given a complex log divisor $I_D$ , one may consider its associated elliptic divisor defined by the condition $I_{\left | D \right |}\otimes \mathbb {C} = I_D \otimes \overline {I}_D$ . In this case, $D[1]$ is automatically co-oriented by the differential of any local generator of $I_D$ . Conversely, given an elliptic divisor $I_{\left | D \right |}$ for which $D[1]$ is co-oriented, there exists a unique complex log divisor $I_{D}$ inducing $I_{\left | D \right |}$ and the given co-orientation.

2.2 Lie algebroids and residue maps

Using the Serre-Swann theorem, one may show that vector fields preserving the divisors from Definition 2.4 define vector bundles and the natural identification of sections of these vector bundles with vector fields makes these vector bundles into Lie algebroids:

As the divisors under consideration have local normal forms, so do have the associated Lie algebroids:

For each of the Lie algebroids above, we obtain a corresponding Lie algebroid de Rham complex on the exterior power of the dual vector bundle. The relation between complex and elliptic tangent bundles translates into the existence of maps between their corresponding de Rham complexes. Indeed, given a complex log divisor, $I_D$ , and associated elliptic divisor, $I_{\left | D \right |}$ , taking the imaginary part of a complex log form is a map $\Im ^* \colon (\Omega ^{\bullet }(\mathcal {A}_D),d_{\mathcal {A}_D}) \rightarrow (\Omega ^{\bullet }(\mathcal {A}_{\left | D \right |}),d_{\mathcal {A}_{\left | D \right |}})$ .

There are several residue maps associated to these Lie algebroids, obtained by taking the coefficients of certain singular generators. We quickly recall the ones we need in this article. Given a smooth elliptic divisor, with D co-orientable, it is explained in [Reference Cavalcanti and Gualtieri9] that there are residue maps:

$$ \begin{align*} \operatorname{\mathrm{Res}}_q : (\Omega^{\bullet}(\mathcal{A}_{\left| D \right|}),d_{\mathcal{A}_{\left| D \right|}}) \rightarrow (\Omega^{\bullet-2}(D),d), \qquad \alpha &\mapsto \iota_D^*(\iota_{r\partial_r\wedge\partial_{\theta}}\alpha)\\ \operatorname{\mathrm{Res}}_r : (\Omega^{\bullet}(\mathcal{A}_{\left| D \right|}),d_{\mathcal{A}_{\left| D \right|}}) \rightarrow (\Omega^{\bullet-1}(S^1ND),d), \qquad \alpha & \mapsto \iota_{r\partial_r}\alpha, \end{align*} $$

where for $\operatorname {\mathrm {Res}}_r$ , $S^1ND$ is the sphere bundle of the normal bundle of D and the Lie algebroid 1-form dual to $\partial _\theta $ is interpreted as fiberwise volume form on the bundle $S^1ND \to D$ .

Let $I_{\left | D \right |}$ be a general elliptic divisor with $D[1]$ co-orientable. The inclusion $\iota _{M\backslash D(2)} : M\backslash D(2) \rightarrow M$ , pulls the elliptic divisor $I_{\left | D \right |}$ back to a smooth elliptic divisor with vanishing locus $D[1]$ . Hence we may define residue maps by composition:

$$ \begin{align*} \operatorname{\mathrm{Res}}_q : \Omega^{\bullet}(\mathcal{A}_{\left| D \right|}) \rightarrow \Omega^{\bullet-2}(D[1]), \qquad \alpha &\mapsto \iota_{D[1]}^*(\iota_{r\partial_r\wedge\partial_{\theta}}(\iota_{M\backslash D(2)}^*\alpha))\\ \operatorname{\mathrm{Res}}_r : \Omega^{\bullet}(\mathcal{A}_{\left| D \right|}) \rightarrow \Omega^{\bullet-1}(S^1ND[1]), \qquad \alpha & \mapsto \iota_{r\partial_r}(\iota^*_{M\backslash D(2)}\alpha), \end{align*} $$

where $S^1ND[1]$ is the sphere bundle of the normal bundle of $D[1]$ . There exists several higher order residue maps, however defining these very quickly becomes a combinatorial task. As we are mainly interested in two-forms, we will define:

Definition 2.8. Let $I_{\left | D \right |}$ be an elliptic divisor, $\omega \in \Omega ^2(\mathcal {A}_{\left | D \right |})$ and let $p \in D[2]$ . Using the coordinates from Remark 2.7 we define

$$ \begin{align*} \operatorname{\mathrm{Res}}_{r_ir_j}\omega(p) = \omega_p(r_i\partial_{r_i},r_j\partial_{r_j}), \quad \operatorname{\mathrm{Res}}_{r_i\theta_j}\omega(p) = \omega_p(r_i\partial_{r_i},\partial_{\theta_j}), \quad \operatorname{\mathrm{Res}}_{\theta_i\theta_j}(\omega)(p) = \omega_p(\partial_{\theta_i},\partial_{\theta_j}). \end{align*} $$

Note that these residues are only well-defined up to sign. Similarly, one may define residue maps for the real log tangent bundle. If $I_Z$ is a smooth real log divisor, we may define

$$ \begin{align*} \operatorname{\mathrm{Res}} : \Omega^{\bullet}(\mathcal{A}_Z) \rightarrow \Omega^{\bullet-1}(Z), \quad \alpha \mapsto \iota^*_Z(\iota_{x\partial_x}\alpha). \end{align*} $$

Again, for a general real log divisor we may define $\operatorname {\mathrm {Res}}^1 : \Omega ^{\bullet }(\mathcal {A}_Z) \rightarrow \Omega ^{\bullet -1}(Z[1])$ by precomposing with the pullback $M\backslash D(2) \hookrightarrow M$ . The higher residues are again somewhat involved to define. For two-forms we define:

(2.1) $$ \begin{align} \operatorname{\mathrm{Res}}^2_{x_ix_j}(\omega)(p) = \omega_p(x_i\partial_{x_i},x_j\partial_{x_j}), \quad p \in Z[2]. \end{align} $$

2.3 Generalized complex structures

In this section we recall the definition of generalized complex structures and focus our attention on the class of stable generalized complex structures called stable. We will recall that these structures are in correspondence with symplectic structures on the elliptic tangent bundle.

Given a manifold M, we form $\mathbb {T}M = TM \oplus T^*M$ . This bundle is endowed with the natural symmetric pairing given by evaluation of forms on vectors.

Definition 2.9. A generalized complex structure on a manifold M, is a pair $(\mathbb {J},H)$ , where $H \in \Omega ^3(M)$ is a closed real 3-form and $\mathbb {J}$ is a complex structure on $\mathbb {T}M$ orthogonal with respect to the natural pairing and whose $+i$ -eigenbundle, $L \subset \mathbb {T}_{\mathbb {C}} M$ is involutive with respect to the Courant–Dorfmann bracket:

$$ \begin{align*} [\![ X+\xi, Y + \eta ]\!]_H = [X,Y] + \mathcal{L}_X\eta - \iota_Yd\xi + \iota_X\iota_YH, \quad X+ \xi,Y + \eta \in \mathbb{T}M. \end{align*} $$

Given a generalized complex structure one may decompose it as a block matrix with respect to the splitting of $\mathbb {T}M$ . The skew-symmetry of $\mathbb {J}$ ensures this decomposition is of the form

$$ \begin{align*} \mathbb{J} = \begin{pmatrix} J & \pi_{\mathbb{J}}^{\sharp} \\ \sigma^{\flat} & -J^*, \end{pmatrix}, \qquad J \in \text{End}(TM), \sigma \in \Omega^2(M), \pi_{\mathbb{J}} \in \mathfrak{X}^2(M). \end{align*} $$

The bivector $\pi _{\mathbb {J}}$ is Poisson. Two generalized complex structures $(\mathbb {J}_1,H_1),(\mathbb {J}_2,H_2)$ are said to be gauge equivalent, if there exists a two-form $B \in \Omega ^2(M)$ such that $dB = H_1-H_2$ and

$$ \begin{align*} \mathbb{J}_2 = \begin{pmatrix} 1 & 0 \\ B^{\flat} & 1 \end{pmatrix} \mathbb{J}_1 \begin{pmatrix} 1 & 0 \\ -B^{\flat} & 1 \end{pmatrix}. \end{align*} $$

If two generalized complex structures $\mathbb {J}_1,\mathbb {J}_2$ are gauge equivalent, then $\pi _{\mathbb {J}_1} = \pi _{\mathbb {J}_2}$ .

The Clifford action of $X + \xi \in \mathbb {T}_{\mathbb {C} }M$ on $\rho \in \wedge ^nT_{\mathbb {C} }M$ is defined via $(X+\xi ) \cdot \rho = \iota _X \rho + \xi \wedge \rho $ . Given a generalized complex structure, one may define its canonical bundle $K \subset \wedge ^{\mathrm {top}}T_{\mathbb {C} }M$ via

$$ \begin{align*} L = \lbrace u \in \mathbb{T}_{\mathbb{C}}M : u\cdot K = 0 \rbrace. \end{align*} $$

The dual bundle $K^*$ has a natural section s, called the anticanonical section which is defined via $s : \Gamma (K) \rightarrow C^{\infty }(M;\mathbb {C} ), \rho \mapsto \rho _0$ .

Example 2.11. Let $(M,J,\pi )$ be a holomorphic Poisson manifold with $\pi = \pi _R + i \pi _I$ and denote by $\pi _I^\sharp \colon T^*M \to TM$ the map associated to $\pi _I$ . Then

$$\begin{align*}\mathbb{J}_{J,\pi} := \begin{pmatrix} -J &4 \pi_I^{\sharp} \\ 0 & J^* \end{pmatrix} \end{align*}$$

defines a generalized complex structure. If furthermore $\wedge ^n\pi (\Gamma (\wedge ^{\mathrm {top}}(T^*M)) \subset C^{\infty }(M;\mathbb {C} )$ defines a complex log divisor, $\mathbb {J}_{J,\pi }$ defines a stable generalized complex structure. Holomorphic Poisson structures with these properties are called log symplectic.

These generalized complex structures can be described as symplectic structures on Lie algebroids:

2.4 Lifting generalized complex structures

Lifting Poisson structures to Lie algebroids is a well-established field of study. However, lifts of generalized complex and Dirac structures to Lie algebroids have not yet received much attention. As we will see, the questions of which structures lift to Lie algebroids and which structures descend from Lie algebroids arise naturally. Before we delve into the details of the theory of lifting, it can be useful to see a few examples that illustrate the issue.

Example 2.13. Consider $\mathbb {C} ^2$ with elliptic divisor $I_{|D|} = \langle |z_1|^2|z_2|^2\rangle $ . Then using polar coordinates, $z_j = r_je^{i\theta _j}$ , the elliptic tangent bundle has a global frame given by

$$\begin{align*}\{r_1\partial_{r_1}, \partial_{\theta_1},r_2\partial_{r_2}, \partial_{\theta_2}\}.\end{align*}$$

If we endow $\mathbb {C} ^2$ with the Poisson structure

$$\begin{align*}\pi = \Im(z_1z_2\partial_{z_1}\wedge\partial_{z_2}) = r_1\partial_{r_1}\wedge\partial_{\theta_2} - r_2\partial_{r_2}\wedge \partial_{\theta_1},\end{align*}$$

we see that the last expression makes sense as a nondegenerate bivector $\pi _{\mathcal {A}_{|D|}}$ on $\mathcal {A}_{|D|}$ , which we view as a lift of $\pi $ to $\mathcal {A}_{|D|}$ as it satisfies the equation

$$\begin{align*}\rho(\pi_{\mathcal{A}_{|D|}}) = \pi.\end{align*}$$

Similarly, the standard complex structure on $\mathbb {C} ^2$ , defined by $I \partial _{x_i} = \partial _{y_i}$ satisfies $I r_i \partial _{r_i} = \partial _{\theta _i}$ . Therefore we can regard the latter as a bundle map $I_{\mathcal {A}_{\left | D \right |}}\colon \mathcal {A}_{\left | D \right |}\to \mathcal {A}_{\left | D \right |}$ satisfying

$$\begin{align*}\rho \circ I_{\mathcal{A}_{\left| D \right|}} = I \circ \rho.\end{align*}$$

This makes $I_{\mathcal {A}_{\left | D \right |}}$ a lift of I to $\mathcal {A}_{\left | D \right |}$ .

2.4.1 Severa class

Let us first shortly study Courant algebroids where we replace the role of the tangent bundle by a general Lie algebroid. Given any Lie algebroid $\mathcal {A}$ and $H \in \Omega ^3(\mathcal {A})$ closed one may define the standard $\mathcal {A}$ -Courant algebroid, in the usual manner. We set $\mathbb {A} = \mathcal {A}\oplus \mathcal {A}^*$ and for $X+\xi ,Y+\eta \in \Gamma (\mathcal {A}\oplus \mathcal {A}^*)$ .

$$ \begin{align*} \left\langle X+\xi,Y+ \eta \right\rangle &= \frac{1}{2}(\eta(X) + \xi(Y)),\\ [\![ X+\xi,Y+\eta ]\!]_H &= [X,Y] + \mathcal{L}_X\eta - \iota_Yd\xi +\iota_X\iota_YH. \end{align*} $$

This notion, which is a particular case of a Lie bialgebroid, can be axiomatised as follows:

Definition 2.14. An exact $\boldsymbol {\mathcal {A}}$ -Courant algebroid is a vector bundle extension

(2.2) $$ \begin{align} 0 \rightarrow \mathcal{A}^* \overset{\pi^*}{\rightarrow} \mathbb{A} \overset{\pi}{\rightarrow} \mathcal{A} \rightarrow 0, \end{align} $$

together with a non-degenerate pairing $\left \langle \cdot ,\cdot \right \rangle $ on $\mathbb {A}$ , and a bracket $[\![ \cdot ,\cdot ]\!]$ on $\Gamma (\mathbb {A})$ satisfying:

• ○ $[\![ e_1,[\![ e_2,e_3 ]\!] ]\!] = [\![ [\![ e_1,e_2 ]\!],e_3 ]\!] + [\![ e_2,[\![ e_1,e_3 ]\!] ]\!]$

• ○ $\pi ([\![ e_1,e_2 ]\!]) = [\pi (e_1),\pi (e_2)]_{\mathcal {A}}$

• ○ $[\![ e_1,fe_2 ]\!] = f[\![ e_1,e_2 ]\!] + \mathcal {L}_{\rho (\pi (e_1))}(f)e_2$

• ○ $\mathcal {L}_{\rho (\pi (e_1))}(\left \langle e_2,e_3 \right \rangle ) = \left \langle [\![ e_1,e_2 ]\!],e_3 \right \rangle + \left \langle e_2,[\![ e_1,e_3 ]\!] \right \rangle $

• ○ $[\![ e_1,e_1 ]\!] = \pi ^*d_{\mathcal {A}}\left \langle e_1,e_1 \right \rangle $ .

Given an exact $\mathcal {A}$ -Courant algebroid, the composition $\rho \hspace {0.1 mm} \circ \hspace {0.1 mm} \pi : \mathbb {A} \rightarrow TM$ together with $\left \langle \cdot ,\cdot \right \rangle $ and $[\![ \cdot ,\cdot ]\!]$ will give $\mathbb {A}$ the structure of a non-exact Courant algebroid.

Just as for ordinary exact Courant algebroids, isotropic splittings $s : \mathcal {A} \rightarrow \mathbb {A}$ of (2.2) are classified by closed three-forms $H \in \Omega ^3(\mathcal {A})$ appearing as the curvature

$$ \begin{align*} \iota_X\iota_YH = s^*[s(X),s(Y)]. \end{align*} $$

The difference between two splittings $s-s'$ , defines a two-form $B \in \Omega ^2(\mathcal {A})$ via $(s-s')(X) = \iota _XB$ . Consequently $[H] \in H^3(\mathcal {A})$ is independent of the isotropic splitting and determines the $\mathcal {A}$ -Courant algebroid structure completely. Hence, this cohomology class is the natural extension of the Severa-class of exact Courant algebroids.

2.4.2 Lifting Dirac structures

The answer to the question on how to lift structures takes different forms depending on what one is trying to lift. Here we will focus on Dirac and generalized complex structures, whose definitions on $\mathbb {A}$ is completely analogous to the definition on $\mathbb {T}M$ :

Definition 2.15. Let $\mathcal {A} \rightarrow M$ be a Lie algebroid. An $\boldsymbol {\mathcal {A}}$ -Dirac structure is a subbundle $E_{\mathcal {A}} \subset \mathbb {A}$ together with a three-form $H \in \Omega ^3_{\mathrm {cl}}(\mathcal {A})$ , such that $E_{\mathcal {A}}$ is isotropic with respect to the pairing and involutive with respect to the bracket on $\mathbb {A}$ .

An $\boldsymbol{\mathcal{A}}$ -generalized complex structure is a pair $(\mathcal {J}_{\mathcal {A}},H_{\mathcal {A}})$ where $H_{\mathcal {A}} \in \Omega ^3_{\mathrm {cl}}(\mathcal {A})$ and $\mathcal {J}_{\mathcal {A}}$ is an orthogonal complex structure $\mathcal {J}_{\mathcal {A}}$ on $\mathbb {A}$ for which $\mathbb {A}^{1,0}$ , the $+i$ -eigenbundle, defines an $\mathbb {A}_{\mathbb {C} }$ -Dirac structure.

Definition 2.16. Let $\mathcal {A} \rightarrow M$ be a Lie algebroid. An $\mathcal {A}$ -Dirac structure $(E_{\mathcal {A}},H_{\mathcal {A}})$ is a lift of a Dirac structure $(E,H)$ on M if $H_{\mathcal {A}} = \rho ^*H$ and $\mathfrak {F}_{\rho }(E_{\mathcal {A}}) = E$ .

Here $\mathfrak {F}_{\rho }$ denotes the Dirac pushforward via the anchor map.

This notion matches the notion of a lift of a Poisson structure, when that is considered as a Dirac structure. Indeed, given a Lie algebroid $\mathcal {A} \rightarrow M$ , a Poisson structure $\pi _{\mathcal {A}}\in \Gamma (\wedge ^2\mathcal {A})$ (that is $[\pi _{\mathcal {A}},\pi _{\mathcal {A}}] =0$ ) and a Poisson structure $\pi \in \mathfrak {X}^2(M)$ one can readily see that $\rho _*\pi _{\mathcal {A}} = \pi $ if and only if $E_{\pi _{\mathcal {A}}} = \mathrm {graph}(\pi _{\mathcal {A}}) \subset \mathbb {A}$ is a lift of $E_{\pi } = \mathrm {graph}(\pi )$ .

For generalized complex structures, however, working only with the push-forward map is too restrictive, so the notion of lift requires some tweaking.

Definition 2.17. Let $\rho \colon \mathcal {A} \to TM$ be a Lie algebroid over M

• ○ $\tilde X+ \tilde \xi \in \mathbb {A}$ is related to $X + \xi \in \mathbb {T}M$ if $\rho (\tilde X) = X$ and $\rho ^*\xi = \tilde \xi $ .

• ○ A generalized complex structure $(\mathcal {J}_{\mathcal {A}},H_{\mathcal {A}})$ on $\mathbb {A}$ is a lift of a generalized complex structure $(\mathcal {J},H)$ on M if whenever $\tilde X+ \tilde \xi \in \mathbb {A}$ is related to $X + \xi \in \mathbb {T}M$ , $\mathcal {J}_{\mathcal {A}} (\tilde X+ \tilde \xi )$ is related to $\mathcal {J} (X + \xi )$ .

Example 2.18. In the situation above, if $J \colon TM \rightarrow TM$ and $J_{\mathcal {A}} \colon \mathcal {A} \to \mathcal {A}$ are complex structures we can promote them to generalized complex structures and ask when $J_{\mathcal {A}}$ is a lift of J. It turns out this happens if and only if the following diagram commutes

which agrees with what one would like to call the lift of a complex structure.

Remark 2.19. In this paper we will work with Lie algebroids $\mathcal {A}$ for which the locus U where $\rho \colon \mathcal {A} \to TM$ is an isomorphism of vector bundles is dense in M. In this case, we see that $\mathcal {J}_{\mathcal {A}}$ is a lift of $\mathcal {J}$ on U if and only if $\mathcal {J}_{\mathcal {A}}|_U= (\rho |_U) \hspace {0.1 mm} \circ \hspace {0.1 mm} \mathcal {J}\hspace {0.1 mm} \circ \hspace {0.1 mm} (\rho |_U)^{-1}$ . If that holds, by continuity $\mathcal {J}_{\mathcal {A}}$ is related to $\mathcal {J}$ in all of M. Therefore we see that the question of whether $\mathcal {J}$ admits a lift to a Lie algebroid with dense isomorphism locus becomes the question of existence of a smooth extension of $(\rho |_U) \hspace {0.1 mm} \circ \hspace {0.1 mm} \mathcal {J}\hspace {0.1 mm} \circ \hspace {0.1 mm} (\rho |_U)^{-1}$ beyond the isomorphism locus.

Example 2.20. It is also good to keep in mind that there are (generalized) complex structures on Lie algebroids, which do not arise as lifts of structures on M. Consider, for example, the elliptic tangent bundle $\mathcal {A}_{\left | D \right |}$ on $\mathbb {C} ^2$ , associated to the ideal $\left \langle r_1^2r_2^2 \right \rangle $ . The following defines a complex structure on $\mathcal {A}_{\left | D \right |}$ :

$$ \begin{align*} z_1\partial_{z_1} \mapsto z_2\partial_{z_2},& \quad \overline{z_1}\partial_{\overline{z_1}} \mapsto \overline{z_2}\partial_{\overline{z_2}},\\ z_2\partial_{z_2} \mapsto -z_1\partial_{z_1},& \quad \overline{z_2}\partial_{\overline{z_2}} \mapsto -\overline{z_1}\partial_{\overline{z_1}}. \end{align*} $$

However, the induced complex structure on $\mathbb {C} ^2\setminus D$ does not extend over D, as we have, for example,

$$ \begin{align*} \partial_{z_1} \mapsto \frac{z_2}{z_1} \partial_{z_2}. \\[-24pt]\end{align*} $$

3.1 Haefliger–Salem classification

The first ingredient of the classification is the following result:

Here we say that $f \in C^{\infty }(B)$ if $\pi ^*f$ is smooth on M. Similarly, we define $\Omega ^{\bullet }(B)$ to correspond to the basic forms on M:

$$\begin{align*}\Omega^{\bullet}_{\mathrm{bas}}(M)=\{\xi \in \Omega^\bullet(M)\colon \iota_X \xi = \mathcal{L}_X \xi =0, \text{ for all infinitesimal generators } X \} \end{align*}$$

When fixing an effective $T^k$ -action on a manifold M it is possible to classify all other such actions which are locally equivalent to M:

Since we will need to get our hands on the cohomology classes prescribed by the proposition, we give a sketch of the proof. Notice that there is no preferred manifold endowed with a torus action locally equivalent to that of M and the class in question measures how different from M another such manifold is.

3.2 Torus actions, divisors and connections

We will lift torus actions to the elliptic tangent bundle in such a way that the associated infinitesimal action becomes free. The required notion we need is the following:

These assumptions allow us to lift to the elliptic tangent bundle:

Proposition 3.9. Let $I_{\left | D \right |}$ be an elliptic divisor compatible with a $T^k$ -action on M. Then there exists an injective Lie algebroid homomorphism $\tilde {a} : \mathfrak {t}^k \rightarrow \mathcal {A}_{\left | D \right |}$ such that the following diagram commutes:

where $a\colon \mathfrak {t}^k \to TM$ is the infinitesimal $T^k$ -action on M.

Proof. The existence of the Lie algebroid map is a local statement, so we can work in neighbourhoods of points.

Given a point p where the action is free, there are two possibilities, either $p\not \in D$ or $p\in D$ . In the first case, $\mathcal {A}$ isomorphic to $TM$ via the anchor map in a neighbourhood of p and the action has a unique lift to $\mathcal {A}_{\left | D \right |}$ . If $p \in D$ , because the action preserves the ideal $\mathcal {A}_{\left | D \right |}$ , then the orbit of p lies within the same stratum as p, say $D[i]$ and because the action is free, we get a rank k subbundle $a(\mathfrak {t}^k) \subset TD[i] \hookrightarrow \mathcal {A}_{|D|}$ , and therefore a has a lift, which is unique by continuity.

Given a point where the action has nontrivial isotropy, say $T^l\subset T^k$ , we can split $T^k = T^l \times T^{k-l}$ . By the previous argument the action of $T^{k-l}$ lifts to $\mathcal {A}_{|D|}$ and the lift is made of a subspace that maps injectively to $TM$ via the anchor. Now we focus on the action of $T^l$ . Since p is fixed by the action of $T^l$ and the action is effective we can linearise it and assume that, in appropriate coordinates (with values in $\mathbb {C} ^m \times \mathbb {R} ^{n-2m}$ ), the action is given by

$$\begin{align*}(\theta_1,\dots, \theta_l)\cdot (z_1,\dots, z_m,x_1,\dots, x_{n-2m}) \mapsto (e^{i(\sum_j a_{1,j}\theta_j)}z_1,\dots, e^{i(\sum_j a_{m,j}\theta_j)}z_m,x_1,\dots,x_{n-2m}), \end{align*}$$

where $m \geq l$ and the matrix $A = (a_{i,j})_{ij}$ has integral coefficients. Because all isotropy groups are connected A has an $l\times l$ submatrix with determinant $\pm 1$ .

Suppose f is a local generator of the elliptic divisor $I_{\left | D \right |}$ . Since D has codimension 2, the locus where f is nonzero is connected and hence we may take f to be non-negative, vanishing only at D. Since f is non-negative, by taking its average over the $T^l$ action, we obtain another nonnegative generator of the ideal, still denoted by f, which is invariant under the $T^l$ action. The invariant polynomials of the $T^l$ -action are generated by $(r_1^2,\ldots ,r_m^2,x_1,\ldots ,x_{n-2m})$ , and hence a result by Schwarz [Reference Schwarz24] implies that f must be of the form $f = g(r_1^2,\ldots ,r_m^2,x_1,\ldots ,x_{n-2m})$ , for some smooth g. Because f has this form, it is immediate that the vector fields $\partial _{\vartheta _1},\ldots ,\partial _{\vartheta _m}$ preserve the ideal generated by f, where we use polar coordinates for $z_i$ : $z_i = r_i e^{i\vartheta _i}$ . Hence $\partial _{\vartheta _i}$ give rise to (non-vanishing) elements of $\mathcal {A}_{\left | D \right |}$ . Because the vector fields generating the $T^k$ -action are given by $A(\partial _{\theta _1}),\ldots ,A(\partial _{\theta _l}) \in \text { span }\langle \partial _{\vartheta _1},\ldots ,\partial _{\vartheta _m}\rangle $ , and A has maximal rank, we see that these vector fields span a rank l-subspace of $\mathcal {A}_{\left | D \right |}$ which over D are in the kernel of the anchor map and hence are independent from the generators corresponding to the $T^{k-l}$ action.

Studying general torus actions goes beyond the scope of this article. For instance, one problem which occurs is that there is too much freedom in choosing a compatible elliptic divisor:

We restrict ourselves to a very well-behaved class of singular torus actions, but still interesting enough to allow for many examples:

Definition 3.11. We say that a $T^k$ -action on a manifold $M^{n}$ is standard if each orbit has a neighbourhood U such that

$$ \begin{align*} U \simeq T^{k-l} \times \mathbb{C}^l \times \mathbb{R}^m \qquad\text{ for some } k,l,m \text{ such that } k+l+m=n, \end{align*} $$

in which the action is given by

$$ \begin{align*} (e^{i\theta_1},\ldots,e^{i\theta_k})&\cdot (\varphi_1,\ldots,\varphi_{k-l}, z_1,\ldots,z_l,x_1,\ldots, x_m) \\&= (e^{i\theta_1}\varphi_1,\ldots,e^{i\theta_{k-l}}\varphi_{k-l},e^{i\theta_{k-l+1}}z_1,\ldots,e^{i\theta_k}z_l,x_1,\ldots, x_m), \end{align*} $$

possibly after changing the generators of $T^k$ .

For general $T^k$ -actions, deriving conditions when they are standard is not simple. However, for full torus actions one can show the following:

The local normal form of a standard $T^k$ -action immediately implies the following:

In particular there is a canonical compatible elliptic divisor associated to a standard $T^k$ -action. Therefore, in what follows when given a standard $T^k$ -action on M, $I_{\left | D \right |}$ will always denote the canonical associated elliptic divisor.

Thanks to Proposition 3.9 and 3.13 a standard $T^k$ -action lifts to the associated Lie algebroid and so does the quotient map. Therefore, we obtain a diagram as follows:

Hence, we obtain a short exact sequence of vector bundles over M:

$$ \begin{align*} 0 \rightarrow \ker d\tilde{\pi} \rightarrow \mathcal{A}_{\left| D \right|} \rightarrow \pi^*\mathcal{A}_{\partial_B} \rightarrow 0 \end{align*} $$

We may quotient by $T^k$ , to obtain the sequence of vector bundles over B:

$$ \begin{align*} 0 \rightarrow B \times \mathfrak{t}^k \rightarrow \mathcal{A}_{\left| D \right|}/T^k \rightarrow \mathcal{A}_{\partial_B} \rightarrow 0 \end{align*} $$

One may think of this sequence as an $\mathcal {A}_{\left | D \right |}$ version of the Atiyah sequence for principal bundles. With this in mind, vector bundle splittings of this sequence give rise to the notion of connections for the $T^k$ -action on M. Concretely, in terms of differential forms we have:

3.3 Smoothness of the curvature

Given a standard $T^k$ -action and a corresponding connection one-form $\Theta \in \Omega ^1(\mathcal {A}_{\left | D \right |};\mathfrak {t}^k)$ , the curvature, $d\Theta $ , is basic and descends to a form in $\Omega ^2(\mathcal {A}_{\partial B};\mathfrak {t}^k)$ . We saw for complex line bundles that there are connections for which the curvature was in fact a smooth form. This is a general phenomenon for standard torus actions. To prove this, we need to understand the cohomology of the log-tangent bundle on the base. Below we use the residue maps introduced in Definition 2.8 and the discussion thereafter.

Proposition 3.18 (Theorem 29 in [Reference Miranda and Scott19],7.24 in [Reference Gualtieri, Li, Pelayo and Ratiu14])

Let $(M,I_Z)$ be a manifold endowed with real log divisor. Then there exists an isomorphism

(3.1) $$ \begin{align} H^{\bullet}(\mathcal{A}_Z) \simeq H^{\bullet}(M) \oplus H^{\bullet-1}(Z[1]) \oplus H^{\bullet-2}(Z[2]) \oplus \cdots, \end{align} $$

where $Z[n]$ denotes the intersection stratification of Z. The maps $H^{\bullet }(\mathcal {A}_Z) \rightarrow H^{\bullet -i}(Z[i])$ are induced by the residue maps $\operatorname {\mathrm {Res}}^i$ , and the map $H^{\bullet }(M) \rightarrow H^{\bullet }(\mathcal {A}_Z)$ is the pullback via the anchor.

Using (3.1) we see that we can associate a well-defined smooth cohomology class to a class in $H^{\bullet }(\mathcal {A}_Z)$ . Explicitly we may also phrase this as follows:

Recall from Lemma 3.13 that the quotient map of a standard $T^k$ -action induces a Lie algebroid submersion $\pi : \mathcal {A}_{\left | D \right |} \rightarrow \mathcal {A}_{\partial B}$ . This map is compatible with the residues in the following manner.

Lemma 3.20. Let $\pi : M \rightarrow B$ be endowed with a standard $T^k$ -action. Then for $i = 1,2$ the following diagram commutes:

As $D[2]$ is a discrete set of points, $\operatorname {\mathrm {Res}}^2_r\omega $ is defined to be $\operatorname {\mathrm {Res}}_{r_1r_2}(\omega )(p)$ . A similar result holds for higher order residues, but is more involved to state.

Using the description of the cohomlogy of $\mathcal {A}_{\partial B}$ , we can show that the curvature of a connection, $d\Theta $ , is cohomologically smooth:

The following example shows that Lemma 3.21 is not a given when the elliptic divisor and action are independent:

3.3.1 Local description of connection one-forms

For future reference we recall the local description for connection one-forms on principal bundles:

Proposition 3.23. Let $\pi : P \rightarrow B$ be a principal G-bundle, and let $h_i : \pi ^{-1}(U_i) \rightarrow U_i \times G$ be local trivializations for $i=1,2$ . Let $\Theta \in \Omega ^1(P;\mathfrak {g})$ be a connection one-form on P. Then

$$ \begin{align*} (h_i^{-1})*\Theta = \omega_{\mathrm{MC}} + \pi^*\alpha_i, \end{align*} $$

with $\omega _{\mathrm {MC}} \in \Omega ^1(G;\mathfrak {g})$ the Maurer-Cartan form, and $\alpha _i \in \Omega ^1(U_i;\mathfrak {g})$ . Moreover, if we let $f_{ji} \in C^{\infty }(U_i\cap U_j;G)$ be the transition functions given by $h_j^{-1} \hspace {0.1 mm} \circ \hspace {0.1 mm} h_i(x,g) = (x,f_{ji}(x)\cdot g)$ , we have

$$ \begin{align*} \alpha_i - \mathrm{Ad}^*_{f_{ji}}\alpha_j = \pi^*d\log f_{ji}. \end{align*} $$

3.4 Cech–de Rham isomorphism

To continue, we need to recall a basic version of the Čech-de Rham isomorphism.

Let M be endowed with a proper group action by some group G. Let $\Omega ^n_{\operatorname {\mathrm {bas}}}$ denote the sheaf of basic differential forms and let $Z^n_{\mathrm {bas}}$ denote the sheaf of closed basic differential forms. Both are sheaves of $C^{\infty }(M)_{\mathrm {bas}}$ -modules and fit into the short exact sequence

(3.2) $$ \begin{align} 0 \rightarrow Z^n_{\mathrm{bas}} \rightarrow \Omega^{n}_{\operatorname{\mathrm{bas}}} \rightarrow Z^{n+1}_{\mathrm{bas}} \rightarrow 0. \end{align} $$

Consider the basic Čech cocycle groups $C_{\operatorname {\mathrm {bas}}}^k(\mathcal {U},\mathcal {F})$ , for which the maps $U_{i_1\ldots i_k} \rightarrow \mathcal {F}$ are invariant with respect to the G-action on $U_{i_1\ldots i_k}$ .

By existence of a G-invariant partition of unity on M, the sheaves $\Omega ^n_{\mathrm {bas}}$ are fine, and hence $H^i_{\operatorname {\mathrm {bas}}}(M,\Omega ^n_{\mathrm {bas}}) = \lbrace 0 \rbrace $ for all $i \geq 1$ . Therefore the connecting morphism of the above exact sequence induces an isomorphism $H^i_{\operatorname {\mathrm {bas}}}(M,Z^{n+1}_{\mathrm {bas}}) \rightarrow H^i_{\operatorname {\mathrm {bas}}}(M,Z^n_{\mathrm {bas}})$ . Applying this consecutively, one obtains the isomorphism between $H^{i+1}_{\operatorname {\mathrm {bas}}}(M,\mathbb {R} ) \rightarrow H^1_{\operatorname {\mathrm {bas}}}(M,Z^i_{\operatorname {\mathrm {bas}}})$ . As the latter corresponds to $H^{i+1}_{dR,\operatorname {\mathrm {bas}}}(M)$ one obtains the desired isomorphism.

We also consider a basic version of the exponential sequence:

(3.3) $$ \begin{align} 0 \rightarrow \mathbb{Z}^n \rightarrow C^{\infty}_{\mathrm{bas}}(M;\mathbb{R}^n) \rightarrow C^{\infty}_{\operatorname{\mathrm{bas}}}(M;T^n) \rightarrow 0. \end{align} $$

The long exact sequence will induce an isomorphism $\check {H}^1_{\operatorname {\mathrm {bas}}}(M,C^{\infty }_{\operatorname {\mathrm {bas}}}(M;T^n)) \rightarrow \check {H}^2_{\mathrm {bas}}(M;\mathbb {Z}^n)$ . Combining this with the Čech-de Rham isomorphism we obtain a map to $H^2_{\mathrm {bas}}(M;\mathbb {Z}^n)$ .

3.5 Connection approach to Haefliger-Salem class

We now have all the theory in place to compare the curvature of a connection for a standard $T^k$ -action with the Haefliger-Salem class:

The description of the Haefliger-Salem class in terms of connections has the following consequence:

The importance of this lemma is that in the classification of standard $T^k$ -actions, it is now possible to pick as a reference an action $M_0 \rightarrow B$ with a flat connection. Using this reference, and a locally equivalent action $M' \rightarrow B$ , we find that the Haefliger-Salem class of $M_0$ with respect to $M'$ is given exactly by $[d\Theta ']$ for some connection $\Theta '$ on $M'$ .

When $M \rightarrow B$ is a principal $T^k$ -action, the reference action is simply the trivial bundle $B \times T^k$ . Contrary to in the principal case, there might be several non locally equivalent standard $T^k$ -actions over the same base with flat connections. We will see examples of this in Section 5.2.1.

5.1 T-duality

In this subsection we will recall the notions of T-duality for principal torus actions. We will be brief and refer to [Reference Bouwknegt, Evslin and Mathai3] and [Reference Cavalcanti and Gualtieri8] for more details.

Definition 5.1. Let M and $\hat {M}$ be principal $T^k$ -bundles with base B and let $H \in \Omega ^3_{T^k}(M),\hat {H} \in \Omega ^3_{T^k}(\hat {M})$ be $T^k$ -invariant closed three-forms. Consider the fibre-product, $M\times _B \hat {M}$ , and the following diagram:

We say that $(M,H)$ and $(\hat {M},\hat {H})$ are T-dual if there exists a $T^{2k}$ -invariant two-form $F \in \Omega ^2_{T^{2k}}(M\times _B\hat {M})$ such that

$$ \begin{align*} dF = p^*H - \hat{p}^*\hat{H}, \end{align*} $$

and the restriction

$$ \begin{align*} F : \mathfrak{t}_M \times \mathfrak{t}_{\hat{M}} \rightarrow \mathbb{R}, \end{align*} $$

where $\mathfrak {t}_M,\mathfrak {t}_{\hat {M}}$ denotes the tangent space to the fibres of p, $\hat {p}$ respectively, is non-degenerate. We call $M \times _B \hat {M}$ the correspondence space.

T-duals always exist, provided the three-form is integral:

T-dual spaces have isomorphic complexes of invariant differential forms:

Theorem 5.3 [Reference Bouwknegt, Evslin and Mathai3]

Let $(M,H)$ and $(\hat {M},\hat {H})$ be T-dual with $dF = p^*H -\tilde {p}^*\hat {H}$ . Then

$$ \begin{align*} \tau :(\Omega^{\bullet}_{T^k}(M),d_H) \rightarrow (\Omega^{\bullet}_{T^k}(\hat{M}),d_{\hat{H}}), \quad \rho \mapsto \int_{T^k}e^F \wedge p^*\rho, \end{align*} $$

is an isomorphism of cochain complexes. Here the integral is over the fibres of $\tilde {p}$ .

5.2 Courant algebroids

T-duality induces an isomorphism of invariant sections of the standard Courant algebroid:

Theorem 5.4 [Reference Cavalcanti and Gualtieri8]

If $(M,H)$ and $(\hat {M},\hat {H})$ are T-dual, then there is an isomorphism of Courant algebroids

$$ \begin{align*} \varphi : (TM \oplus T^*M)/T^k \rightarrow (T\hat{M} \oplus T^*\hat{M})/T^k, \end{align*} $$

such that $\tau (v\cdot \rho ) = \varphi (v) \cdot \tau (\rho )$ , for all $v\in (TM \oplus T^*M)/T^k$ and $\rho \in \Omega ^{\bullet }(M)$ .

The strength of this description is that it immediately allows one to transport structures:

We are now ready to introduce the notion of T-duality for standard torus actions:

Definition 5.6. Let $M,\hat {M}$ be two manifolds endowed with standard $T^k$ -actions over the same base B and let $I_{\left | D \right |},I_{\left | \hat {D} \right |}$ be the induced elliptic divisors. Furthermore, suppose we are given $H \in \Omega ^3_{\mathrm {cl}}(\mathcal {A}_{\left | D \right |}), \hat {H} \in \Omega ^3_{\mathrm {cl}}(\mathcal {A}_{\left | \hat {D} \right |})$ closed and $T^k$ -invariant. We consider the following diagram:

We say that $(M,H)$ and $(\hat {M},\hat {H})$ are T-dual if there exists a $T^{2k}$ -invariant two-form $F \in \Omega ^2(\mathcal {A}_{\left | D \right |} \times _B \mathcal {A}_{\left | \hat {D} \right |})$ such that $p^*H-\hat {p}^*\hat {H} = dF$ and

(5.1) $$ \begin{align} F : \mathfrak{t}_{M} \otimes \mathfrak{t}_{\hat{M}} \rightarrow \mathbb{R}, \end{align} $$

is non-degenerate.

Let us expand on this definition. First $\mathcal {A}_{\left | D \right |}\times _B \mathcal {A}_{\left | \hat {D} \right |}$ is the Lie algebroid induced by viewing $M \times _B \hat {M}$ as an $(\mathcal {A}_{\left | D \right |}\boxtimes \mathcal {A}_{\left | \hat {D} \right |})$ -adapted subspace of $M \times \hat {M}$ . The two-form F is a form for this Lie algebroid, and can in practice be thought of as the restriction of a two-form $\tilde {F} \in \Omega ^2(\mathcal {A}_{\left | D \right |}\boxtimes \mathcal {A}_{\left | \hat {D} \right |})$ . Here $\mathfrak {t}_{M_i}$ denotes the kernel of the Lie algebroid submersion $\mathcal {A}_{\left | D \right |}\times _B \mathcal {A}_{\left | \hat {D} \right |} \rightarrow \mathcal {A}_{\left | D_i \right |}$ induced by $p_i$ .

A connection one-form $\Theta \in \Omega ^1(\mathcal {A}_{\left | D \right |};\mathfrak {t})$ induces an isomorphism $\mathcal {A}_{\left | D \right |}/S^1 \simeq \mathfrak {t} \oplus \mathcal {A}_{\partial B}$ , therefore, we obtain a decomposition

$$ \begin{align*} \Omega^3_{T^k}(\mathcal{A}_{\left| D \right|}) = \sum_{i=0}^3\Omega^i(\mathcal{A}_{\partial B},\wedge^{3-i}\mathfrak{t}^*). \end{align*} $$

The T-duality conditions forces the component of H in $\Omega ^0(\mathcal {A}_{\partial B};\wedge ^3 t^*)$ to vanish, and the component in $\Omega ^1(\mathcal {A}_{\partial B};\wedge ^2 t^*)$ to be exact. When considering geometric structures (up to gauge equivalence), it is the cohomology class $[H]$ , rather than the form itself, which is of importance. Hence we can impose that we consider H which only have two terms in this decomposition:

(5.2) $$ \begin{align} H = \left\langle \hat{c},\Theta \right\rangle + h, \end{align} $$

for some $\hat {c} \in \Omega ^2(\mathcal {A}_{\partial B};\mathfrak {t}^*)$ and $h \in \Omega ^3(\mathcal {A}_{\partial B})$ . Using this decomposition, and the relation of the curvature with the Haefliger-Salem class, we can now address the question of existence of T-duals.

Proof. Let $M_0 \rightarrow B$ , be the reference torus action of M. Using Proposition 3.4, with respect to $M_0$ and $\hat {M}$ , the class $[\hat {c}] \in H^2(B;\mathbb {Z} ^k)$ corresponds to a manifold $\hat {M} \rightarrow B$ endowed with a locally equivalent action. Moreover, there exists a connection $\hat {\Theta } \in \Omega ^1(\mathcal {A}_{\left | \hat {D} \right |};\mathfrak {t}^k)$ for which $d\hat {\Theta } = \hat {c}$ . Now define

$$ \begin{align*} \hat{H} = \left\langle c,\hat{\Theta} \right\rangle+h \in \Omega^3(\mathcal{A}_{\left| \hat{D} \right|}), \end{align*} $$

with $c = d\Theta $ , for some connection $\Theta \in \Omega ^1(M;\mathfrak {t}^k)$ . Now $(M,H)$ and $(\hat {M},\hat {H})$ are T-dual because

$$ \begin{align*} p^*H-\tilde{p}^*\hat{H} = \left\langle \hat{c},\Theta \right\rangle - \left\langle c,\hat{\Theta} \right\rangle = \left\langle d\hat{\Theta},\Theta \right\rangle - \left\langle d\Theta,\hat{\Theta} \right\rangle = - d\left\langle \Theta,\hat{\Theta} \right\rangle, \end{align*} $$

and as $\Theta $ and $\hat {\Theta }$ are connection-one-forms, we obtain that $F =- \left \langle \Theta ,\hat {\Theta } \right \rangle $ satisfies the requirements of T-duality.

5.2.1 Topology change from singularities

The singularities of the torus actions provide a new source of topology change for T-duals:

Proposition 5.11. Let $M,\hat {M}$ be manifolds with standard $T^k$ -actions over the same contractible base B. Then $(M,H=0)$ and $(\hat {M},\hat {H}=0)$ are T-dual.

Proof. Because the base of M and $\hat {M}$ is contractible there exists flat connections $\Theta ,\hat {\Theta }$ on M respectively $\hat {M}$ . Then, as in the proof of Proposition 5.8 (the restriction of) $F = -\left \langle \Theta ,\hat {\Theta } \right \rangle $ provides a T-dual between $(M,0)$ and $(\hat {M},0)$ .

In [Reference Orlik and Raymond20] a classification of four-manifolds with effective $T^2$ -actions and connected isotropy groups appears. All such manifolds are of the following form:

• ○ $\#n (S^2\times S^2)$ , $n \in \mathbb {N}$

• ○ $\#n (\mathbb {C} P^2) \# m(\overline {\mathbb {C} P^2})$ , $n,m \in \mathbb {N}$

• ○ $S^4$

All of these manifolds have a contractible base, so we conclude by the above Theorem that if two have the same base they are T-dual. The only invariant of the base is the number of corners, which can be read of from the Euler characteristic of the total space. For instance $2\mathbb {C}P^2,\mathbb {C}P^2\# \overline {\mathbb {C} P^2}$ and $S^2\times S^2$ are all T-dual. The common base of these manifolds is contractible, and hence the actions are necessarily not locally equivalent. For if the actions were locally equivalent, Theorem 3.1 would imply that these spaces are all diffeomorphic.

Just as in principal T-duality, we obtain an isomorphism of differential complexes. The proof is omitted as it is identical with only the minor change that instead of developing the operation of push-forward/fiber integration we use the equivalent operation of ‘interior product with a top degree multi-vector field’ which is readily understood in the Lie algebroid context.

Proposition 5.12. Let $M,\hat {M}$ be manifolds endowed with standard $T^k$ -actions over the same base B. Let $H \in \Omega ^3_{\mathrm {cl}}(\mathcal {A}_{\left | D \right |}),\hat {H} \in \Omega ^3_{\mathrm {cl}}(\mathcal {A}_{\left | \hat {D} \right |})$ , and let $F \in \Omega ^2(\mathcal {A}_{\left | D \right |} \times _B \mathcal {A}_{\left | \hat {D} \right |})$ provide a T-dual. Then

(5.3) $$ \begin{align} \tau : \Omega^{\bullet}_{T^k}(\mathcal{A}_{\left| D \right|}) \rightarrow \Omega^{\bullet}_{T^k}(\mathcal{A}_{\left| \hat{D} \right|}), \quad \tau(\rho) = \iota_{X_1\wedge \cdots \wedge X_n} (e^F\wedge \rho), \end{align} $$

is an isomorphism of differential complexes $(\Omega ^{\bullet }_{T^k}(M;\mathcal {A}_{\left | D \right |}),d_{H})$ and $(\Omega ^{\bullet }_{T^k}(\hat {M};\mathcal {A}_{\left | \hat {D} \right |}),d_{\hat {H}})$ . Here $X_1,\ldots ,X_n \in \Gamma (\mathcal {A}_{\left | D \right |})$ are the infinitesimal generators of the torus action on M.

Moreover, we may consider the standard $\mathcal {A}_{\left | D \right |}$ -Courant algebroid, $\mathcal {A}_{\left | D \right |} \oplus \mathcal {A}_{\left | D \right |}^*$ , and obtain Theorem 5.4 in our setting. Again, the proof is identical to the principal case and thus omitted.

Theorem 5.13. Let $M,\hat {M}$ be manifolds endowed with standard $T^k$ -actions over the same base B. Let $H \in \Omega ^3_{\mathrm { cl}}(\mathcal {A}_{\left | D \right |}),\hat {H} \in \Omega ^3_{\mathrm {cl}}(\mathcal {A}_{\left | \hat {D} \right |})$ , and assume that $(M,H)$ and $(\hat {M},\hat {H})$ are T-dual. Then there is an isomorphism between Courant algebroids $(\mathcal {A}_{\left | D \right |} \oplus \mathcal {A}_{\left | \hat {D} \right |}^*)/T^k \rightarrow M$ and $(\mathcal {A}_{\left | D \right |} \oplus \mathcal {A}_{\left | \hat {D} \right |}^*)/T^k \rightarrow \hat {M}$ .

Using this isomorphism we are able to transport geometric structures on $\mathcal {A}_{\left | D \right |} \rightarrow M$ to geometric structures on $\mathcal {A}_{\left | \hat {D} \right |} \rightarrow \hat {M}$ :

Theorem 5.14. Let $(M,H)$ and $(\hat {M},\hat {H})$ be T-dual. Then any Dirac, generalized complex or SKT structure on $\mathcal {A}_{\left | D \right |} \rightarrow M$ invariant under the torus action will be transported to a structure of the same kind on $\mathcal {A}_{\left | D \right |} \rightarrow \hat {M}$ .

However, there is one key difference compared to the case of principal T-duality. Given a $T^k$ -invariant generalized complex structure on M which lifts to a generalized complex structure on $\mathcal {A}_{\left | D \right |} \rightarrow M$ , we can transport it to a generalized complex structure on $\mathcal {A}_{\left | \hat {D} \right |} \rightarrow \hat {M}$ . However, a prior there is no guarantee that it again descends to a generalized complex structure on M. We illustrate this phenomenon by the following examples:

Example 5.15 ( $\mathbb {\mathbb {C} }$ )

Consider $M = \mathbb {C} $ with the standard $S^1$ -action, and endow it with the elliptic symplectic form $d\log r \wedge d\theta = \frac {dx \wedge dy}{x^2+y^2}$ . If we let $\hat {M} = \mathbb {C} $ denote another another copy of $\mathbb {C} $ , we have that $F= d\theta \wedge d\hat {\theta }$ provides a T-dual. Under this T-duality, the elliptic symplectic structures $e^{id\log r \wedge d\theta }$ is send to spinor $d\log \hat {z}$ . This spinor corresponds to the complex structure on $\mathcal {A}_{\left | D \right |} = \lbrace r\partial _r,\partial _{\theta } \rbrace $ defined by $r\partial _r \mapsto \partial _{\theta }$ . It is immediate to see that this complex structure is a lift of the standard complex structure on $\mathbb {C} $ . Hence we conclude that the elliptic symplectic form, which is not a generalized complex structure, is T-dual to the standard complex structure.

The next example will illustrate that T-duality for standard torus actions is very sensitive to the specific $F \in \Omega ^2(\mathcal {A}_{\left | D \right |} \times _B \mathcal {A}_{\left | \hat {D} \right |})$ chosen:

Example 5.16 ( $\mathbb {\mathbb {C} }^2$ )

Consider $M = \mathbb {C} ^2$ with coordinates $(z_1,z_2)$ and $\hat {M} = \mathbb {C} ^2$ with coordinates $(\hat {z}_1,\hat {z}_2)$ . Endow both with the standard $T^2$ -action. There are ample choices for a T-duality between $(M,0)$ and $(\hat {M},0)$ . Indeed, any of the following two-forms will suffice:

• ○ $F_{1,\pm } = d\theta _1 \wedge d\hat {\theta }_1 \pm d\theta _2 \wedge d\hat {\theta }_2$ .

• ○ $F_{2,\pm } = -d\theta _1 \wedge d\hat {\theta }_2 \pm d\theta _2 \wedge d\hat {\theta }_1$ .

Although each of these forms satisfy the conditions necessary of T-duality, the behaviour with respect to specific generalized complex structures is quite different.

We consider the stable generalized complex structure given by $\rho = z_1z_2 + dz_1 \wedge dz_2$ . The corresponding elliptic symplectic form is $\omega = d\log r_1 \wedge d\theta _2 + d\theta _1 \wedge d\log r_2$ with spinor $\rho = 1 + id\log r_1 \wedge d\theta _2 + id\theta _1 \wedge d\log r_2 - d\log r_1 \wedge d\theta _1 \wedge d\log r_2 \wedge d\theta _2$ .

By Theorem 5.14 we know that T-duality with any of these forms will provide generalized complex structures on the elliptic tangent bundle. However, a direct computation will show that only $F_{2,+}$ provides a generalized complex structure on M. In that case we have that the T-dual generalized complex structure is given by:

$$ \begin{align*} \rho_{2,+} = d\log z_1 \wedge d\log z_2. \end{align*} $$

One immediately checks that $\rho _{2,+}$ is a lift of the standard complex structure on $\mathbb {C} $ . In conclusion, this elliptic symplectic structure on $\mathbb {C} ^2$ is T-dual to the standard complex structure on $\mathbb {C}^2$ .

The story for the real part of $\rho $ is quite different. Consider the elliptic symplectic form $\omega = d\log r_1 \wedge d\log r_2 - d\theta _1 \wedge d\theta _2$ , with spinor

$$ \begin{align*} \rho = 1 + i d\log r_1 \wedge d\log r_2 - id\theta_1 \wedge d\theta_2 + d\theta_1 \wedge d\theta_2 \wedge d\log r_1 \wedge d\log r_2. \end{align*} $$

Again, one can perform a direct computation showing that only $F_{1,-}$ will provide a generalized complex structure on M. In that case the T-dual is given by:

$$ \begin{align*} -ie^{-i(d\log r_1 \wedge d\log r_2 + d\hat{\theta}_1 \wedge d\hat{\theta_2})} \end{align*} $$

In conclusion we see that $\rho $ is T-dual to itself.

6.1 Blow-ups in generalized complex geometry

The contents of this subsection appear in [Reference Bailey, Cavalcanti and Duran2], we recall them for completeness.

As we do not assume any holomorphicity of our manifolds, more data is necessary in constructing a blow-up:

As shown in [Reference Bailey, Cavalcanti and Duran2], a submanifold Y admits a holomorphic ideal if and only if the normal bundle $NY$ admits a complex structure.

Theorem 6.2 [Reference Bailey, Cavalcanti and Duran2]

Given a submanifold Y and a holomorphic ideal $I_Y$ , there exists a, unique up to isomorphism, manifold $\tilde {M}$ together with a map $p : \tilde {M} \rightarrow M$ such that $p^*I_Y$ is a divisor satisfying the following universal property: For any smooth map $f : X \rightarrow M$ , such that $f^*I_Y$ is a divisor, there exists a unique $\tilde {f} : X \rightarrow \tilde {M}$ such that the following diagram commutes:

We call $\tilde {M}$ the blow-up of $\boldsymbol{I}_{\boldsymbol{Y}}$ in $\boldsymbol{M}$ , and $p : \tilde {M} \rightarrow M$ the blow-down map. The natural manifolds one can blow-up in generalized complex geometry are the following:

Definition 6.3 [Reference Bailey, Cavalcanti and Duran2]

Let $(M,\mathbb {J},H)$ be a generalized complex manifold. A submanifold Y is called generalized Poisson if

$$ \begin{align*} \mathbb{J} N^*Y = N^*Y. \end{align*} $$

An immediate consequence of this definition is that Y is a Poisson submanifold of the underlying Poisson structure $\pi _{\mathbb {J}}$ . Moreover, if Q is a holomorphic Poisson submanifold and $\mathbb {J}_{Q}$ is the associated generalized complex structure, a submanifold Y is a generalized Poisson submanifold if and only if it is a holomorphic Poisson submanifold.

By definition, the generalized complex structure induces a complex structure on the normal bundle of a generalized Poisson submanifold. Hence:

Given that a generalized Poisson submanifold gives rise to a holomorphic ideal, and hence a canonical blow-up, the next step is to determine when the generalized complex structure lifts to the blow-up. For this we need the following notion:

Definition 6.5. A Lie algebra $\mathfrak {g}$ is said to be degenerate if the map $\Lambda ^3(\mathfrak {g}) \rightarrow \mathrm {Sym}^2(\mathfrak {g})$

$$ \begin{align*} x\wedge y \wedge z \mapsto [x,y] \wedge z, \end{align*} $$

vanishes.

Given a generalized Poisson submanifold $Y \subset (M,\mathbb {J})$ , it is in particular a Poisson submanifold and hence one obtains a fibre-wise Lie algebra structure on $N^*Y$ . This Lie algebra structure captures whether the generalized complex structure lifts to the blow-up:

We may blow-up components of the degeneracy locus of a stable generalized complex structure:

Proof. First we note that Y is a union of symplectic leaves of the underlying Poisson structure $\pi _{\mathbb {J}}$ , and hence a Poisson submanifold. For every $p \in Y$ , there exists a neighbourhood U, with a complex structure and a holomorphic Poisson structure Q inducing the generalized complex structure (up to gauge equivalence). We have that $Y \cap U$ , can be written as the zero-set of some of the complex coordinates, and hence it is a complex submanifold of U. Therefore, $Y \cap U$ is a holomorphic Poisson submanifold with respect to the holomorphic Poisson structure Q, and consequently a generalized Poisson submanifold of $\mathbb {J}$ on U. As being a generalized Poisson submanifold is a point-wise condition, this shows that Y is a generalized Poisson submanifold of M.

Using a tubular neighbourhood of Y, we may transport $\pi _{\mathbb {J}}$ to a Poisson structure on $NY$ which we will still denote in the same fashion. The Lie algebra structure on $N^*Y$ is then defined by

$$ \begin{align*} [s,t] := \pi_{\mathbb{J}}(d\hat{s},d\hat{t}), \text{for } s,t \in \Gamma(N^*Y), \end{align*} $$

where $\hat {s},\hat {t}$ denotes the fibre-wise linear function associated to the section. Or, $[s,t] = \pi ^{\sharp }_{\mathbb {J}}(d\hat {s})(d\hat {t})$ .

Because $\pi _{\mathbb {J}}$ is an elliptic symplectic structure it lifts to an element of $\Gamma (\wedge ^2\mathcal {A}_{\left | D \right |})$ . In particular, this shows that the image of $\pi _{\mathbb {J}}^{\sharp }$ lands in $\mathcal {A}_{\left | D \right |}$ . But as Y is contained in a stratum of the divisor D, this in particular implies that $\operatorname {\mathrm {Im}} \pi _{\mathbb {J}}^{\sharp } \cap NY = \lbrace 0 \rbrace $ . This shows that the Lie algebra structure on $N^*Y$ is trivial.

We conclude that irreducible components of the degeneracy locus of a stable generalized complex structure may be blown-up in a generalized complex fashion.

6.2 Blowing up for the elliptic tangent bundle

There is yet another explanation why one can blow up irreducible components of the degeneracy locus of a stable generalized complex structure. Namely, the elliptic tangent bundle is very well-behaved under blow-ups. To state a result independent of a generalized complex structure, we need that the holomorphic ideal is adapted to the elliptic ideal:

Note that this definition in particular implies that $Y \subset D$ . Given such a holomorphic ideal, we may blow-up the elliptic divisor to obtain:

Theorem 6.9. Let $(M,I_{\left | D \right |})$ be a manifold endowed with an elliptic divisor, and let $Y \subset D(k)$ be an irreducible component. Assume that we are given a holomorphic ideal $I_Y$ for Y adapted to $I_{\left | D \right |}$ . Then there exists an elliptic ideal $I_{\left | \tilde {D} \right |}$ on the blow-up $\tilde {M}$ such that the blow-down map induces a fibre-wise isomorphism:

$$ \begin{align*} p_* : (\mathcal{A}_{\left| \tilde{D} \right|})_x \overset{\simeq}{\longrightarrow} (\mathcal{A}_{\left| D \right|})_{p(x)}, \end{align*} $$

for all $x \in \tilde {M}$ .

Proof. To study the blow-down map we need to use local coordinates describing it. Given $l\geq k$ let $x \in Y \cap D[l]$ , let $U = \mathbb {C} ^l \times \mathbb {R} ^m$ be a local chart with coordinates $(z_1,\ldots ,z_l,x_1,\ldots ,x_m)$ , with the $z_i$ as in Definition 6.1. Let $\tilde {U} = \tilde {\mathbb {C} }^l \times \mathbb {R} ^m$ , and $p = (p',\text {Id}) : \tilde {U} \rightarrow U$ be the blow-down map. The manifold $\tilde {\mathbb {C} }^l$ has a cover of l-coordinate charts given by

$$ \begin{align*} (v_1,\ldots,v_{i-1},z_i,v_{i+1},\ldots,v_l) &\leftrightarrow (z_i \cdot (v_1,\ldots,v_{i-1},1,v_{i+1},\ldots,v_l),\\ &[v_1:\ldots:v_{i-1}:1:v_{i+1}:\ldots:v_l]). \end{align*} $$

In these coordinate charts the blow-down map is given by

$$ \begin{align*} p' : (v_1,\ldots,v_{i-1},z_i,v_{i+1},\ldots,v_l) &\mapsto (z_iv_1,\ldots,z_iv_{i-1},z_i,z_iv_{i+1},\ldots,z_iv_l). \end{align*} $$

With these coordinates at hand, we can consider the local form of $p^*I_{\left | D \right |}$ . Note that this does not provide an elliptic divisor yet, however $I_{\left | D \right |} := \sqrt {p^*I_{\left | D \right |}}$ does. Because a vector field preserves $p^*I_{\left | D \right |}$ if and only if it preserves $\sqrt {p^*I_{\left | D \right |}}$ we find that the blow-down map induces a Lie algebroid map $p_* : \mathcal {A}_{\left | \tilde {D} \right |} \rightarrow \mathcal {A}_{\left | D \right |}$ .

To check that the blow-down map p induces a fibre-wise isomorphism, it suffices to show that it pulls-back a volume element $(\mathcal {A}_{\left | D \right |})_{p(x)}$ to a volume element of $(\mathcal {A}_{\left | \tilde {D} \right |})_x$ . In the above coordinates, the following would define a local volume form on $(\mathcal {A}_{\left | D \right |})_x$ :

$$ \begin{align*} \Omega = d\log r_1 \wedge d\theta_1 \wedge \cdots \wedge d\log r_l \wedge d\theta_l \wedge dx_1 \wedge \cdots \wedge dx_m. \end{align*} $$

Therefore

$$ \begin{align*} p^*\Omega = d\log \tilde{r}_1 \wedge d\hat{\theta}_1 \cdots \wedge d\log \tilde{r}_l \wedge d\hat{\theta}_l \wedge d\log r_{l+1} \wedge d\theta_{l+1} \wedge \cdots d\log r_k \wedge d\theta_k \wedge dx_1 \wedge \cdots dx_m, \end{align*} $$

where the $(\tilde {r}_j,\hat {\theta }_j)$ are the polar coordinates corresponding to $(v_1,\ldots ,v_{i-1},z_i,v_{i+1},\ldots ,v_l)$ . We conclude that p indeed induces the fibrewise isomorphism as required.

Given a generalized complex structure on any of the Lie algebroids $\mathcal {A}_{\left | D \right |}$ may lift it to a structure of the same kind on the blown-up Lie algebroid. But similarly as for T-duality, it might happen that said structure does not descend to give a generalized complex structure on $\tilde {M}$ . However, when the holomorphic ideal is induced by the generalized complex structure, this will always be the case:

We conclude that the point of view of blowing-up using the elliptic tangent bundle results in the same blow-up as in [Reference Bailey, Cavalcanti and Duran2]. However, Theorem 6.9 can be used in a larger context: for instance, elliptic symplectic forms not corresponding to generalized complex structures may be blown-up using this result.

6.3 T-duality and blow-up

We now study how T-duality interacts with blow-ups. First we recall that toric actions are preserved under blow-ups:

Lemma 6.14. If $M^{2n}$ admits a standard $T^n$ action with quotient map $\pi \colon M \to B$ , and $x \in M$ is a fixed point, let $\tilde {M}$ be either blow up of M at p, that is $\tilde {M} = M\#\mathbb {C} P^n$ or $M\#\overline {\mathbb {C} P^n}$ , with blow-down map $q\colon \tilde {M} \to M$ . Then $\tilde {M}$ admits a standard $T^n$ -action for which the q is equivariant, and the orbit space on $\tilde {M}$ is $\tilde {B}$ , the real blow-up of B at $\pi (x)$ . These maps are related by the following diagram

The fact that the blow-down map induces a fibre-wise isomorphism between elliptic tangent bundles immediately implies the following:

Proposition 6.15 (T-duality commutes with blow-up)

Let $M^{2n}$ and $\hat {M}^{2n}$ be endowed with standard $T^n$ -actions over the same base B, and let $I_{\left | D \right |},I_{\left | \hat {D} \right |}$ denote the induced elliptic divisors. Let $H \in \Omega ^3_{\mathrm {cl}}(\mathcal {A}_{\left | D \right |}), \hat {H} \in \Omega ^3_{\mathrm {cl}}(\mathcal {A}_{\left | \hat {D} \right |})$ , and $F\in \Omega ^2(\mathcal {A}_{\left | D \right |} \times _B \mathcal {A}_{\left | \hat {D} \right |})$ provide a T-dual between $(M,H)$ and $(\hat {M},\hat {H})$ .

Let $x\in M, \hat {x} \in \hat {M}$ be fixed points for the action with $\pi (x) = \hat {\pi }(\hat {x})$ , and let $\tilde {M}$ and $\tilde {\hat {M}}$ denote either blow-up of M and $\hat {M}$ . Let $q : \tilde {M} \rightarrow M$ , $\hat {q} : \tilde {\hat {M}} \rightarrow \hat {M}$ denote the blow-down maps. Then we have a canonical T-duality between the blow-ups:

Consequently, the following diagram commutes:

7.1 Basic examples

We start by considering simple examples of generalized complex structures. The analysis heavily depends on the local description in Example 5.16. So recall from there that any of the following two-forms:

• ○ $F_{1,\pm } = d\theta _1 \wedge d\hat {\theta }_1 \pm d\theta _2 \wedge d\hat {\theta }_2$ .

• ○ $F_{2,\pm } = -d\theta _1 \wedge d\hat {\theta }_2 \pm d\theta _2 \wedge d\hat {\theta }_1$ .

provides a T-duality from $(\mathbb {C} ,0)$ to itself.

Example 7.2 ( $\mathbb {C}P^2$ )

We consider $\mathbb {C}P^2$ in the usual fashion as $(\mathbb {C} ^3\backslash \lbrace 0 \rbrace )/\mathbb {C} $ with torus action $(\lambda _1,\lambda _2)\cdot [z_0:z_1:z_2] = [e^{i\lambda _1}z_0:e^{i\lambda _2}z_1:z_2]$ . Consider the complex log forms $\tilde {\zeta _1} = d\log z_0 - d\log z_2$ and $\tilde {\zeta _2} = d\log z_1 - d\log z_2$ . As these are basic with respect to the $\mathbb {C} ^*$ -action, they descend to complex log forms $\zeta _1,\zeta _2$ . We let $\Theta _1 = \Im ^* \zeta _1,\Theta _2 = \Im ^* \zeta _2$ denote the imaginary parts. Then $(\Theta _1,\Theta _2)$ provides a connection-one form for the $T^2$ -action on $\mathbb {C} P^2$ .

Consider another copy of $\mathbb {C}P^2$ , and denote the same connection one-form by $(\hat {\Theta }_1,\hat {\Theta }_2)$ . Similarly to Example 5.16 we now have that any of the following two-forms provides a self T-dual for $(\mathbb {C} P^2,0)$ .

$$ \begin{align*} F_{1,\pm} &= \Theta_1 \wedge \hat{\Theta}_1 \pm \Theta_2 \wedge \hat{\Theta}_2,\\ F_{2,\pm} &= - \Theta_1 \wedge \hat{\Theta}_2 \pm \Theta_2 \wedge \hat{\Theta}_1. \end{align*} $$

We will now spell out the effect of T-duality on the generalized complex structures on $\mathbb {C} P^2$ for the 2-forms $F_{2,+}$ and $F_{1,-}$ .

Consider the holomorphic Poisson structure $\tilde {\pi } = z_0z_1\partial _{z_0}\wedge \partial _{z_1}$ . Because this is $\mathbb {C} ^*$ -invariant it induces a holomorphic Poisson structure $\pi $ on $\mathbb {C} P^2$ , which is moreover log symplectic. One can show, as in Example 5.16, that $F_{2,+}$ provides a T-dual between the corresponding stable generalized complex structure and the standard complex structure on $\mathbb {C} P^2$ . Similarly, consider the holomorphic Poisson structure $\tilde {\pi _2} = iz_0z_1\partial _{z_0}\wedge \partial _{z_1}$ , and the induced generalized complex structure $\rho _2$ . Then $F_{1,-}$ , as in Example 5.16, will provide a T-dual between $\rho _2$ and itself.

We may now blow-up $\mathbb {C}P^2$ to obtain more examples of stable generalized complex structures, which are T-dual:

7.2 Non-zero residue elliptic symplectic structures

Another class of structures which behave well with respect to T-duality are elliptic symplectic forms with non-zero elliptic residue. These structures admit a $\mathbb {C} ^*$ -invariant normal form:

Lemma 7.7 [Reference Witte25]

Let $(M,I_{\left | D \right |})$ be a manifold with a smooth elliptic divisor, and let $\omega \in \Omega ^2(\mathcal {A}_{\left | D \right |})$ be an elliptic symplectic form with $\lambda := \operatorname {\mathrm {Res}}_q(\omega ) \neq 0$ . Then there exists a canonical flat connection on $\nu _D$ and a tubular neighbourhood of D such that

(7.1) $$ \begin{align} \omega = \lambda\rho \wedge \Theta + p^*\omega_D. \end{align} $$

Here $\rho ,\Theta $ are the connection-one forms corresponding to the flat connection, as defined in 3.16 and $\omega _D = \iota ^*_D\omega \in \Omega ^2(D)$ .

Conversely, given any complex line bundle $L \rightarrow D$ , $\nabla $ a flat connection on L, $\lambda \neq 0$ and $\omega _D \in \Omega ^2(D)$ symplectic, (7.1) defines an elliptic symplectic form on L.

The form $\omega $ is $S^1$ -invariant, and hence we can consider its T-dual. It will appear via the following lemma:

Proof. Let $\hat {\Theta }$ denote the same connection one-form, on another copy of $L \rightarrow M$ . Then $F = \Theta \wedge \hat {\Theta }$ provides the desired T-dual. We have that, using $\tau $ from Proposition 5.12 that

$$ \begin{align*} \tau(e^{i\omega}) = i\lambda(\rho + i\Theta) \wedge e^{i\omega_D}. \end{align*} $$

Locally, this corresponds to the spinor given by $i\lambda dz \wedge e^{i\omega _D}$ , which is precisely the generalized complex structure from Lemma 7.8.