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Singularity of a Holomorphic Map Jing Zhang State University of New York, Albany

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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 H i (X, Z) → H i (Y, Z) is an isomorphism for 0≤i≤n-2.

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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.

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Let D be an effective big divisor on a compact connected complex manifold. We assume that X is projective. D is big if h 0 (X, O(nD))≥ cn d, where c>0 is a constant and d is the dimension of X.

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Question : Let D be an effective big divisor on an irreducible smooth projective variety X. Given a point P 0 on X, is a general divisor passing through P 0 in |nD| is smooth for sufficiently large n?

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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 P 0 be a point on Y. Let π: X’→ X be the blow up of X at P 0. 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.

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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.

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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.

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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.

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Let U be an open subset of C n and f a holomorphic map from U to C m. Then the Jacobian matrix of f is Jf=(əf i /əz j ) 0≤i≤m, 0≤j≤ n where z 1,…, z n are local coordinates.

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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.

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Definition : Let f: X→Y be a holomorphic map between two complex manifolds of dimension n and m, where f=(f 1,..., f m ), each f i 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.

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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.

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More precisely, let X j = {xϵX, rank Jf(x)=j}. Then dimf(X j )≤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.

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Definition. The dimension of the vector space Op/ (Əf 1 /Əz 1,..., Əf m /Əz n ) over the complex field C is called the Milnor number of the holomorphic map f at the point P, where (Əf 1 /Əz 1,..., Əf m /Əz n ) is the ideal generated by all partial derivatives Əf 1 /Əz 1,..., Əf m /Əz n in Op.

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Let U be an open subset in C n and V an open subset of C m. Let f: (U, 0)→ (V, 0) be a holomorphic map such that f(z)=(f 1 (z),..., f m (z)), where z=(z 1,..., z m ) are the local coordinates.

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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 z i, there is a positive integer N i, such that z i Ni is contained in the ideal (Əf 1 /Əz 1,..., Əf m /Əz n ), and Əf i /Əz j (0)=0 for all i=1,..., m, j=1,..., n.

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If h 0 (X, O X (nD))>0 for some n>0 and X is normal, choose a basis f 0, f 1, …, f m of the vector space H 0 (X, O X (nD)), it defines a rational map Φ |nD| from X to the projective space P m by sending a point x on X to (f 0 (x), f 1 (x), …, f m (x)) in P m. 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.

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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 X j ={x ϵ X j, rank(Jf(x))=j}. If dimX j ≤ j-1 for all 0

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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 Y j =X j ∩Y is less than j and dimY 0 =0, then the general member of |nD| passing through a fixed point P 0 on Y\Y 0 is a smooth divisor on Y. Here at every point y in Y j, the Jacobian matrix of f has rank j.

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