AI 中文总结
本文研究Calabi-Yau流形与全纯向量丛的联合形变，在Fujiki流形等条件下证明无阻碍性，并构造有阻碍的例子，推广了Li-Pan和Iacono-Manetti定理，回答了Felten的两个问题。
AI 中文摘要
本文研究对$(X,E)$的联合形变，其中$X$是紧致复流形，$E\rightarrow X$是全纯向量丛。我们证明：若$X$是具有挠典范丛的Fujiki流形且$H^2(X,\textrm{End}^0E)=0$（其中$\textrm{End}^0 E$表示无迹自同态丛），则$(X,E)$具有无阻碍形变。当典范丛平凡时，Fujiki假设可被三个弱$\partial\bar\partial$-条件替代。我们构造了满足这些条件的例子，其Frölicher谱序列在$E_1$处不退化。这些结果为Li-Pan和Iacono-Manetti的定理提供了非Kähler推广。在没有此消没假设的情况下，我们同时得到无阻碍和有阻碍的对。在Thomas的例子中，我们证明该对是无阻碍的，尽管该丛在流形固定时具有有阻碍形变。另一方面，我们证明每个维数至少为三的严格射影Calabi-Yau流形都容许一个具有有阻碍联合形变的简单丛。我们还在复环面上的Hermitian平坦丛上构造了有阻碍的对。这些结果回答了Felten提出的两个问题。
英文摘要
In this paper, we study joint deformations of pairs $(X,E)$, where $X$ is a compact complex manifold and $E\rightarrow X$ is a holomorphic vector bundle. We prove that $(X,E)$ has unobstructed deformations if $X$ is a Fujiki manifold with torsion canonical bundle and $H^2(X,\textrm{End}^0E)=0$, where $\textrm{End}^0 E$ denotes the trace-free endomorphism bundle. When the canonical bundle is trivial, the Fujiki assumption can be replaced by three weak $\partial\bar\partial$-conditions. We construct examples satisfying these conditions whose Frölicher spectral sequences do not degenerate at $E_1$. These results provide non-Kähler extensions of the theorems of Li--Pan and Iacono--Manetti. Without this vanishing assumption, we obtain both unobstructed and obstructed pairs. In Thomas's example, we prove unobstructedness of the pair, although the bundle has obstructed deformations with the manifold fixed. On the other hand, we prove that every strict projective Calabi--Yau manifold of dimension at least three admits a simple bundle with obstructed joint deformations. We also construct obstructed pairs with Hermitian flat bundles on complex tori. These results answer two questions raised by Felten.
Comments43 pages. With an appendix joint with Hisashi Kasuya. Comments are very welcome!
