Rigid currents on compact hyperkahler manifolds
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909973971206144 |
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| author | Sibony, Nessim Soldatenkov, Andrey Verbitsky, Misha |
| author_facet | Sibony, Nessim Soldatenkov, Andrey Verbitsky, Misha |
| contents | A rigid cohomology class on a complex manifold is a class that is represented by a unique closed positive current. The positive current representing a rigid class is also called rigid. For a compact Kahler manifold $X$ all eigenvectors of hyperbolic automorphisms acting on $H^{1,1}(X)$ that have non-unit eigenvalues are rigid classes. Such classes are always parabolic, namely, they belong to the boundary of the Kahler cone and have vanishing volume. We study parabolic $(1,1)$-classes on compact hyperkahler manifolds with $b_2 \geq 7$. We show that a parabolic class is rigid if it is not orthogonal to a rational vector with respect to the BBF form. This implies that a general parabolic class on a hyperkahler manifold is rigid. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_11362 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Rigid currents on compact hyperkahler manifolds Sibony, Nessim Soldatenkov, Andrey Verbitsky, Misha Algebraic Geometry Complex Variables Differential Geometry 53C26, 14J42, 32U40, 37F80 A rigid cohomology class on a complex manifold is a class that is represented by a unique closed positive current. The positive current representing a rigid class is also called rigid. For a compact Kahler manifold $X$ all eigenvectors of hyperbolic automorphisms acting on $H^{1,1}(X)$ that have non-unit eigenvalues are rigid classes. Such classes are always parabolic, namely, they belong to the boundary of the Kahler cone and have vanishing volume. We study parabolic $(1,1)$-classes on compact hyperkahler manifolds with $b_2 \geq 7$. We show that a parabolic class is rigid if it is not orthogonal to a rational vector with respect to the BBF form. This implies that a general parabolic class on a hyperkahler manifold is rigid. |
| title | Rigid currents on compact hyperkahler manifolds |
| topic | Algebraic Geometry Complex Variables Differential Geometry 53C26, 14J42, 32U40, 37F80 |
| url | https://arxiv.org/abs/2303.11362 |