Question. Are there smooth complex surfaces of general type with an irregularity $q = 1$ and 
Euler characteristic $3$. 


If the answer is yes, what is known about the geometry of such surfaces? Are there some explicit constructions? 



Algebraic surfaces of general type
If you mean *holomorphic* Euler characteristic equal to $3$, i.e. $p\_g=3$, then the answer is *yes*.


For some explicit constructions, look at the paper by Takahashi


[Certain algebraic surfaces of general type with irregularity one and their canonical mappings](https://projecteuclid.org/journals/tohoku-mathematical-journal/volume-50/issue-2/Certain-algebraic-surfaces-of-general-type-with-irregularity-one-and/10.2748/tmj/1178224978.full), Tohoku Math. J. (2) Volume **50**, Number 2 (1998), 261-290.


For any value of $p\_g \geq 2$, the author shows the existence of minimal surfaces of general type with $K^2=3p\_g$ and $q=1$ and studies their canonical mappings.


These surfaces are the minimal resolution of a relative quartic hypersurface (having at most rational double points as singularities) in a $\mathbf{P}^2$-bundle over an elliptic curve.


If you mean instead *topological* Euler characteristic, Noether formula gives $$K^2=12 \chi(\mathcal{O}) -3.$$


On the other hand, by Bogomolov-Miyaoka-Yau inequality one has $K^2 \leq 9 \chi(\mathcal{O})$, so the only possibility is $\chi(\mathcal{O})=1$, i.e. $p\_g=q=1$, $K^2=9$.


As far as I know, the existence of such a surface was announced by Cartwright and Steger. They used a ball-quotient construction (based also on computer calculations), similar to the one used by Prasad and Yeung in order to classify the fake projective planes. I do not think that a more explicit geometric construction is currently known.


