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The goal of this article is twofold. On one hand, we study the subvarietiesof projective varieties which possess partially ample normal bundle; we provethat they are G2 in the ambient space. This generalizes results of Hartshorneand B\u{a}descu-Schneider. We work with the cohomological partial amplenessintroduced by Totaro.  On the other hand, we define the concept of a partially ample subvariety,which generalizes the notion of an ample subvariety introduced by Ottem. Weprove that partially ample subvarieties enjoy the stronger G3 property.Moreover, we present an application to a connectedness problem posed byFulton-Hansen and Hartshorne. The results are illustrated with examples.

Subvarieties with q-ample normal bundle and q-ample subvarieties