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If $X$ is a smooth variety over the complex numbers, you can use the topological Euler characteristic of the support $X(\mathbb{C})$ of the scheme. The most efficient way to compute it is to use Chern classes and the Poincaré-Hopf theorem: the Euler characteristic is the degree of the top Chern class. Chern classes can be defined purely algebraically (in homology) as it is done for example in Chapter 3 of Fulton's book "Intersection Theory".

When $X$ is singular, there are many nonequivalent generalizations of the Euler characteristic. One efficient approach is again to work in the spirit of the Poincaré-Hopf theorem by first defining Chern classes for singular varieties. This provides a purely algebraic treatment. I will give a tour of the 4 most famous examples:

-The Chern-MacPherson class (also known as the Chern-Schwartz-MacPherson class ) is probably the most important. It can be defined on the Chow group of a variety using constructible functions. It enjoys beautiful functorial properties under proper maps and it is compatible with specializationspecializations. The existence of the Chern-MacPherson class realizes a conjecture of Deligne and Grothendieck of 1969 as the unique natural transformation between the covariant functor of constructible functionfunctions and the usual covariant functor of $\mathbb{Z}$-homology such that it maps the characteristic function of a manifold to its (homological) Chern class. The Chern-MacPherson class computecomputes the topological Euler characteristic of a (possibly singular) variety.

-The Chern-Mather-class enters the definition of the Chern-MacPherson class. It is relevant to compute the stringy Euler characteristic stringy invariants. It has nice properties under proper birational maps and can be simply defined using a Nash-blow-up.

These two classes (MacPherson and Mather) can be generalized to scheme by considering the support of the scheme. This makes calculations much more easy, but obviously one looses a lot of the information carried by the scheme. The next two definitions are much more "scheme friendly" but they have less understood functorial properties.

-The Chern-Fulton and the Chern-Fulton-Johnson classes are both extremely sensitive to the scheme structure. They coincide in the case of local proper intersections. In general they don't satisfy the "inclusion-exclusion principle" which is satisfies by the topological Euler characteristic.

References

For a friendly introduction with references to the appropriate literature, I would recommend this short lecture notes by Paolo Aluffi. The general theorytreatment is discussed in Fulton's book Intersection Theory (specially section s 4.2.6, 4.2.9 and 19.1.7). For applicationapplications to motivic integration and stringy invariantinvariants see for example this review or this one.

If $X$ is a smooth variety over the complex numbers, you can use the topological Euler characteristic of the support $X(\mathbb{C})$ of the scheme. The most efficient way to compute it is to use Chern classes and the Poincaré-Hopf theorem: the Euler characteristic is the degree of the top Chern class. Chern classes can be defined purely algebraically (in homology) as it is done for example in Chapter 3 of Fulton's book "Intersection Theory".

When $X$ is singular, there are many nonequivalent generalizations of the Euler characteristic. One efficient approach is again to work in the spirit of the Poincaré-Hopf theorem by first defining Chern classes for singular varieties. This provides a purely algebraic treatment. I will give a tour of the 4 most famous examples:

-The Chern-MacPherson class (also known as the Chern-Schwartz-MacPherson class ) is probably the most important. It can be defined on the Chow group of a variety using constructible functions. It enjoys beautiful functorial properties under proper maps and it is compatible with specialization. The existence of the Chern-MacPherson class realizes a conjecture of Deligne and Grothendieck of 1969 as the unique natural transformation between the covariant functor of constructible function and the usual covariant functor of $\mathbb{Z}$-homology such that it maps the characteristic function of a manifold to its (homological) Chern class. The Chern-MacPherson class compute the topological Euler characteristic of a (possibly singular) variety.

-The Chern-Mather-class enters the definition of the Chern-MacPherson class. It is relevant to compute the stringy Euler characteristic. It has nice properties under proper birational maps and can be simply defined using a Nash-blow-up.

These two classes (MacPherson and Mather) can be generalized to scheme by considering the support of the scheme. This makes calculations much more easy, but obviously one looses a lot of the information carried by the scheme. The next two definitions are much more "scheme friendly" but they have less understood functorial properties.

-The Chern-Fulton and the Chern-Fulton-Johnson classes are both extremely sensitive to the scheme structure. They coincide in the case of local proper intersections. In general they don't satisfy the "inclusion-exclusion principle" which is satisfies by the topological Euler characteristic.

References

For a friendly introduction with references to the appropriate literature, I would recommend this short lecture notes by Paolo Aluffi. The general theory is discussed in Fulton's book Intersection Theory (specially section 4.2.6, 4.2.9 and 19.1.7). For application to motivic integration and stringy invariant see this or this.

If $X$ is a smooth variety over the complex numbers, you can use the topological Euler characteristic of the support $X(\mathbb{C})$ of the scheme. The most efficient way to compute it is to use Chern classes and the Poincaré-Hopf theorem: the Euler characteristic is the degree of the top Chern class. Chern classes can be defined purely algebraically (in homology) as it is done for example in Chapter 3 of Fulton's book "Intersection Theory".

When $X$ is singular, there are many nonequivalent generalizations of the Euler characteristic. One efficient approach is again to work in the spirit of the Poincaré-Hopf theorem by first defining Chern classes for singular varieties. This provides a purely algebraic treatment. I will give a tour of the 4 most famous examples:

-The Chern-MacPherson class (also known as the Chern-Schwartz-MacPherson class ) is probably the most important. It can be defined on the Chow group of a variety using constructible functions. It enjoys beautiful functorial properties under proper maps and it is compatible with specializations. The existence of the Chern-MacPherson class realizes a conjecture of Deligne and Grothendieck of 1969 as the unique natural transformation between the covariant functor of constructible functions and the usual covariant functor of $\mathbb{Z}$-homology such that it maps the characteristic function of a manifold to its (homological) Chern class. The Chern-MacPherson class computes the topological Euler characteristic of a (possibly singular) variety.

-The Chern-Mather-class enters the definition of the Chern-MacPherson class. It is relevant to compute stringy invariants. It has nice properties under proper birational maps and can be simply defined using a Nash-blow-up.

These two classes (MacPherson and Mather) can be generalized to scheme by considering the support of the scheme. This makes calculations much more easy, but obviously one looses a lot of the information carried by the scheme. The next two definitions are much more "scheme friendly" but they have less understood functorial properties.

-The Chern-Fulton and the Chern-Fulton-Johnson classes are both extremely sensitive to the scheme structure. They coincide in the case of local proper intersections. In general they don't satisfy the "inclusion-exclusion principle" which is satisfies by the topological Euler characteristic.

References

For a friendly introduction with references to the appropriate literature, I would recommend this short lecture notes by Paolo Aluffi. The general treatment is discussed in Fulton's book Intersection Theory (specially section s 4.2.6, 4.2.9 and 19.1.7). For applications to motivic integration and stringy invariants see for example this review or this one.

Improved formatting, small precisions
Source Link
JME
JME
  • 3k
  • 25
  • 30

If $X$ is a smooth variety over the complex numbers, you can use the topological Euler characteristic of the support $X(\mathbb{C})$ of the scheme. The most efficient way to compute it is to use Chern classes and the Poincaré-Hopf theorem: the Euler characteristic is the degree of the top Chern class. Chern classes can be defined purely algebraically (in homology) as it is done for example in Chapter 3 of Fulton's book "Intersection Theory".

When $X$ is singular, there are many nonequivalent generalizations of the Euler characteristic. One efficient approach is again to work in the spirit of the Poincaré-Hopf theorem by first defining Chern classes for singular varieties. This provides a purely algebraic treatment. I will give a tour of the 4 most famous examples:

-The Chern-MacPherson class (also known as the Chern-Schwartz-MacPherson class ) is probably the most important. It can be defined on the Chow group of a variety using constructible functions. It enjoys beautiful functorial properties under proper maps and is it is compatible with specialization. The existence of the Chern-MacPherson class realizes a conjecture of Deligne and Grothendieck of 1969 as the unique natural transformation between the covariant functor of constructible function and the usual covariant functor of $\mathbb{Z}$-homology such that it maps the characteristic function of a manifold to its (homological) Chern class. The Chern-MacPherson class compute the topological Euler characteristic of a (possibly singular) variety.

-The Chern-Mather-class enters the definition of the Chern-MacPherson class. It is relevant to compute the stringy Euler characteristic. It has nice properties under proper birational maps and can be simply defined using a Nash-blow-up.

These two classes (MacPherson and Mather) can be generalized to scheme by considering the support of the scheme. This makes calculations much more easy, but obviously one looses a lot of the information carried by the scheme. The next two definitions are much more "scheme friendly" but they have less understood functorial properties.

-The Chern-Fulton and the Chern-Fulton-Johnson classes are both extremely sensitive to the scheme structure. They coincide in the case of local proper intersections. In general they don't satisfy the "inclusion-exclusion principle" which is satisfies by the topological Euler characteristic.

References

For a friendly introduction with references to the appropriate literature, I would recommend this short lecture notes by Paolo Aluffi. The general theory is discussed in Fulton's book Intersection Theory (specially section 4.2.6, 4.2.9 and 19.1.7). For application to motivic integration and stringy invariant see this or this.

If $X$ is a smooth variety over the complex numbers, you can use the topological Euler characteristic of the support $X(\mathbb{C})$ of the scheme. The most efficient way to compute it is to use Chern classes and the Poincaré-Hopf theorem: the Euler characteristic is the degree of the top Chern class. Chern classes can be defined purely algebraically (in homology) as it is done for example in Chapter 3 of Fulton's book "Intersection Theory".

When $X$ is singular, there are many nonequivalent generalizations of the Euler characteristic. One efficient approach is again to work in the spirit of the Poincaré-Hopf theorem by first defining Chern classes for singular varieties. This provides a purely algebraic treatment. I will give a tour of the 4 most famous examples:

-The Chern-MacPherson class is probably the most important. It can be defined on the Chow group of a variety using constructible functions. It enjoys beautiful functorial properties under proper maps and is compatible with specialization. The existence of the Chern-MacPherson class realizes a conjecture of Deligne and Grothendieck. The Chern-MacPherson class compute the topological Euler characteristic of a (possibly singular) variety.

-The Chern-Mather-class enters the definition of the Chern-MacPherson class. It is relevant to compute the stringy Euler characteristic. It has nice properties under proper birational maps.

These two classes (MacPherson and Mather) can be generalized to scheme by considering the support of the scheme. This makes calculations much more easy, but obviously one looses a lot of the information carried by the scheme. The next two definitions are much more "scheme friendly" but they have less understood functorial properties.

-The Chern-Fulton and the Chern-Fulton-Johnson classes are both extremely sensitive to the scheme structure. They coincide in the case of local proper intersections. In general they don't satisfy the "inclusion-exclusion principle" which is satisfies by the topological Euler characteristic.

References

For a friendly introduction with references to the appropriate literature, I would recommend this short lecture notes by Paolo Aluffi. The general theory is discussed in Fulton's book Intersection Theory (specially section 4.2.6, 4.2.9 and 19.1.7). For application to motivic integration and stringy invariant see this or this.

If $X$ is a smooth variety over the complex numbers, you can use the topological Euler characteristic of the support $X(\mathbb{C})$ of the scheme. The most efficient way to compute it is to use Chern classes and the Poincaré-Hopf theorem: the Euler characteristic is the degree of the top Chern class. Chern classes can be defined purely algebraically (in homology) as it is done for example in Chapter 3 of Fulton's book "Intersection Theory".

When $X$ is singular, there are many nonequivalent generalizations of the Euler characteristic. One efficient approach is again to work in the spirit of the Poincaré-Hopf theorem by first defining Chern classes for singular varieties. This provides a purely algebraic treatment. I will give a tour of the 4 most famous examples:

-The Chern-MacPherson class (also known as the Chern-Schwartz-MacPherson class ) is probably the most important. It can be defined on the Chow group of a variety using constructible functions. It enjoys beautiful functorial properties under proper maps and it is compatible with specialization. The existence of the Chern-MacPherson class realizes a conjecture of Deligne and Grothendieck of 1969 as the unique natural transformation between the covariant functor of constructible function and the usual covariant functor of $\mathbb{Z}$-homology such that it maps the characteristic function of a manifold to its (homological) Chern class. The Chern-MacPherson class compute the topological Euler characteristic of a (possibly singular) variety.

-The Chern-Mather-class enters the definition of the Chern-MacPherson class. It is relevant to compute the stringy Euler characteristic. It has nice properties under proper birational maps and can be simply defined using a Nash-blow-up.

These two classes (MacPherson and Mather) can be generalized to scheme by considering the support of the scheme. This makes calculations much more easy, but obviously one looses a lot of the information carried by the scheme. The next two definitions are much more "scheme friendly" but they have less understood functorial properties.

-The Chern-Fulton and the Chern-Fulton-Johnson classes are both extremely sensitive to the scheme structure. They coincide in the case of local proper intersections. In general they don't satisfy the "inclusion-exclusion principle" which is satisfies by the topological Euler characteristic.

References

For a friendly introduction with references to the appropriate literature, I would recommend this short lecture notes by Paolo Aluffi. The general theory is discussed in Fulton's book Intersection Theory (specially section 4.2.6, 4.2.9 and 19.1.7). For application to motivic integration and stringy invariant see this or this.

Source Link
JME
JME
  • 3k
  • 25
  • 30

If $X$ is a smooth variety over the complex numbers, you can use the topological Euler characteristic of the support $X(\mathbb{C})$ of the scheme. The most efficient way to compute it is to use Chern classes and the Poincaré-Hopf theorem: the Euler characteristic is the degree of the top Chern class. Chern classes can be defined purely algebraically (in homology) as it is done for example in Chapter 3 of Fulton's book "Intersection Theory".

When $X$ is singular, there are many nonequivalent generalizations of the Euler characteristic. One efficient approach is again to work in the spirit of the Poincaré-Hopf theorem by first defining Chern classes for singular varieties. This provides a purely algebraic treatment. I will give a tour of the 4 most famous examples:

-The Chern-MacPherson class is probably the most important. It can be defined on the Chow group of a variety using constructible functions. It enjoys beautiful functorial properties under proper maps and is compatible with specialization. The existence of the Chern-MacPherson class realizes a conjecture of Deligne and Grothendieck. The Chern-MacPherson class compute the topological Euler characteristic of a (possibly singular) variety.

-The Chern-Mather-class enters the definition of the Chern-MacPherson class. It is relevant to compute the stringy Euler characteristic. It has nice properties under proper birational maps.

These two classes (MacPherson and Mather) can be generalized to scheme by considering the support of the scheme. This makes calculations much more easy, but obviously one looses a lot of the information carried by the scheme. The next two definitions are much more "scheme friendly" but they have less understood functorial properties.

-The Chern-Fulton and the Chern-Fulton-Johnson classes are both extremely sensitive to the scheme structure. They coincide in the case of local proper intersections. In general they don't satisfy the "inclusion-exclusion principle" which is satisfies by the topological Euler characteristic.

References

For a friendly introduction with references to the appropriate literature, I would recommend this short lecture notes by Paolo Aluffi. The general theory is discussed in Fulton's book Intersection Theory (specially section 4.2.6, 4.2.9 and 19.1.7). For application to motivic integration and stringy invariant see this or this.