  The geometric properties of the critical fluctuations in abelian gauge
theories such as the Ginzburg-Landau model are analyzed in zero background
field. Using a dual description, we obtain scaling relations between exponents
of geometric and thermodynamic nature. In particular we connect the anomalous
scaling dimension $\eta$ of the dual matter field to the Hausdorff dimension
$D_H$ of the critical fluctuations, {\it which are fractal objects}. The
connection between the values of $\eta$ and $D_H$, and the possibility of
having a thermodynamic transition in finite background field, is discussed.
