On a 2-step nilmanifold with abelian complex structure, there exists an invariant $(1,0)$-form ρ such that dρ is type-$(1,1)$. It acts on the kernel of ρ by contraction. When this contraction map is non-degenerate, for any given infinitesimal generalized complex deformation ${\mathrm{\Gamma }}_1$ we construct a solution $(\mathrm{\Gamma },\overrightarrow{\partial })$ for the extended Maurer-Cartan equation. It amounts to identifying the obstruction $\overrightarrow{\partial }$ for ${\mathrm{\Gamma }}_1$ to be integrable, and constructing the deformation Γ when the obstruction vanishes. As a consequence of our explicit solutions for $(\mathrm{\Gamma },\overrightarrow{\partial })$, we prove that on any real six-dimensional 2-step nilmanifold with abelian complex structure, when the contraction map is non-degenerate, every infinitesimal generalized complex deformation sufficiently close to zero is integrable. We also show that in all dimensions, if the contraction map is skew-Hermitian, then every infinitesimal generalized complex deformation sufficiently close to zero is integrable. Moreover the differential graded algebra controlling the generalized deformation of the underlying abelian complex structure is quasi-isomorphic to the one after deformation.

在具有阿贝尔复数结构的两步尼尔流形上，存在一个不变量 $(1,0)$-形成ρ使得dρ是类型-$(1,1)$. 它通过收缩作用于ρ的核。当这个收缩图是非退化的，对于任何给定的无穷小广义复变形${\mathrm{\Gamma }}_1$ 我们构建一个解决方案 $(\mathrm{\Gamma },\overrightarrow{\partial })$对于扩展的 Maurer-Cartan 方程。它相当于识别障碍$\overrightarrow{\partial }$ 为了 ${\mathrm{\Gamma }}_1$可积，并构造障碍物消失时的变形Γ。作为我们明确解决方案的结果$(\mathrm{\Gamma },\overrightarrow{\partial })$，我们证明了在任何具有阿贝尔复数结构的实六维 2-step nilmanifold 上，当收缩图是非退化的时，每一个足够接近零的无穷小广义复形变形都是可积的。我们还表明，在所有维度上，如果收缩图是偏赫米特的，那么每个足够接近零的无穷小广义复变形都是可积的。此外，控制底层阿贝尔复结构广义变形的微分分级代数与变形后的代数是准同构的。