The resolved star-formation relation in nearby active galactic nuclei

Aims. We present an analysis of the relation between the star formation rate (Sigma(SFR)) surface density (Sigma(SFR)) and mass surface density of molecular gas (Sigma(H2)), commonly referred to as the Kennicutt-Schmidt (K-S) relation, on its intrinsic spatial scale, i.e. the size of giant molecular clouds (similar to 10-150 pc), in the central, high-density regions of four nearby low-luminosity active galactic nuclei (AGN). These are AGN extracted from the NUclei of GAlaxies (NUGA) survey. This study investigates the correlations and slopes of the K-S relation, as a function of spatial resolution and of the different (CO)-C-12 emission lines used to trace Sigma(H2), and tests its validity in the high-density central regions of spiral galaxies. Methods. We used interferometric IRAM (CO)-C-12(1-0) and (CO)-C-12(2-1) and SMA (CO)-C-12(3-2) emission line maps to derive Sigma(H2) and HST-H alpha images to estimate Sigma(SFR). Results. Each galaxy is characterized by a distinct molecular SF relation on spatial scales between 20 to 200 pc. The K-S relations can be sublinear, but also superlinear, with slopes ranging from similar to 0.5 to similar to 1.3; slopes are generally superlinear on spatial scales >100 pc and sublinear on smaller scales. Depletion times range from similar to 1 and 2 Gyr, which is compatible with results for nearby normal galaxies. These findings are valid independently of which transition -(CO)-C-12(1-0), (CO)-C-12(2-1), or (CO)-C-12(3-2) - is used to derive Sigma(H2). Because of either star-formation feedback, the lifetime of clouds, turbulent cascade, or magnetic fields, the K-S relation might be expected to degrade on small spatial scales (< 100 pc). However, we find no clear evidence of this, even on scales as small as similar to 20 pc, and this might be because of the higher density of GMCs in galaxy centers that have to resist higher shear forces. The proportionality between Sigma(H2) and Sigma(SFR) found between 10 and 100 M-circle dot pc(-2) is valid even at high densities, similar to 10(3) M-circle dot pc(-2). However, by adopting a common CO-to-H-2 conversion factor (alpha(CO)), the central regions of the NUGA galaxies have higher Sigma(SFR) for a given gas column than those expected from the models, with a behavior that lies between the mergers or high-redshift starburst systems and the more quiescent star-forming galaxies, assuming that the first ones require a lower value of alpha(CO).