We derive the star formation history for several regions of the LMC, using deep near-infrared data from the VISTA near-infrared YJKs survey of the Magellanic system (VMC). The regions include three almost-complete 1.4 sqdeg tiles located 3.5 deg away from the LMC centre in distinct directions. To this dataset, we add two 0.036 sqdeg subregions inside the 30 Doradus tile. The SFH is derived from the simultaneous reconstruction of two different CMDs, using the minimization code StarFISH. The distance modulus (m-M)_0 and extinction Av is varied within intervals 0.2 and 0.5 mag wide, respectively, within which we identify the best-fitting star formation rate SFR(t), age-metallicity relation (AMR), (m-M)_0 and Av. Our results demonstrate that VMC data, due to the combination of depth and little sensitivity to differential reddening, allow the derivation of the space-resolved SFH of the LMC with unprecedented quality compared to previous wide-area surveys. In particular, the data clearly reveal the presence of peaks in the SFR(t) at ages log(t/yr)=9.3 and 9.7, which appear in most of the subregions. The most recent SFR is found to vary greatly from subregion to subregion, with the general trend of being more intense in the innermost LMC, except for the tile next to the N11 complex. In the bar region, the SFR seems remarkably constant over the time interval from 8.4 to 9.7. The AMRs, instead, turn out to be remarkably similar across the LMC. The fields studied so far are fit extremely well by a single disk of inclination 26.2+-2.0 deg, position angle of the line of nodes 129.1+-13.0 deg, and distance modulus of 18.470+-0.006 mag (random errors only) up to the LMC centre.
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