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Cohomologies of generalized complex manifolds and nilmanifolds. (English) Zbl 1362.57046
For a compact differentiable manifold $$M$$ of dimension $$2n$$, the vector bundle $$TM\oplus T^*M$$ endowed with the natural symmetric pairing $$\langle X+\xi\mid Y+\eta\rangle=\frac12(\xi(Y)+\eta(X))$$, defines the action $$(X+\xi)\cdot\rho=i_X\rho+\xi\wedge\rho$$ on differential forms $$\wedge^\bullet T^*M$$. If $$[X+\xi,Y+\eta]=[X,Y]+\mathcal{L}_X\eta-\mathcal{L}_Y\xi-\frac12{d}(\imath_X\eta-\imath_Y\xi)$$ is the Courant bracket on the space $$C^\infty(TX\oplus T^*M)$$, then a generalized complex structure on $$M$$ is an endomorphism $$\mathcal{J}\in\text{End}(TX\oplus T^*M)$$ such that $$\mathcal{J}^2=-1$$ and the $$i$$-eigenbundle $$L\subset (TX\oplus T^*M)\otimes\mathbb C$$ is involutive with respect to the Courant bracket. A generalized complex structure $$\mathcal{J}$$ on $$M$$ of dimension $$2n$$ defines a decomposition of complex differential forms $$\wedge^\bullet T^*M\otimes\mathbb C=\bigoplus_{j=-n}^nU^j$$ with the bi-differential $$\mathbb Z$$-graded complex $$(\mathcal{U},\partial,\overline\partial)$$. The cohomologies $$GH^\bullet_\partial=\frac{\ker\partial}{\text{im}\partial}$$, $$GH^\bullet_{\overline\partial}=\frac{\ker\overline\partial}{\text{im}\overline\partial}$$, and $$GH^\bullet_{BC}=\frac{\ker\partial\cap\ker\overline\partial}{\text{im}\partial\overline\partial}$$, $$GH^\bullet_{A}=\frac{\ker\partial\overline\partial}{\text{im}\partial+\text{im}\overline\partial}$$ are the generalized Dolbeault and the generalized Bott-Chern and Aeppli cohomologies, respectively. For sufficiently small $$\varepsilon\in C^\infty(\wedge^2L^*)$$, the small deformation $$L_\varepsilon=(1+\varepsilon)L\subset(TX\oplus T^*M)\otimes\mathbb C$$ defines the endomorphism $$\mathcal{J}_\varepsilon\in\text{End}(TX\oplus T^*M)$$ whose $$i$$-eigenbundle and $$-i$$-eigenbundle are $$L_\varepsilon$$ and $$\overline L_\varepsilon$$, respectively. Then $$\mathcal{J}_\varepsilon$$ is a generalized complex structure if and only if $$\varepsilon$$ satisfies the Maurer-Cartan equation $$d_L\varepsilon+\frac12[\varepsilon,\varepsilon]=0$$.
In this paper, the authors investigate generalized cohomologies of a nilmanifold $$M =\Gamma\setminus G$$, that is, a compact quotient of a real simply connected nilpotent Lie group $$G$$ by a discrete co-compact subgroup $$\Gamma$$, with left-invariant generalized complex structures on $$M$$, equivalently, linear generalized complex structures on the Lie algebra $$\mathfrak{g}$$ of $$G$$. There is a generalized complex decomposition also at the level of the Lie algebra, namely, $$\wedge^\bullet\mathfrak{g}^*=\bigoplus_J\mathfrak{U}^j$$, and a bi-differential $$\mathbb Z$$-graded sub-complex $$(\mathfrak{U}^\bullet,\partial,\overline\partial)\mapsto(\mathcal{U}^\bullet,\partial,\overline\partial)$$. It induces the map $$GH^\bullet_{\overline\partial}(\mathfrak{g})\to GH^\bullet_{\overline\partial}(M)$$ in cohomology, which is in fact always injective. The authors show that for a nilmanifold $$M =\Gamma\setminus G$$ with a left-invariant generalized complex structure $$\mathcal{J}$$ and the Lie algebra $$\mathfrak{g}$$ of $$G$$, if the isomorphism $$GH^\bullet_{\overline\partial}(\mathfrak{g})\cong GH^\bullet_{\overline\partial}(\Gamma\setminus G)$$ holds on the original generalized complex structure $$\mathcal{J}$$, then the same isomorphism holds on the deformed generalized complex structure $$\mathcal{J}_{\varepsilon(t)}$$ for sufficiently small $$t$$. They also show that if the isomorphism $$GH^\bullet_{\overline\partial}(\mathfrak{g})\cong GH^\bullet_{\overline\partial}(\Gamma\setminus G)$$ holds on the original generalized complex structure $$\mathcal{J}$$, then any sufficiently small deformation of the generalized complex structure is equivalent to a left-invariant generalized complex structure $$\mathcal{J}_{\varepsilon}$$ with $$\varepsilon\in C^\infty(\wedge^2\mathfrak{L}^*)$$ satisfying the Maurer-Cartan equation.

##### MSC:
 57T15 Homology and cohomology of homogeneous spaces of Lie groups 53D18 Generalized geometries (à la Hitchin) 32G07 Deformations of special (e.g., CR) structures
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