 
Summary: arXiv:0909.4690v4[math.AG]19Jul2010
DEFORMATIONS OF KšAHLER MANIFOLDS WITH NON
VANISHING HOLOMORPHIC VECTOR FIELDS
JAUME AMOR ŽOS, M `ONICA MANJARŽIN, MARCEL NICOLAU
Abstract. We study compact Kšahler manifolds X admitting nonvanishing
holomorphic vector fields, extending the classical birational classification of
projective varieties with tangent vector fields to a classification modulo de
formation in the Kšahler case, and biholomorphic in the projective case. We
introduce and analyze a new class of tangential deformations, and show that
they form a smooth subspace in the Kuranishi space of deformations of the
complex structure of X. We extend Calabi's theorem on the structure of com
pact Kšahler manifolds X with c1(X) = 0 to compact Kšahler manifolds with
nonvanishing tangent fields, proving that any such manifold X admits an arbi
trarily small tangential deformation which is a suspension over a torus; that is,
a quotient of F Ś Cs fibering over a torus T = Cs/. We further show that ei
ther X is uniruled or, up to a finite Abelian covering, it is a small deformation
of a product F Ś T where F is a Kšahler manifold without tangent vector fields
and T is a torus. A complete classification when X is a projective manifold,
in which case the deformations may be omitted, or when dim X s + 2 is
also given. As an application, it is shown that the study of the dynamics of
