Asymptotic expansion of the variation of the Quillen metric and its moment map interpretation
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915585426718720 |
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| author | Eum, Kiyoon |
| author_facet | Eum, Kiyoon |
| contents | In Kähler geometry, the Donaldson-Fujiki moment map picture interprets the scalar curvature of a Kähler metric as a moment map on the space of compatible almost complex structures on a fixed symplectic manifold. In this paper, we generalize this picture using the framework of equivariant determinant line bundles. Given a prequantization $P=(L,h,\nabla)$ of a compact symplectic manifold $(M,ω)$, let $\mathcal{G}=\mathrm{Aut}(P)$. We construct for each $k\in\mathbb{N}$ a $\mathcal{G}$-equivariant determinant line bundle $λ^{(k)}\rightarrow\mathcal{J}_{int}$ on the space of integrable compatible almost complex structures, equipped with the $\mathcal{G}$-invariant Quillen metric. The curvature form of $λ^{(k)}$ admits an asymptotic expansion whose coefficients yield a sequence of $\mathcal{G}$-invariant closed two-forms $Ω_j$ on $\mathcal{J}_{int}$ and corresponding moment maps $μ_j:\mathcal{J}_{int}\rightarrow C^\infty(M)$. Each $μ_j$ arises from the asymptotic expansion of the variation of the log of the Quillen metric with respect to Kähler potentials, keeping the complex structure fixed. This provides a natural generalization of the Donaldson-Fujiki moment map interpretation of scalar curvature. Moreover, we show that $μ_j$ coincide with the $Z$-critical equations introduced by Dervan-Hallam, and we state a generalization of Fujiki's fiber integral formula. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_25456 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic expansion of the variation of the Quillen metric and its moment map interpretation Eum, Kiyoon Differential Geometry Complex Variables 58J52, 32Q15 In Kähler geometry, the Donaldson-Fujiki moment map picture interprets the scalar curvature of a Kähler metric as a moment map on the space of compatible almost complex structures on a fixed symplectic manifold. In this paper, we generalize this picture using the framework of equivariant determinant line bundles. Given a prequantization $P=(L,h,\nabla)$ of a compact symplectic manifold $(M,ω)$, let $\mathcal{G}=\mathrm{Aut}(P)$. We construct for each $k\in\mathbb{N}$ a $\mathcal{G}$-equivariant determinant line bundle $λ^{(k)}\rightarrow\mathcal{J}_{int}$ on the space of integrable compatible almost complex structures, equipped with the $\mathcal{G}$-invariant Quillen metric. The curvature form of $λ^{(k)}$ admits an asymptotic expansion whose coefficients yield a sequence of $\mathcal{G}$-invariant closed two-forms $Ω_j$ on $\mathcal{J}_{int}$ and corresponding moment maps $μ_j:\mathcal{J}_{int}\rightarrow C^\infty(M)$. Each $μ_j$ arises from the asymptotic expansion of the variation of the log of the Quillen metric with respect to Kähler potentials, keeping the complex structure fixed. This provides a natural generalization of the Donaldson-Fujiki moment map interpretation of scalar curvature. Moreover, we show that $μ_j$ coincide with the $Z$-critical equations introduced by Dervan-Hallam, and we state a generalization of Fujiki's fiber integral formula. |
| title | Asymptotic expansion of the variation of the Quillen metric and its moment map interpretation |
| topic | Differential Geometry Complex Variables 58J52, 32Q15 |
| url | https://arxiv.org/abs/2510.25456 |