不变形式计算复幂零流形的Dolbeault上同调
Invariant forms compute the Dolbeault cohomology of complex nilmanifolds
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中文总结 AI 辅助
该研究证明赋予左不变复结构的紧幂零流形的左不变形式到Dolbeault复形的包含映射在各双次数上诱导上同调同构，还解决了Hasegawa、Angella等人的相关猜想，明确了三类不变量的计算方式与独立性。
中文摘要 AI 辅助
我们证明，赋予左不变复结构$J$的紧幂零流形$M$的左不变形式到其Dolbeault复形的包含映射在各双次数上诱导上同调同构，从而解决了一个长期存在的猜想。作为推论，我们证明$J$的小形变仍为不变量，这正是Hasegawa所猜想的。我们还证明Bott--Chern不变量、Aeppli不变量和Frölicher不变量可通过不变形式计算，且与格无关，从而解决了Angella关于Bott--Chern上同调的一个猜想。
英文摘要
We prove that the inclusion of left-invariant forms into the Dolbeault complex of a compact nilmanifold $M$ endowed with a left-invariant complex structure $J$ induces an isomorphism in cohomology in every bidegree, settling a long-standing conjecture. As consequences, we show that small deformations of $J$ are still invariant, as conjectured by Hasegawa. We also prove that Bott--Chern, Aeppli, and Frölicher invariants are computed by invariant forms and are independent of the lattice, settling a conjecture of Angella on Bott--Chern cohomology.
