Singularity of a Holomorphic Map

Jing ZhangState 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 n-dimensional 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 D-dimension 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 derivativesƏf1/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 zi

Ni 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!