  The VISTA Magellanic Cloud (VMC, PI M.R. Cioni) survey is collecting deep
Ks-band time-series photometry of the pulsating variable stars hosted by the
system formed by the two Magellanic Clouds (MCs) and the "bridge" connecting
them. In this paper we present the first results for Classical Cepheids, from
the VMC observations of two fields in the Large Magellanic Cloud (LMC). The VMC
Ks-band light curves of the Cepheids are well sampled (12-epochs) and of
excellent precision. We were able to measure for the first time the Ks
magnitude of the faintest Classical Cepheids in the LMC (Ks\sim17.5 mag), which
are mostly pulsating in the First Overtone (FO) mode, and to obtain FO
Period-Luminosity (PL), Period-Wesenheit (PW), and Period-Luminosity-Color
(PLC) relations, spanning the full period range from 0.25 to 6 day. Since the
longest period Cepheid in our dataset has a variability period of 23 day, we
have complemented our sample with literature data for brighter F Cepheids. On
this basis we have built a PL relation in the Ks band that, for the first time,
includes short period pulsators, and spans the whole range from 1.6 to 100 days
in period. We also provide the first ever empirical PW and PLC relations using
the (V-Ks) color and time-series Ks photometry. The very small dispersion
(\sim0.07 mag) of these relations makes them very well suited to study the
three-dimensional (3D) geometry of the Magellanic system. The use of "direct"
(parallax- and Baade-Wesselink- based) distance measurements to both Galactic
and LMC Cepheids, allowed us to calibrate the zero points of the PL, PW, and
PLC relations obtained in this paper, and in turn to estimate an absolute
distance modulus of (m-M)0=18.46\pm0.03 for the LMC. This result is in
agreement with most of the latest literature determinations based on Classical
Cepheids.
