Timeline for Examples of quasi-negative but not negative holomorphic sectional curvature
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9 events
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| Jan 12, 2017 at 16:01 | comment | added | diverietti | Thanks for the remark on the bisectional curvature. But I am not convinced that it suggests the holomorphic sectional curvature tends to be zero in a lot of points: think at the bidisk with the product Poincaré metric... | |
| Jan 12, 2017 at 15:59 | comment | added | diverietti | The existence of a projective hypersurface of high degree cutting the abelian variety along a smooth surface and containing the fixed elliptic curve is guaranteed by the generalized Bertini's theorem contained in "Bertini theorems for hypersurface sections containing a subscheme" by Kleiman and B. Altman. | |
| Jan 10, 2017 at 8:06 | comment | added | Bo_Y | First, I thought there are no abelian varieties whose theta divisor can contain elliptic curves, I might be wrong here. I would like to know how to pick the degree of your hypersurface to ensure it contains elliptic curves. Second, in your example, assume other property you mentioned holds, we can see the bisectional curvature of the induced metric has lots of zeros in the following sense: given any point, and any direction $u$, there exists a direction $v$ such that $R_{u \bar{u} v \bar{v}}=0$, this can be derived since second segre form $c_1^2-c_2$ is 0. | |
| Jan 9, 2017 at 13:40 | comment | added | diverietti | @Bo_Y Hi. I am thinking t the abelian variety as embedded in some $\mathbb P^n$. Then I cut it with a projective hypersurface, which is an ample divisor. The intersection is thus an ample divisor in the abelian variety. If smooth, its canonical bundle is just the restriction of the line bundle defined by this divisor onto itself, by adjunction. Therefore it is ample. As for your second question, I actually don't know, I should think about it. Why do you ask? | |
| Jan 8, 2017 at 14:19 | comment | added | Bo_Y | Sorry if I said anything wrong, try to understand the example you proposed. the 3-dim abelian variety $Y$ has to be isogenous to the produce of $E$ and an abelian surface. Could you give some details on how to choose a high degree hypersurface to ensure its insection with $Y$ is a surface $X$ of $K_X>0$, and can we always realize such a $X$ as some Theta divisor of $Y$? | |
| Jan 5, 2017 at 9:27 | history | edited | diverietti | CC BY-SA 3.0 |
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| Jan 3, 2017 at 10:17 | history | edited | diverietti | CC BY-SA 3.0 |
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| Jan 2, 2017 at 10:38 | history | edited | diverietti | CC BY-SA 3.0 |
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| Jan 2, 2017 at 10:32 | history | asked | diverietti | CC BY-SA 3.0 |