Kunneth formula for Hessian manifolds
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918494683004928 |
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| author | Osipov, Pavel |
| author_facet | Osipov, Pavel |
| contents | We study Dolbeault--Koszul cohomology $H^{p,q}(M)$ of flat affine manifolds. We proove a Künneth formula \[ H^{p,q}(M\times N) \cong \bigoplus_{i,j} H^{i,j}(M)\otimes H^{p-i,q-j}(N) \] for flat affine manifolds $M,N$ with at least one compact. For compact manifolds we also give a proof via Hodge theory on flat affine manifolds, analogous to the classical Künneth formula for Dolbeault cohomology.
We apply this formula to Hessian manifolds. A Hessian metric $g$ defines a class $[g]\in H^{1,1}(M)$, and metrics in the same class differ by $Dα$ for a closed $1$-form $α$. Using the Künneth formula we describe all Hessian metrics on products, on products with hyperbolic manifolds, and on manifolds admitting a flat Riemannian metric. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_10743 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Kunneth formula for Hessian manifolds Osipov, Pavel Differential Geometry We study Dolbeault--Koszul cohomology $H^{p,q}(M)$ of flat affine manifolds. We proove a Künneth formula \[ H^{p,q}(M\times N) \cong \bigoplus_{i,j} H^{i,j}(M)\otimes H^{p-i,q-j}(N) \] for flat affine manifolds $M,N$ with at least one compact. For compact manifolds we also give a proof via Hodge theory on flat affine manifolds, analogous to the classical Künneth formula for Dolbeault cohomology. We apply this formula to Hessian manifolds. A Hessian metric $g$ defines a class $[g]\in H^{1,1}(M)$, and metrics in the same class differ by $Dα$ for a closed $1$-form $α$. Using the Künneth formula we describe all Hessian metrics on products, on products with hyperbolic manifolds, and on manifolds admitting a flat Riemannian metric. |
| title | Kunneth formula for Hessian manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2605.10743 |