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We present high-resolution observations of the 880 $\mu$m (rest-frame FIR)continuum emission in the z$=$4.05 submillimeter galaxy GN20 from the IRAMPlateau de Bure Interferometer (PdBI). These data resolve the obscured starformation in this unlensed galaxy on scales of0.3$^{\prime\prime}$$\times$0.2$^{\prime\prime}$ ($\sim$2.1$\times$1.3 kpc).The observations reveal a bright (16$\pm$1 mJy) dusty starburst centered on thecold molecular gas reservoir and showing a bar-like extension along the majoraxis. The striking anti-correlation with the HST/WFC3 imaging suggests that thecopious dust surrounding the starburst heavily obscures the rest-frameUV/optical emission. A comparison with 1.2 mm PdBI continuum data reveals noevidence for variations in the dust properties across the source within theuncertainties, consistent with extended star formation, and the peak starformation rate surface density (119$\pm$8 M$_{\odot}$ yr$^{-1}$ kpc$^{-2}$)implies that the star formation in GN20 remains sub-Eddington on scales down to3 kpc$^2$. We find that the star formation efficiency is highest in the centralregions of GN20, leading to a resolved star formation law with a power lawslope of $\Sigma_{\rm SFR}$ $\sim$ $\Sigma_{\rm H_2}^{\rm 2.1\pm1.0}$, and thatGN20 lies above the sequence of normal star-forming disks, implying that thedispersion in the star formation law is not due solely to morphology or choiceof conversion factor. These data extend previous evidence for a fixed starformation efficiency per free-fall time to include the star-forming medium on$\sim$kpc-scales in a galaxy 12 Gyr ago.

The kiloparsec-scale star formation law at redshift 4: wide-spread, highly efficient star formation in the dust-obscured starburst galaxy GN20