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In the literature there are two different definitions of polarized K3 surfaces.

1) A polarized K3 is the data $(X,\omega)$. Where $X$ is a K3 surface and $\omega$ is an ample class in $H^2(X,\mathbb{Z})$.

2) A polarized K3 is the data of a K3 surface $X$ together with a polarization on the Hodge decomposition $H^2(X,\mathbb{Z})$, which is a pairing $H^2(X,\mathbb{Z}) \times H^2(X,\mathbb{Z}) \rightarrow \mathbb{Z}$. Such a polarization is given by the intersection pairing: $$(v,w) \mapsto \int_X v \wedge w\;.$$

This, together with an observation from Huybrechts lecture notes on K3 surfaces page 40:

...it is not the intersection pairing that defines a polarization, but the pairing that is obtained from it by changing the sign of the intersection pairing for an ample class.

justifies the following question.

Question: How does the choice of an ample class $\omega$ comes into the definition of the intersection pairing?

It is clear that we need such a class to define a polarization on lower degree cohomology groups, for example on $H^1(X,\mathbb{Z})$. And the well definition of the intersection pairing is related to the existence of an ample class (i.e. the projectivity of $X$). But I don't see how the intersection pairing would change changing the choice of the ample class.

It is possible (or even probable) that I'm making a huge mess out of nothing, but I'm quite confused. Thank you in advance!

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1 Answer 1

up vote 5 down vote accepted

The intersection pairing on $H^2$ is non-degenerate (by Poincare duality) and is symmetric (we are considering even-dimensional cohomology), but it is not positive definite. One can "change the sign" of the intersection pairing to make it positive definite, but the change in sign is not uniform (i.e. there is not a single sign you have to change it by).

Roughly speaking, in order to define a polarization on $H^2$, you have to decompose $H^2$ into a direct sum of the span of the ample class and the primitive cohomology, and consider each one separately. For more details, see this MO answer. (So your second definition of polarization is not correct; a polarization is not just given by the intersection pairing; it is a modification of the intersection pairing to achieve certain positivity properties, and the modification depends on the choice of ample class.)

If you want to learn more about this, the keywords are primitive cohomology, Lefschetz decomposition, and Hodge--Riemann bilinear relations. Unfortunately wikipedia doesn't seem to give much detail on these topics; Griffiths and Harris has a brief sketch near the end of their discussion Hodge theory in Chapter 0.

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Thank you very much, now I see it! Unfortunately I didn't encounter the Lefschetz decomposition while googling around. I have a stupid follow up question: "Do all the polarizations of an algebraic K3 surface arise as intersection pairing with signs changed?". On one hand I would say no, because there are polarizable non-algebraic K3s. But on the other hand this would contradict the choice of a polarized algebraic K3 as a pair $(X,\omega)$ for $\omega$ ample. Again I'm a bit confused. Thank you! :) –  Giovanni De Gaetano Nov 26 '12 at 15:30

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