We show that given an embedding of an Enriques manifold of index d in a large enough projective space, there will exist embedded multiple structures with conormal bundle isomorphic to the trace zero module of the universal covering map, the universal cover being either a hyperkähler or a Calabi–Yau manifold. We then show that these multiple structures (also known as d-ropes) can be smoothed to smooth hyperkähler or Calabi–Yau manifolds respectively. Hence we obtain a flat family of hyperkähler (or Calabi–Yau) manifolds embedded in the same projective space which degenerates to an embedded d-rope structure on the given Enriques manifold of index d. The above shows that these d-rope structures on the embedded Enriques manifold are points of the Hilbert scheme containing the fibres of the above family. We show that they are smooth points of the Hilbert scheme when \(d=2\).

我们表明，给定索引d的 Enriques 流形嵌入到足够大的射影空间中，将存在嵌入多个结构，其共正规丛同构于通用覆盖图的迹零模块，通用覆盖是 hyperkähler 或 a卡拉比-丘流形。然后我们证明这些多重结构（也称为d-绳索）可以分别平滑以平滑 hyperkähler 或 Calabi-Yau 流形。因此，我们获得了嵌入同一射影空间中的 hyperkähler（或 Calabi-Yau）流形的平面族，该流形退化为指数d的给定 Enriques 流形上的嵌入d绳结构。以上表明这些d嵌入 Enriques 流形上的绳结构是包含上述家族纤维的希尔伯特方案的点。我们证明当\(d=2\)时，它们是希尔伯特方案的平滑点。