  Context. This is the fourth paper in a series showing the results of planet
population synthesis calculations.
  Aims. Our goal in this paper is to systematically study the effects of
important disk properties, namely disk metallicity, mass and lifetime on
fundamental planetary properties.
  Methods. For a large number of protoplanetary disks we calculate a population
of planets with our core accretion formation model including planet migration
and disk evolution.
  Results. We find a large number of correlations: Regarding the planetary
initial mass function, metallicity, disk mass and disk lifetime have different
roles: For high [Fe/H], giant planets are more frequent. For high disk masses,
giant planets are more massive. For long disk lifetimes, giant planets are both
more frequent and massive. At low metallicities, very massive giant planets
cannot form, but otherwise giant planet mass and metallicity are uncorrelated.
In contrast, planet masses and disk gas masses are correlated. The sweet spot
for giant planet formation is at 5 AU. In- and outside this distance, higher
planetesimals surface densities are necessary. Low metallicities can be
compensated by high disk masses, and vice versa, but not ad infinitum. At low
metallicities, giant planets only form outside the ice line, while at high
metallicities, giant planet formation occurs throughout the disk. The extent of
migration increases with disk mass and lifetime and usually decreases with
metallicity. No clear correlation of metallicity and the semimajor axis of
giant planets exists because in low [Fe/H] disks, planets start further out,
but migrate more, whereas for high [Fe/H] they start further in, but migrate
less. Close-in low mass planets have a lower mean metallicity than Hot
Jupiters.
  Conclusions. The properties of protoplanetary disks are decisive for the
properties of planets, and leave many imprints.
