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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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