Intersection theory.

*(English)*Zbl 0541.14005If the intersection of two subvarieties \(U\) and \(X\) of an ambient space \(Y\) is to be studied by elementary computation, the most convenient situation is to have one of the two, \(U\) say, in parameter representation and the other, \(X\), given by equations \((\ldots = 0)\) in terms of coordinates on \(Y\). One then substitutes the parameters into the equations. This introduces an asymmetry in the study of the intersection, which is fundamental in intersection theories. Another remark is on excess dimension. If, for instance, \(U\) and \(X\) are a line and a plane in \(Y=\mathbb{P}^3\), one expects an intersection point, but if \(U\subset X\), the intersection point (to be viewed as limit of the intersection point of a slightly perturbed situation) is undetermined. In general, one obtains a rational equivalence class of the proper dimension on \(U\cap X\).

Both these aspects are reflected in the basic construction of the book, which associates to the given situation of a regular imbedding \(i: X\to Y\) of codimension \(d\) and a morphism \(f: V\to Y\) \((V\) plays the role of the parameter space of above), a rational equivalence class \(X._YV\) of \((k-d)\)-cycles on \(W=f^{-1}(X)\), where \(k=\dim V\). It is obtained by conceiving the normal cone \(C\) to \(W\) in \(V\) as a cycle on the pull-back \(N\) to \(W\) of the normal bundle to \(X\) in \(Y\), i.e., by respecting the multiplicities of its components, and intersecting this cycle with the zero section of \(N\) (this is the \(\ldots = 0\) of above); the intersection cycle then represents the intersection class \(X._YV\). This associates to the imbedding \(i\) a collection of homomorphisms of the rational equivalence classes \(i^!: A_kY' \to A_{k-d}(X\times_YY')\) for morphisms \(Y'\to Y\). Such a collection is called a bivariant class and the formalization of the theory is called bivariant intersection theory.

If the imbedding of \(W\) in \(V\) is regular of codimension \(d'\), one has an excess intersection formula \(X.V=c_{d-d'}(N/C)\cap [W]\) (denoting Chern class of the vector bundle \(N/C\) over \(W)\). If \(V\) is a subvariety of \(Y\) and \(Z\) is a given closed subscheme of \(W=X\cap V,\) one can write the intersection class \(X.V\) as the sum of a certain class on \(Z\) and a residual class \(\in A_{k-d}(R)\) on the “residual subscheme” \(R\) to \(Z\) in \(W\) and the formula is called the residual intersection formula. By blowing up, one may assume \(Z\) to be a divisor on \(V\); then \(R\) is such that \(W=Z\cup R\) and the sheaf of ideals \(\mathcal I(W)=\mathcal I(Z)\cdot \mathcal I(R)\) on \(V\). Both the excess intersection formula and the residual intersection formula are improved in the bivariant formulation.

There are further chapters on applications, such as to Schubert calculus, to the Riemann-Roch theorem for singular varieties (and without quasi-projectivity assumption), etc., on other cycle equivalences and, in case of varieties over \(\mathbb{C}\), comparison with homology cycle classes, on the degeneracy loci of a homomorphism of vector bundles, on correspondences, on the link of intersection theory with higher K-theory, etc.

Two appendices recapitulate the prerequisites of commutative algebra and of algebraic geometry; Chern classes are treated in a separate chapter.

Each chapter has a summary and many historical remarks are given at the end of the chapters. There is an extensive bibliography (“only a sampling”…).

Both these aspects are reflected in the basic construction of the book, which associates to the given situation of a regular imbedding \(i: X\to Y\) of codimension \(d\) and a morphism \(f: V\to Y\) \((V\) plays the role of the parameter space of above), a rational equivalence class \(X._YV\) of \((k-d)\)-cycles on \(W=f^{-1}(X)\), where \(k=\dim V\). It is obtained by conceiving the normal cone \(C\) to \(W\) in \(V\) as a cycle on the pull-back \(N\) to \(W\) of the normal bundle to \(X\) in \(Y\), i.e., by respecting the multiplicities of its components, and intersecting this cycle with the zero section of \(N\) (this is the \(\ldots = 0\) of above); the intersection cycle then represents the intersection class \(X._YV\). This associates to the imbedding \(i\) a collection of homomorphisms of the rational equivalence classes \(i^!: A_kY' \to A_{k-d}(X\times_YY')\) for morphisms \(Y'\to Y\). Such a collection is called a bivariant class and the formalization of the theory is called bivariant intersection theory.

If the imbedding of \(W\) in \(V\) is regular of codimension \(d'\), one has an excess intersection formula \(X.V=c_{d-d'}(N/C)\cap [W]\) (denoting Chern class of the vector bundle \(N/C\) over \(W)\). If \(V\) is a subvariety of \(Y\) and \(Z\) is a given closed subscheme of \(W=X\cap V,\) one can write the intersection class \(X.V\) as the sum of a certain class on \(Z\) and a residual class \(\in A_{k-d}(R)\) on the “residual subscheme” \(R\) to \(Z\) in \(W\) and the formula is called the residual intersection formula. By blowing up, one may assume \(Z\) to be a divisor on \(V\); then \(R\) is such that \(W=Z\cup R\) and the sheaf of ideals \(\mathcal I(W)=\mathcal I(Z)\cdot \mathcal I(R)\) on \(V\). Both the excess intersection formula and the residual intersection formula are improved in the bivariant formulation.

There are further chapters on applications, such as to Schubert calculus, to the Riemann-Roch theorem for singular varieties (and without quasi-projectivity assumption), etc., on other cycle equivalences and, in case of varieties over \(\mathbb{C}\), comparison with homology cycle classes, on the degeneracy loci of a homomorphism of vector bundles, on correspondences, on the link of intersection theory with higher K-theory, etc.

Two appendices recapitulate the prerequisites of commutative algebra and of algebraic geometry; Chern classes are treated in a separate chapter.

Each chapter has a summary and many historical remarks are given at the end of the chapters. There is an extensive bibliography (“only a sampling”…).

Reviewer: J. H.de Boer (Nijmegen)

##### MSC:

14C17 | Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry |

14-02 | Research exposition (monographs, survey articles) pertaining to algebraic geometry |

14F99 | (Co)homology theory in algebraic geometry |