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$p$-K\"ahler and balanced structures on nilmanifolds with nilpotent complex structures
Differential Geometry · Submitted 23 Dec 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Finally, we show that, contrarily to the K\"ahler case, on compact complex manifolds there is no relation between the existence of balanced metrics and the degeneracy step of the Fr\"olicher spectral sequence.
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The abstract, sentence by sentence
Let $(X,J)$ be a nilmanifold with a left-invariant nilpotent complex structure. We study the existence of $p$-K\"ahler structures (which include K\"ahler and balanced metrics) on $X$. More precisely, we determine an optimal $p$ such that there are no $p$-K\"ahler structures on $X$. Finally, we show that, contrarily to the K\"ahler case, on compact complex manifolds there is no relation between the existence of balanced metrics and the degeneracy step of the Fr\"olicher spectral sequence. More precisely, on balanced manifolds the degeneracy step can be arbitrarily large.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $(X,J)$ be a nilmanifold with a left-invariant nilpotent complex structure. | background 0.96 | 0.07 | 0.02 | asserted 0.81 |
| s1 | We study the existence of $p$-K\"ahler structures (which include K\"ahler and balanced metrics) on $X$. | contribution 0.99 | 0.80 | 0.06 | asserted 0.79 |
| s2 | More precisely, we determine an optimal $p$ such that there are no $p$-K\"ahler structures on $X$. | result 0.62 | 0.94 | 0.55 | asserted 1.00 |
| s3 | Finally, we show that, contrarily to the K\"ahler case, on compact complex manifolds there is no relation between the existence of balanced metrics and the degeneracy step of the Fr\"olicher spectral sequence. | result 0.95 | 0.80 | 0.41 | asserted 1.00 |
| s4 | More precisely, on balanced manifolds the degeneracy step can be arbitrarily large. | result 0.95 | 0.81 | 0.22 | asserted 0.99 |
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Original abstract
Let $(X,J)$ be a nilmanifold with a left-invariant nilpotent complex structure. We study the existence of $p$-K\"ahler structures (which include K\"ahler and balanced metrics) on $X$. More precisely, we determine an optimal $p$ such that there are no $p$-K\"ahler structures on $X$. Finally, we show that, contrarily to the K\"ahler case, on compact complex manifolds there is no relation between the existence of balanced metrics and the degeneracy step of the Fr\"olicher spectral sequence. More precisely, on balanced manifolds the degeneracy step can be arbitrarily large.
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1 arXiv paper cite this in 4 sentences
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Cites it as support 1 paper
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[ref] For every $n≥2$, the nilmanifold $X^4n-2$ is balanced.
Fr\"olicher spectral sequence of compact complex manifolds with special Hermitian metrics · The Frölicher spectral sequence of balanced manifolds · 0.96
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