  We present the first fully dynamic connectivity data structures for geometric
intersection graphs achieving constant query time and sublinear amortized
update time for most types of geometric objects in 2D. Our data structures can
answer connectivity queries between two objects, as well as "global"
connectivity queries (e.g., deciding whether the entire graph is connected).
Previously, the data structure by Afshani and Chan (ESA'06) achieved such
bounds only in the special case of axis-aligned line segments or rectangles but
did not work for arbitrary line segments or disks, whereas the data structures
by Chan, P\u{a}tra\c{s}cu and Roditty (FOCS'08) worked for more general classes
of geometric objects but required $n^{\Omega(1)}$ query time and could not
handle global connectivity queries. Specifically, we obtain new data structures
with $O(1)$ query time and amortized update time near $n^{4/5}$, $n^{7/8}$, and
$n^{20/21}$ for axis-aligned line segments, disks, and arbitrary line segments
respectively. Besides greatly reducing the query time, our data structures also
improve the previous update times for axis-aligned line segments by Afshani and
Chan (from near $n^{10/11}$ to $n^{4/5}$) and for disks by Chan,
P\u{a}tra\c{s}cu, and Roditty (from near $n^{20/21}$ to $n^{7/8}$).
