Symplectic Invariance of Sweeping Families of Rational Curves and Surfaces
-- Jason Starr, September 28, 2018

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By Koll&aacuter and Ruan, uniruledness of complex projective manifolds is 
symplectically invariant and deformation invariant for projective, 
smooth families over a connected base scheme (maybe reducible, maybe 
mixed characteristic, but avoiding an explicit list of "bad primes").  I 
will prove existence of a "sweeping" family of rational surfaces under a 
hypothesis of positivity of certain Gromov-Witten invariants.  This 
hypothesis is symplectically invariant and deformation invariant away 
from "bad primes".  In particular, there are sweeping families of 
rational surfaces in every complex projective manifold that is 
symplectically deformation equivalent to a projective homogeneous space.