49 research outputs found

### On compact splitting complex submanifolds of quotients of bounded symmetric domains

In the current article our primary objects of study are compact complex
submanifolds of quotient manifolds of irreducible bounded symmetric domains by
torsion free discrete lattices of automorphisms. We are interested in the
characterization of the totally geodesic submanifolds among compact splitting
complex submanifolds, i.e. under the assumption that the tangent sequence
splits holomorphically over the submanifold.Comment: Accepted for publication in SCIENCE CHINA Mathematic

### Rigidity of irreducible Hermitian symmetric spaces of the compact type under K"ahler deformation

We study deformations of irreducible Hermitian symmetric spaces $S$ of the
compact type, known to be locally rigid, as projective-algberaic manifolds and
prove that no jump of complex structures can occur. For each $S$ of rank $\ge
2$ there is an associated reductive linear group $G$ such that $S$ admits a
holomorphic $G$-structure, corresponding to a reduction of the structure group
of the tangent bundle. $S$ is characterized as the unique simply-connected
compact complex manifold admitting such a $G$-structure which is at the same
time integrable. To prove the deformation rigidity of $S$ it suffices that the
corresponding integrable $G$-structures converge.
We argue by contradiction using the deformation theory of rational curves.
Assuming that a jump of complex structures occurs, cones of vectors tangent to
degree-1 rational curves on the special fiber $X_0$ are linearly degenerate,
thus defining a proper meromorphic distribution $W$ on $X_0$. We prove that
such $W$ cannot possibly exist. On the one hand, integrability of $W$ would
contradict the fact that $b_2(X)=1$. On the other hand, we prove that $W$ would
be automatically integrable by producing families of integral complex surfaces
of $W$ as pencils of degree-1 rational curves. For the verification that there
are enough integral surfaces we need a description of generic cones on the
special fiber. We show that they are in fact images of standard cones under
linear projections. We achieve this by studying deformations of normalizations
of Chow spaces of minimal rational curves marked at a point, which are
themselves Hermitian symmetric, irreducible except in the case of
Grassmannians

### Ax-Schanuel for Shimura varieties

We prove the Ax-Schanuel theorem for a general (pure) Shimura variety

### Nonexistence of holomorphic submersions between complex unit balls equivariant with respect to a lattice and their generalizations

In this article we prove first of all the nonexistence of holomorphic
submersions other than covering maps between compact quotients of complex unit
balls, with a proof that works equally well in a more general equivariant
setting. For a non-equidimensional surjective holomorphic map between compact
ball quotients, our method applies to show that the set of critical values must
be nonempty and of codimension 1. In the equivariant setting the line of
arguments extend to holomorphic mappings of maximal rank into the complex
projective space or the complex Euclidean space, yielding in the latter case a
lower estimate on the dimension of the singular locus of certain holomorphic
maps defined by integrating holomorphic 1-forms. In another direction, we
extend the nonexistence statement on holomorphic submersions to the case of
ball quotients of finite volume, provided that the target complex unit ball is
of dimension m>=2, giving in particular a new proof that a local biholomorphism
between noncompact m-ball quotients of finite volume must be a covering map
whenever m>=2. Finally, combining our results with Hermitian metric rigidity,
we show that any holomorphic submersion from a bounded symmetric domain into a
complex unit ball equivariant with respect to a lattice must factor through a
canonical projection to yield an automorphism of the complex unit ball,
provided that either the lattice is cocompact or the ball is of dimension at
least 2

### Birationality of the tangent map for minimal rational curves

For a uniruled projective manifold, we prove that a general rational curve of
minimal degree through a general point is uniquely determined by its tangent
vector. As applications, among other things we give a new proof, using no Lie
theory, of our earlier result that a holomorphic map from a rational
homogeneous space of Picard number 1 onto a projective manifold different from
the projective space must be a biholomorphic map.Comment: AMS-tex, 14 pages, Dedicated to Yum-Tong Siu on his 60th birthda

### Proper holomorphic maps between bounded symmetric domains with small rank differences

In this paper we study the rigidity of proper holomorphic maps $f\colon
\Omega\to\Omega'$ between irreducible bounded symmetric domains $\Omega$ and
$\Omega'$ with small rank differences: $2\leq \text{rank}(\Omega')<
2\,\text{rank}(\Omega)-1$. More precisely, if either $\Omega$ and $\Omega'$
have the same type or $\Omega$ is of type~III and $\Omega'$ is of type~I, then
up to automorphisms, $f$ is of the form $f=\imath\circ F$, where $F = F_1\times
F_2\colon \Omega\to \Omega_1'\times \Omega_2'$. Here $\Omega_1'$, $\Omega_2'$
are bounded symmetric domains, the map $F_1\colon \Omega \to \Omega_1'$ is a
standard embedding, $F_2: \Omega \to \Omega_2'$, and $\imath\colon
\Omega'_1\times \Omega'_2 \to \Omega'$ is a totally geodesic holomorphic
isometric embedding. Moreover we show that, under the rank condition above,
there exists no proper holomorphic map $f: \Omega \to \Omega'$ if $\Omega$ is
of type~I and $\Omega'$ is of type~III, or $\Omega$ is of type~II and $\Omega'$
is either of type~I or III. By considering boundary values of proper
holomorphic maps on maximal boundary components of $\Omega$, we construct
rational maps between moduli spaces of subgrassmannians of compact duals of
$\Omega$ and $\Omega'$, and induced CR-maps between CR-hypersurfaces of mixed
signature, thereby forcing the moduli map to satisfy strong local
differential-geometric constraints (or that such moduli maps do not exist), and
complete the proofs from rigidity results on geometric substructures modeled on
certain admissible pairs of rational homogeneous spaces of Picard number 1