1 Introduction

A significant fraction of the old stars we now observe in galaxies appear to have formed in Starbursts. Therefore, understanding violent star formation becomes crucial if we want to understand the star formation history of the Universe. 30 Doradus in the LMC is the nearest and best studied example of a massive starburst cluster and thus it has become a sort of "Rosetta Stone'' for deciphering the physics of violent star formation processes (Walborn 1991; Selman et al. 1999b (Paper III) and references therein).

Although 30 Dor has been the subject of intensive observational effort from the ground and space there are still a number of critical problems that remain unsolved. Perhaps the most burning open problem is the issue of mass segregation first raised by Malumuth & Heap (1995) that has important implications for our understanding of the process of cluster formation in general (Clarke 2001).

This Paper is the fourth of a series devoted to a comprehensive study of the 30 Doradus starburst cluster (NGC 2070). We can summarize the central results of the previous papers of this series as follows:

• The Initial Mass Function (IMF) of NGC 2070 between 3 and 120 $M_\odot$ is very well represented by a Salpeter power-law. The total "photometric'' mass of the cluster - inside 20 pc - from 0.1 $M_\odot$ to 120 $M_\odot$ is $M_{\rm {phot}}\sim 3 \times 10^5$ $M_\odot$;

• The star-formation history of the cluster shows three distinct peaks centered 5 Myrs, 2.5 Myrs, and <1.5 Myrs ago with no evidence for a spatial segregation of ages;

• The density distribution of stars between 20 and 60 $M_\odot$ is a power-law of slope of -2.85. This means that the total mass increases very slowly with radius (r^0.15 or even logarithmically);

• There is some evidence for mass segregation revealed by the fact that the most massive stars appear to be more concentrated toward the center. However, there is a distinct "ring'' of massive stars located at about 6 pc from the center of the cluster (R136). This ring contains stars of the three age groups described above.

The purpose of the present Paper is to study the stellar kinematics of the brightest stars in the cluster using the NTT observations described in Paper II of this series (Bosch et al. 1999). We use the data to investigate the presence of dynamical mass segregation in the cluster, and to attempt to constrain the lower end of the IMF from the comparison between the photometric and the dynamical masses.