Hodge diamond of complete intersections

Question

Suppose we have a smooth complete intersection of hypersurfaces with degrees $d_1,...,d_r$ in some $\mathbb{P}^N$. This should be a surface and in certain situations a surface of general type. What can one say about the Hodge diamond? Or what is its Grothendieck group ?

Answer 1

This question is addressed in Appendix I of Topological Methods in Algebraic Geometry. Let $V_n^{d_1, \ldots, d_r}$ denote the complete intersection of $r$ generic hypersurfaces of degrees $d_1, \ldots d_r$ in $\mathbb{P}^{n + r}$. Let

$$ \chi_y(V_n^{d_1,\ldots, d_r}) = \sum_{p,q \geq 0} (-1)^qh^{p,q}(V_n^{d_1,\ldots, d_r})y^p = \sum_{p \geq 0} \chi^p(V_n^{d_1,\ldots, d_r})y^p $$

where $y$ is an indeterminate and $h^{p,q}$ are the Hodge numbers and

$$ \chi^p(V_n^{d_1,\ldots, d_r}) = \sum_{q \geq 0} (-1)^qh^{p,q}(V_n^{d_1,\ldots, d_r}). $$

Then Theorem 22.1.1 of the above reference says that

$$ \sum_{n \geq 0} \chi_y(V_n^{d_1,\ldots, d_r}) z^{n+r} = \frac{1}{(1-z)(1 + zy)}\prod_{i=1}^r\frac{(1 + zy)^{d_i}-(1-z)^{d_i}}{(1+zy)^{d_i}+y(1-z)^{d_i}}. $$

This let's you compute the numbers $\chi^p(V_n^{d_1,\ldots, d_r})$ which aren't exactly the Hodge numbers. However, the next Theorem makes it possible to find the actual Hodge numbers from this data. Thereom 22.1.2 in the same section says that

$$ h^{p,q}(V_n^{d_1,\ldots, d_r}) = \delta_{p,q} \enspace \enspace \text{for} \enspace \enspace p + q \neq n, $$

$$ \chi^p(V_n^{d_1,\ldots, d_r}) = (-1)^{n-p}h^{p,n-p}(V_n^{d_1,\ldots, d_r}) + (-1)^p \enspace \enspace \text{for} \enspace \enspace 2p \neq n $$

and

$$ \chi^m(V_n^{d_1,\ldots, d_r}) = (-1)^mh^{m,m}(V_n^{d_1,\ldots, d_r}) \enspace \enspace \text{for} \enspace \enspace 2m = n $$

Answer 2

The Grothendieck group/ring tensor $Q$ is the same as the Severi-Chow group/ring tensor $Q$, which varies a lot when you vary your hypersurfaces. This happens already in the case of algebraic surfaces in $P^3$.

Answer 3

I have written a small (Python) program which computes Hodge numbers of hypersurfaces;

It is very easy to modify to work for complete intersections, too.

It contains references (to Hirzebruch and Deligne) in the comments section, and also a couple of examples in dimensions two and three:

Link, at the GitLab