Singularity of a Holomorphic Map
Jing Zhang
State University of New York, Albany
Projective varieties are studied by linear
systems of their divisors and hyperplane
sections. For example, Lefschetz theorem
on hyperplane sections: Let X be an ndimensional submanifold of a projective
space and H a hyperplane such that H
intersects X with a complex manifold Y,
then the inclusion homomorphism
Hi (X, Z) → Hi (Y, Z)
is an isomorphism for 0≤i≤n-2.
Classical Bertini’s theorem states that a
general hyperplane section of an irreducible
smooth projective variety over an
algebraically closed field is smooth and
irreducible. In fact, given any point P on X, a
general hyperplane section passing through P
is irreducible and smooth.
Let D be an effective big divisor on a compact
connected complex manifold. We assume that
X is projective. D is big if
h0(X, O(nD))≥ cnd,
where c>0 is a constant and d is the
dimension of X.
Question: Let D be an effective big divisor
on an irreducible smooth projective variety X.
Given a point P0 on X, is a general divisor
passing through P0 in |nD| is smooth for
sufficiently large n?
Example.
Suppose that the dimension of
X is at least 3. Let D be an effective big
divisor on X. Let Y=X-D and P0 be a point on Y.
Let π: X’→ X be the blow up of X at P0. Then
the pull back π*D is a big divisor on X’. But
any effective big divisor D’ linearly equivalent
to π*D on X' passing through a point on the
exceptional divisor E is not smooth. It
contains E and another component G.
For the convenience, we will use the
following definition of base locus. Let F be
an effective divisor on X. We say that F is a
fixed component of linear system L if
E>F for all E in L. F is the fixed part of a
linear system if every irreducible component
of F is a fixed component of the system and F
is maximal with respect to the order ≥. If F is
the fixed part of L, then every element E in
the system can be written in the form E=E'+F.
We say that E' is the variable (or movable)
part of E.
A point x in X is a base point of the linear
system if x is contained in the supports
of variable parts of all divisors in the
system. The set of all base points of L is a
closed subset of the linear system L
(viewed as a projective space) called the
base locus of L.
Bertini’s theorem:
If κ(D, X)≥2, then
the variable part of a general member of the
complete linear system |D| is irreducible
and smooth away from the singular locus
of X and the base locus of |D|. Here the Ddimension κ(D, X) is the maximal dimension
of the image of the rational map defined by
|nD| for all n>0.
Let U be an open subset of Cn and f a
holomorphic map from U to Cm. Then
the Jacobian matrix of f is
Jf=(əf i/əz j) 0≤i≤m, 0≤j≤ n
where z1,…, zn are local coordinates.
Definition:
Let f: X → Y
be a
holomorphic map between two complex
manifolds. Its rank at a point P on X is its
rank of Jacobian at P. The rank of f is
defined to be the maximal rank of its
Jacobian on X.
Definition:
Let f: X→Y be a
holomorphic map between two complex
manifolds of dimension n and m, where
f=(f1 ,..., fm), each fi is a holomorphic
function on X. A point P in X is a critical
point if Jf(P) is not of maximal rank. It is a
singular critical point if
əf i/əz j(P)=0
for all i=1,..., m, j=1,..., n, that is, Jf(P) is a
zero matrix.
If f is proper and surjective holomorphic map,
then we have Sard's Theorem: There is a
nowhere dense analytic subset S of M such
that f has maximal rank at any point of X-S
and f(S) is a nowhere dense analytic subset
of Y.
More precisely, let
Xj = {xϵX, rank Jf(x)=j}.
Then dimf(Xj)≤j.
Notice that if f is not an algebraic
morphism and not proper, then even though
f(S) has Lebesgue measure zero in Y, it is very
complicated and might be dense in Y.
Definition.
The dimension of the vector
space
Op/ (Əf1/Əz1,..., Əf m/Əzn)
over the complex field C is called the Milnor
number of the holomorphic map f at the
point P, where (Əf1/Əz1,..., Əf m/Əzn) is the
ideal generated by all partial derivatives
Əf1/Əz1,..., Əf m/Əzn in Op.
Let U be an open subset in Cn and V an open
subset of Cm. Let f: (U, 0)→ (V, 0) be a
holomorphic map such that
f(z)=(f1(z),..., fm(z)),
where
z=(z1,..., zm )
are the local
coordinates.
Theorem. The origin 0
is an isolated
singular critical point of f if and only if
(1) the Milnor number is finite and not zero
or
(2) if and only if for every coordinate
function zi, there is a positive integer Ni, such
that ziNi is contained in the ideal
(Əf1/Əz1,..., Əf m/Əzn),
and Əf i/Əzj(0)=0 for all i=1,..., m, j=1,..., n.
If h0(X, OX(nD))>0 for some n>0 and X is
normal, choose a basis f0, f1, …, fm of the
vector space H0(X, OX(nD)), it defines a
rational map Φ|nD| from X to the
projective space Pm by sending a point x on
X to (f0(x), f1(x), …, fm(x)) in Pm. Φ|nD| is a
morphism if |nD| has no base locus, but
may have fixed components. In this case, in
fact, we replace Φ|nD| by Φ|nD-F|, where F is
the fixed part.
Theorem. Let X
be an irreducible smooth
projective variety of dimension d and D an
effective big divisor on X such that f= Φ|nD|
defines a birational morphism. Let
Xj ={x ϵ Xj, rank(Jf(x))=j}.
If dimXj≤ j-1 for all 0<j<d and dimX0≤0, then
for any point P0 on X\X0, the movable part of
a general member of L is smooth, where L is
the linear system of effective divisors passing
through P0 in |nD|.
Theorem.
Let X be a smooth complete
variety with an effective divisor D and
f=Φ|nD| for sufficiently large n. If the
dimension of Yj=Xj∩Y is less than j and
dimY0=0, then the general member of |nD|
passing through a fixed point P0 on Y\Y0 is a
smooth divisor on Y. Here at every point y in
Yj, the Jacobian matrix of f has rank j.
Thank you!