Timeline for What kind of line bundles have Chern class of Hodge type (2,0) or (0,2)?
Current License: CC BY-SA 2.5
6 events
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| Jan 31, 2020 at 13:06 | comment | added | u184 | About than 10 years after I have a question: Are there known examples of such surfaces? In particular, say X is a K3 surface then $h^{2,0}=h^{0,2}=1$. Are there K3s with $H^{2,0}\oplus H^{0,2}$ generated by two integral classes? | |
| Jun 9, 2010 at 4:08 | history | edited | Tim Perutz | CC BY-SA 2.5 |
Corrected uniqueness
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| Jun 9, 2010 at 3:47 | history | edited | Tim Perutz | CC BY-SA 2.5 |
Struck out wrong genericity statement.
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| Jun 9, 2010 at 3:45 | comment | added | Tim Perutz | Ah yes, you're right, these subspaces depend only on the complex structure. Apologies for the rash afterthought. What I had in mind was that, as one varies the (conformal class of the) Riemannian metric on a 4-manifold, one can realize arbitrary variations in the splitting of the harmonic 2-forms into (anti-)self-dual parts. So, if both summands of the splitting are non-zero, they generically miss all the lattice points. But my attempt to adapt this to Kaehler metrics was misguided... | |
| Jun 9, 2010 at 3:11 | comment | added | David Treumann | I don't think I've understood it yet, but this is a cool interpretation. Tell me more about your catch: presumably to talk about whether or not a harmonic differential form is integral you are using the identification $\mathcal{H}^{0,2} + \mathcal{H}^{2,0} = H^{0,2} + H^{2,0}$. Isn't the right hand side, as a subspace of $H^2(X;C)$, independent of the Kahler form and even the Kahler class chosen? | |
| Jun 8, 2010 at 21:38 | history | answered | Tim Perutz | CC BY-SA 2.5 |