Surfaces with irregularity zero

algebraic-geometry

I was wondering if there is a classification of algebraic surfaces with irregularity zero (i.e $H^1(\mathcal{O}_X)=0$). If not, can you help me come up with examples other than (weak) del Pezzos, hypersurfaces in $\mathbb{P}^3$, branched covers of $\mathbb{P}^2$? I shall list them below:

I) (Ariyan Javanpeykar) Any simply connected surface, e.g., fake projective planes or smooth complete intersections in $\mathbb{P}^n$.

Any comments are appreciated. Thanks.

Best Answer

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Hi! By the Enriques classification we can see that for any Kodaira dimension we can find minimal surfaces with vanish irregularity. For example you can see the Beauville's book of surfaces or the known as the four authors book of surfaces, compact complex surfaces.

There are many examples of surfaces with irregularity zero, complete intersections, K3 surfaces, every Enriques surfaces, and Surfaces of general type as the Horikawa surfaces.

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