The VMC survey IV. The LMC star formation history and disk geometry from four VMC tiles

We derive the star formation history (SFH) for several regions of the Large Magellanic Cloud (LMC), using deep near-infrared data from the VISTA near-infrared YJK(s) survey of the Magellanic system (VMC). The regions include three almost-complete 1.4 deg(2) tiles located similar to 3.5 degrees away from the LMC centre in distinct directions. They are split into 21.0' x 21.5' (0.12 deg(2)) subregions, and each of these is analysed independently. To this dataset, we add two 11.3' x 11.3' (0.036 deg(2)) subregions selected based on their small and uniform extinction inside the 30 Doradus tile. The SFH is derived from the simultaneous reconstruction of two different colour-magnitude diagrams (CMDs), using the minimization code StarFISH together with a database of partial models representing the CMDs of LMC populations of various ages and metallicities, plus a partial model for the CMD of the Milky Way foreground. The distance modulus (m-M)(0) and extinction A(V) is varied within intervals similar to 0.2 and similar to 0.5 mag wide, respectively, within which we identify the best-fitting star formation rate SFR(t) as a function of lookback time t, age-metallicity relation (AMR), (m-M)(0) and A(V). 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) similar or equal to 9.3 and 9.7, which appear in most of the subregions. The most recent SFR(t) 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(t) seems remarkably constant over the time interval from log(t/yr) similar or equal to 8.4 to 9.7. The AMRs, instead, turn out to be remarkably similar across the LMC. Thanks to the accuracy in determining the distance modulus for every subregion - with typical errors of just similar to 0.03 mag - we make a first attempt to derive a spatial model of the LMC disk. The fields studied so far are fit extremely well by a single disk of inclination i = 26.2 +/- 2.0 degrees, position angle of the line of nodes theta(0) = 129.1 +/- 13.0 degrees, and distance modulus of (m-M)(0) = 18.470 +/- 0.006 mag (random errors only) up to the LMC centre. We show that once the (m-M)(0) values or each subregion are assumed to be identical to those derived from this best-fitting plane, systematic errors in the SFR(t) and AMR are reduced by a factor of about two.