# A crib sheet for surfaces

- \(q(X)\) (irregularity) is the Hodge number \(h^{0,1}\),
- \(p_g(X)\) (geometric genus) is the Hodge number \(h^{0,2}\),
- \(\chi(X)\) (holomorphic Euler characteristic) is the alternating
sum \(h^{0,2}-h^{0,1}+h^{0,0}\),

## Characteristic classes and intersection form

A complex surface has characteristic numbers \(c_1^2\), \(c_2\) and \(p_1\). The class \(c_2\) equals the topological Euler characteristic of \(X\). Since \(TX\otimes\mathbf{C}\cong TX\oplus (TX)^*\) we get \[p_1(X)=-c_2(TX\otimes\mathbf{C})=c_1^2(X)-2c_2(X).\] The Hirzebruch signature theorem tells us that the signature is given by \(\sigma=p_1/3=(2c_2-c_1^2)/3\). We write \(b^+\) and \(b^-\) for the dimensions of the positive and negative eigenspaces of the intersection form.

One can relate the invariants \(q\), \(p_g\) and \(\chi\) to the Chern numbers \(c_2\), \(c_1^2\) and the number \(b^+\) as follows.

- By Hirzebruch-Riemann-Roch, we have \(\chi=\frac{1}{12}(c_1^2+c_2)\)
(which equals the Todd class of \(X\)). This is called
*Noether's formula*. - By the Hodge index theorem, the negative part of the intersection
form is of dimension \(h^{1,1}-1\), so we have \(b^+=2p_g+1\),

## Examples

### \(\mathbf{CP}^2\)

The complex projective plane has \(q=p_g=0\), \(\chi=1\), \(b^+=1\), \(c_1^2=9\) and \(c_2=3\). The Hodge diamond is:

### \(K3\)

A K3 surface has \(q=0\), \(p_g=1\), \(\chi=2\), \(b^+=3\), \(c_1^2=0\) and \(c_2=24\). The Hodge diamond is:

### Hypersurface of degree \(d\) in \(\mathbf{CP}^3\).

Generalising the previous two examples (\(d=1,4\)), the hypersurface of degree \(d\) in \(\mathbf{CP}^3\) is simply-connected (by the Lefschetz hyperplane theorem) so \(q=0\), and has \(c_1^2=d(d-4)^2\), \(c_2=d(d^2-4d+6)\) (see e.g. here for how to do this computation). This means \[\chi=\frac{1}{6}d(d^2-6d+11)=p_g+1.\] For example, the Hodge diamond of the quintic surface (\(c_1^2=\chi=5\), \(p_g=4\), \(c_2=55\)) is therefore:

## Minimal surfaces of general type

### Geography

The characteristic numbers of minimal surfaces of general type satisfy some further inequalities:

- Noether's inequality: \[p_g\leq \frac{1}{2}c_1^2+2.\] When \(q=0\), we have \(\frac{1}{12}(c_1^2+c_2)=\chi=p_g+1\) and this inequality becomes \[c_2\leq 5c_1^2+36.\]
- The Bogomolov-Miyaoka-Yau inequality: \[c_1^2\leq 3c_2.\]
- \(c_1^2+c_2\equiv 0\mod 12\) (by Noether's formula).
- \(c_1^2\) and \(c_2\) are positive.

### (Numerically) Godeaux surfaces (\(p_g=q=0\), \(c_1^2=1\))

### Horikawa surfaces (on the Noether line)

These have been studied by Horikawa. See this summary for much more information about these surfaces.

### Ball quotients (on BMY line)

Yau proved that surfaces on the BMY line are all uniformised by the complex hyperbolic ball.

**Update:** See also this page of Pieter Belmans and Johan Commelin
for an interactive complex surface explorer!