How do different definitions of "degree" coincide?

I've recently read about a number of different notions of "degree." Reading over Javier Álvarez' excellent answer for the thousandth time finally prompted me to ask this question:

How exactly do the following three notions of "degree" coincide?

(1) Algebraic Topology. Let $f\colon X \to Y$ be a continuous map between compact connected oriented $n$-manifolds.

Wikipedia tells me that $H_n(X) \cong H_n(Y) \cong \mathbb{Z}$, and that a choice of orientations for $X$ and $Y$ amount to choices of generators $[X], [Y]$ for $H_n(X), H_n(Y)$, respectively. We then define $\deg f$ via $$f_*([X]) = (\deg f)[Y].$$

(2) Differential Topology. Let $f\colon X \to Y$ be a smooth map between oriented $n$-manifolds, where $X$ is compact and $Y$ is connected.

Let $y \in Y$ be a regular value of $f$ (which exists by Sard's Theorem), let $D_xf\colon T_xX \to T_yY$ denote the derivative (a.k.a. pushforward), and define $$(\deg f)_y = \sum_{x \in f^{-1}(y)}\text{sgn}(\det D_xf).$$ It can be shown that $(\deg f)_y$ is independent of the choice of $y \in Y$, so we can talk meaningfully about a single quantity $\deg f = (\deg f)_y$.

(3) Riemann Surfaces. Let $f\colon X \to Y$ be a holomorphic map between compact connected Riemann surfaces.

For $x \in X$, we let $\text{mult}_x(f)$ denote the multiplicity of $f$ at $x \in X$. For $y \in Y$, we define $$(\deg f)_y = \sum_{x \in f^{-1}(y)} \text{mult}_x(f).$$ As in (2), it can be shown that $(\deg f)_y$ is independent of the choice of $y \in Y$. (Does this generalize to arbitrary complex manifolds?)

Thoughts: As was mentioned in my topology class last semester (and also on Wikipedia), there is this concept of "local homology" which lets us compute (1) as a sum of "local degrees."

I imagine that in the case of (2), each of these local degrees is, in fact, equal to $\text{sgn}(\det D_xf)$ because $f$ is a local diffeomorphism at each regular point $x$. I also imagine that in the case of (3), each of these local degrees is, in fact, equal to $\text{mult}_x(f)$ because the degree of $\mathbb{S}^n \to \mathbb{S}^n$, $z \mapsto z^k$ is $k$. (Does this also mean that $f$ is not regular at any point where $\text{mult}(f) \geq 2$? This would make sense, but what is the proof?)

This all seems correct in my head, but I would really like more details if possible.

First things first, thank you very much for your appraisal of my other answer on consequences of degree concepts, mostly for differentiable manifolds. I have in fact expanded it posting another answer listing the applications of degree theory in complex algebraic geometry using the definitions explained below. Indeed I have also spent a great deal of time trying to understand all connections among "degrees" as much as possible, so let me try to complete a little bit your list and my digression with the, in my view, most important, geometric and unifying notion of degree: that coming from complex algebraic geometry.

(4) ALGEBRAIC GEOMETRY OF COMPLEX PROJECTIVE VARIETIES, that is to say, complex submanifolds $X$ of, or embeddings into, the complex projective space $\mathbb{CP}^n$. Any non-singular projective $n$-variety is isomorphic, in the algebraic category, to a subvariety of $\mathbb{CP}^{2n+1}$ [Shafarevich vol. I, 5.4 Th.9]. By Chow's theorem [Griffiths-Harris, p.167][Mumford Cor.4.6] any complex manifold seen as $k$-submanifold of $\mathbb{CP}^n$ is algebraic, i.e. any real $2k$-submanifold of $\mathbb{RP}^{2n}$ which admits a complex structure is actually given as the zero locus of a system of homogeneous polynomials (all varieties coming from manifolds are smooth, but there are singular varieties which are not manifolds though!). In particular any real closed orientable surface admits a complex structure and so is a complex projective algebraic curve, thus including case (3) of compact Riemann surfaces [Miranda, Th.IV.1.9]; higher dimensional manifolds may not always admit complex structures, a necessary and sufficient condition is the Newlander-Nirenberg theorem [Kobayashi-Nomizu vol. II, Appx.8][Voisin vol. I, sec.2.2.3]: vanishing of the Nijenhuis tensor for an almost-complex structure. By Lefschetz's principle this is essentially enough for dealing with the general case of abstract varieties (integral separated schemes of finite type over an algebraically closed field $k$) embedded as subschemes of $\mathbb{P}^n_k$.

All the following definitions of degree are proved to be equivalent to each other so one can pick any of them as starting definition and get the rest as interesting theorems. (We always talk about nonsingular irreducible complex projective curves, surfaces, hypersurfaces, varieties... etc., and thus compact Riemann surfaces, except explicit mention of the contrary). I shall only explain in detail the original classical geometric notions of degree and mention the rest.

• A. If $X$ is a hypersurface of $\mathbb{CP}^n$, i.e. $\dim X=n-1$, by [Hartshorne, Exercise I.2.8] it is given by the zero locus, $X=Z(f)$, of an irreducible homogeneous polynomial $f\in S_d$ of algebraic degree $d$, i.e. any monomial summand $a_{k_0\dots k_n}x_0^{k_0}\cdots x_n^{k_n}$ in $f$ has degree $d=\sum_i k_i$, where all such monomials generate the abelian group $S_d$, all of which make the ring of polynomials a graded ring $\mathbb{C}[x_0,\dots,x_n]=\bigoplus_{d=0}^\infty S_d$. So any such $f$ has a canonical associated (algebraic) degree, so

Degree of a hypersurface as the algebraic degree of its defining homogeneous polynomial: $$\deg X_{n-1}=\deg Z(f):=\deg(f)=d,\;\;\; f\in S_d$$

• B. If $k:=\dim X_k< n-1$, let $L_r\cong\mathbb{CP}^r$ be a generic (i.e. in general position) linear variety (linear projective vector subspace) of $\mathbb{CP}^n$ of dimension $r\leq n-k-1$. The projecting cone, $C(X_k, L_r)$, of $X_k$ from "vertex" $L_r$ is defined to be the joint locus of the subspaces $L_{r+1}$ that join the given $L_r$ with each point of $X_k$ (this generalizes the intuitive "cone" of lines obtained by projecting from a point). By [Beltrametti et al., sec.3.4.5] the projecting cone is also a, possibly reducible, algebraic variety of dimension $r+k+1$ (which justifies the upper bound of r at the beginning). For a generic $L_{n-k-2}$ the projecting cone of $X_k$ is thus a, possibly reducible, hypersurface as in A. above (if it is irreducible it is the case A. if it is reducible then its defining zero locus polynomial is reducible but has nevertheless well defined degree). Call

Degree of a subvariety as the degree of the generic projecting cones which are hypersurfaces: $$\deg X_k:=\max\limits_{L\in\mathbb{Gr}(n-k-2,\mathbb{CP}^n)}\{\deg C(X_k, L)\}, $$ where $C(X_k, L_{n-k-2})=Z(g)\,\vert\, g\in S_p$.

where $L$ is an element of the Grasssmannian of the required dimension. The degree of a variety $X_r\subset\mathbb{CP}^n$ is thus defined to be the degree of the generic hypersurface-projecting-cone; this is proved to be well-defined as this max deg is constant for a dense Zariski-open subset of the Grassmannian, cf. [Harris, Exercise 18.2]. For example if $L_0$ is a generic point in $\mathbb{CP}^3$ and $X_1$ a spatial algebraic curve, for each point of $X_1$ there is only one line joining it with $L_0$. Moving along all such points of the curve we obtain a cone swept by the joining lines with the fixed $L_0$, cone which is an algebraic surface, thus the zero locus of an homogeneous polynomial in projective space. So we are calling the degree of the spatial curve the degree of its projecting cone generic surface polynomial. Note that a curve in projective space is generically given by the intersection of two surfaces of possibly different degrees, $X_1=Z(h_1,h_2)\subset\mathbb{CP}^3$, so it has no canonical unique polynomial degree as is the case for plane curves. It is also important to remember that any nonsingular algebraic curve (thus Riemann surface) is isomorphic to a smooth spatial curve in $\mathbb{CP}^3$ [Hartshorne, Cor.IV.3.6][Shafarevich vol. I, sec.5.4 Cor.2] and birational to a plane curve with at most node singularities [Hartshorne, Cor.IV.3.11]. Note also that the first definition A. above is necessary, since the projecting cone of a hypersurface cannot be defined due to the constraint $r\leq n-k-1$. So what we have done is defining, for any lower dimensional variety, associated hypersurfaces which have generically well-defined polynomial degree.

• C. The "vertex" $L_r$ of a generic projecting cone $C(X_k, L_r)$ of a variety $X_k$ is given by $n-r$ linearly independent linear equations: $L_r=Z(h_1,\dots,h_{n-r})$ where $h_i$ are linear forms which define hyperplanes $H_i=Z(h_i)\cong\mathbb{CP}^{n-1}$ within $\mathbb{CP}^n$, so that $L_r=\bigcap_{i=1}^{n-r} H_i$. Projecting cones take their name from the fact that they define a generalized projection of a variety to a linear subspace (e.g. projecting from a point a spatial curve into a plane): the projection [Shafarevich vol. I, sec.4.4 Ex.1], with center or vertex $L_r$, is the rational map $\pi_{L_r}(x):=[h_1(x):\dots :h_{n-r}(x)]$ which is a regular morphism on the Zariski-open set $\mathbb{CP}^n\setminus L_r$. Therefore its restriction to any variety disjoint from the vertex, $\pi_{L_r}\vert_{X_k}:X_k\rightarrow\mathbb{CP}^{n-r-1}$ is a regular map of it to a projective subspace. Take any linear variety disjoint from $L_r$ as representative, i.e. $\mathbb{CP}^{n-r-1}\cong L'_{n-r-1}\subset\mathbb{CP}^n$ such that $L_r\cap L'_{n-r-1}=\varnothing$, which is always possible by [Beltrametti et al., Th.3.3.8] (since $\dim L_r\cap L'_{n-r-1}\geq r+(n-r-1)-n=-1$ so they do not intersect necessarily). Now for every point $x\in\mathbb{CP}^n\setminus L_r$, in particular $X_k$, there is a unique $L''_{r+1}$ passing through the vertex $L_r$ and $x$ by elementary dimension counting. The locus of all these generators $L''_{r+1}$ is just the generic projecting cone $C(X_k,L_r)$ for generic center $L_r$!. Each generator intersects $L'_{n-r-1}$ in a unique point (solution of a system of $n-(r+1)+n-(n-r-1)=n$ linear equations) which corresponds to $\pi_{L_r}(x)$ through the isomorphism with $\mathbb{CP}^{n-r-1}$. Therefore, given a generic linear subspace $L_r$ we can regularly (rationally if $L_r\cap X_k\neq\varnothing$) project any variety $X_k\subseteq\mathbb{CP}^n$ to a lower dimensional generic linear subspace $L'_{n-r-1}\cong\mathbb{CP}^{n-r-1}$ by intersecting the projecting cone with it, $C(X_k,L_r)\cap L'_{n-r-1}$, and calling $\pi_{L_r}(X_k)\subset \mathbb{CP}^{n-r-1}$ the projection of $X_k$ from $L_r$ to $L'_{n-r-1}$. The case $r=0$ is the classical projection from a point into a hyperplane (like our spatial curve projected to a plane curve). Therefore for any $X_k$, projecting from a generic center $L_{n-k-2}$, we obtain a, possibly reducible, variety $\bar{X}_k$ in $\mathbb{CP}^{k+1}$ as projection; since this comes from the intersection of the hypersurface $C(X_k,L_{n-k-2})$ with a linear variety $L'_{k+1}$, by the projective dimension theorem [Hartshorne, Th.I.7.2] every of its irreducible components has dimension $\geq (n-1)+(k+1)-n=k$. In fact, if $\dim X_k\geq 2$ by repeated application of Bertini's theorem [Hartshorne, Th.II.8.18], any such intersection is generically not only smooth but connected and thus irreducible, thus any generic such projection is a hypersurface $\bar{X}_k\subset\mathbb{CP}^{k+1}$ (generically reducible for $X_1$ a curve) and so it has a defining zero locus irreducible homogeneous polynomial $q\in\mathbb{C}[x_0,...,x_{k+1}]$ with well-defined algebraic degree (if $X_1$ is a curve then each of its irreducible components after intersecting will be points solution of a reducible polynomial). This is equivalent to B. since the intersection of a generic projecting cone hypersurface with a generic linear variety has the same polynomial degree as the cone (solve as many variables as possible from the linear system defining the linear variety and substitute in the homogeneous polynomial of the cone; each of its equal-degree monomials produce new monomials in less variables but of the same degree as the original, so one gets a new homogeneous polynomial in less variables, i.e. a hypersurface in a lower-dimensional projective space). This shows a geometric construction for the theorem of the birational equivalence of any projective algebraic set of dimension $k$ with a hypersurface in $\mathbb{CP}^{k+1}$, cf. [Beltrametti et al., sec.2.6.11] and [Hartshorne, Prop.I.4.9].

Degree of a $k$-subvariety of $\mathbb{CP}^{n}$ as the polynomial degree of the, possibly reducible, hypersurface obtained by generically projecting to $\mathbb{CP}^{k+1}$, i.e. intersecting the projecting cone with a suitable generic linear subspace: $$\deg X_k:=\deg \pi_{L}(X_k)=\deg C(X_k,L_{n-k-2})\cap L'_{k+1},$$ for generic $L\in\mathbb{Gr}(n-k-2,\mathbb{CP}^n)$ and $L'\in\mathbb{Gr}(n-k-2,\mathbb{CP}^n)$.

• D. Now take the projected variety hypersurface $\pi_{L_{n-k-2}}(X_k)= \bar{X}_k \subsetneq \mathbb{CP}^{k+1}$, possibly reducible, and project it again with vertex a generic point $\bar{L}_0\in\mathbb{CP}^{k+1}$ disjoint from $\bar{X}_k$, onto a generic hyperplane $\bar{L}_k\in\mathbb{Gr}(n-1,\mathbb{CP}^{k+1})$. It is a standard exercise to prove that any projection from vertex $L_r$ can be decomposed into a sequence of projections from $r+1$ points $L_{0(0)}\dots L_{0(r)}$ spanning $L_r$, so everything done in C. above can be interpreted as projecting our $k$-variety down from successive $n-k-1$ points to a projective $(k+1)$-space where it becomes a hypersurface, so that one can talk about a generic degree. Therefore, now we are just stopping our chain of projections when we get the surjection $\pi_{L_{n-k-1}}|_{X_k}:X_k\twoheadrightarrow\mathbb{CP}^k$ which comes from projecting from generic center $L_{n-k-1}$ onto generic linear variety of the same dimension $L'_k$ (each projection from a point reduces by 1 the dimension of the projective space into which we are projecting, so we need $n-k$ independent generic points). By construction the projection onto $L'_k\cong\mathbb{CP}^k$ is generically a finite map, since each projection of a hypersurface from a generic point is a line which intersects it in a finite number of points (by the projective dimension theorem), the fiber of the projected point, and this is an equivalent condition for finiteness of a morphism for projective varieties [Harris, Lemma 14.8]. Now, we defined above the degree of $X_k$ to be the degree of its hypersurface projected model into $\mathbb{CP}^{k+1}$, so another projection from a generic point $\bar{L}_{0(n-k)}\in\mathbb{CP}^{k+1}$ onto generic $\mathbb{CP}^k\cong\bar{L}_k \subset \mathbb{CP}^{k+1}$ comes from a line joining the point with each point of $\bar{X}_k$; as the vertex is generic, this line intersects $\bar{X}_k$ in a finite number of points which is no other than the degree of its zero locus defining homogeneous polynomial $\bar{X}_k=Z(q)$! (parametrize the straight line by $[x_0(t):...:x_{k+1}(t)]$ so that the intersection points are the finite number of roots of $g(t)=0$, which are $\deg(g)$ in number by the fundamental theorem of algebra). It is not hard to convince oneself that the generic projection $\pi_{L_{n-k-1}}(X_k)$ has the same number of points in its generic fiber as that last component projection which brings it down to $\mathbb{CP}^k$, since up to $\mathbb{CP}^{k+1}$ the hyperplanes are higher dimensional than $X_k$. (It is surjective because a line and a hypersurface always intersect in projective space). The number of points in a general fiber is called the degree of the map. It is the same Brower-Kronecker degree of a continuous mapping in Differential Topology, but in the complex case it coincides with the number of pre-images, for complex structure fixes orientation and regular maps=holomorphic maps preserve it because any complex linear transformation (e.g. the Jacobians of the map) are never negative, cf. [Dubrovin et al., Th.13.4.2]. Thus:

Degree of a variety $X_k\subset\mathbb{CP}^n$ as the number of pre-images of a generic fiber (i.e. degree of a regular or rational map) of the generic finite surjective projection map $\pi_{\Lambda}:X_k\twoheadrightarrow\mathbb{CP}^k$: $$\deg X_k:=\deg\pi_{\Lambda}=\#\,\pi_{\Lambda}^{-1}(x),$$ for generic $x\in\mathbb{CP}^k,\; \Lambda\in\mathbb{Gr}(n-k-1,\mathbb{CP}^n).$

• E. Following [Beltrametti et al., Prop.3.4.8] let us go back to B. or C. above, our projection of $X_k$ into a hypersurface of $\mathbb{CP}^{k+1}$ via generic center $L_{n-k-2}\subset\mathbb{CP}^{n}$. Take a generic line $l_1\subset\mathbb{CP}^{n}$ not contained in $L_{n-k-2}$, so that the linear space $\operatorname{Join}(L_{n-k-2}, l_1)=\langle L_{n-k-2}, l_1 \rangle$ is a generic $L'_{n-k}$ because this is just a projecting cone, thus having dimension $(n-k-2)+(1)+1$ (cf. beginning of B. above). It is clear that any such generic linear $(n-k)$-space can be obtained in this way by generically decomposing it into a line and a linear $n-k-2$-subspace contained in it. Now, the intersection of a $k$-variety with a generic hyperplane has irreducible components of dimension $k-1$ [Shafarevich vol. I, sec.6.2], thus $X_k\cap L'_{n-k}$ consists generically of a finite number of points. This number of points is constant in a dense Zariski-open subset of the Grassmannian $\mathbb{Gr}(n-k,\mathbb{CP}^{n})$, cf. [Harris, Ex.18.2]. This can be readily proved by noticing that it is the number of points in the generic fiber of $\pi_{\Lambda}$ with center a generic hyperplane $\Lambda\subset L'_{n-k}$ seen in D. above. To see this, note that our generic $L'_{n-k}$ can be thought as a projecting cone with center $\mathbb{CP}^{n-k-1}\cong\Lambda\subset L'_{n-k}$, and the fiber of $\pi_{\Lambda}$, which is finite by D., is by construction the intersection of the projecting cone with the variety. Therefore $\#\, (X_k\cap L'_{n-k})=\deg \pi_{\Lambda}$, showing equivalence with all the previous notions. It is worth mentioning that many (most) classical treatments define degree as this finite number of points of a generic intersection with a linear space of dimension the codimension of the variety. An independent proof is [Mumford, Th.5.1] where it is shown that generic linear $(n-k)$-subspaces meeting transversaly our variety $X_k$, do so in a common number of points: the degree. (This relates to definitions in differential topology of intersections of submanifolds meeting properly, i.e. $T_pX_k\cap T_pL'_{n-k}={0}$ and $T_pX_k\oplus T_pL'_{n-k}=T_p\mathbb{CP}^n$).

Degree of a subvariety as the number of points of intersection (generically transversal) with a generic codimensional linear variety (i.e., $\dim L=n-\dim X$): $$ \deg X_k := \# (X_k\cap L_{n-k})= \# (X_k\cap C(X_k, \Lambda_{n-k-1})),$$ for generic $L\in\mathbb{Gr}(n-k,\mathbb{CP}^n)$, $\Lambda\in\mathbb{Gr}(n-k-1,\mathbb{CP}^n)$.

• F. The projection map of def. D. is a dominant rational map and as such defines a pullback inclusion $\pi_{\Lambda}^\ast:K(\mathbb{CP}^k)\hookrightarrow K(X_k)$ by $f\mapsto f\circ\pi_{\Lambda}$ for any rational (meromorphic) function $f\in \mathbb{C}(x_0,…,x_k)$, which is a homomorphism of $\mathbb{C}$-algebras. In fact by [Hartshorne, Th.I.4.4] this establishes a contravariant equivalence of categories between the category of complex projective varieties and dominant rational maps and the category of finitely generated field extensions of $\mathbb{C}$, thus birational equivalent varieties have isomorphic function fields. Now by [Harris, Prop.7.16], the transcendence degree of the finite field extension is the number of points in the generic fiber of D., showing equivalence to all previous definitions.

Degree as the transcendence degree of the finite field extension of the function field of projective space with respect to the function field of the variety, generically projected to it. $$\deg X_k:=[K(\mathbb{CP}^k):K(X_k)],$$ for generic $\pi_{\Lambda}^\ast :K(\mathbb{CP}^k) \hookrightarrow K(X_k),\; \Lambda\in\mathbb{Gr}(n-k-1,\mathbb{CP}^n).$

• G.

Degree as $\dim X!$ times the leading coefficient of the Hilbert polynomial of the variety: $$P_{X}(t)=\frac{\deg X}{\dim X!}t^{\dim X}+\dots,$$ where $P_{X}(t):=\dim_{\mathbb{C}}(\mathbb{C}[X]\cap S_t),\, t\gg 0.$ is the dimension of the $t$-graded homogeneous part of the coordinate ring of the variety, which is proved to be a polynomial in $t$ for large grading [Shafarevich vol. II, sec.4.2].

• H.

Degree as coefficient in homology $H_{2k}(\mathbb{CP}^n,\mathbb{Z})$ $$[X_k]=(\deg X_k)\cdot [L_k],$$ or integration coefficient in de Rham cohomology $H^{2k}_{dR}(\mathbb{CP}^n,\mathbb{C})$ $$\langle [X_k],\omega\rangle=\deg X_k\cdot\langle[L_k],\omega \rangle\Leftrightarrow \int_{X_k}\omega=\deg X_k\cdot\int_{L_k}\omega.$$

• I.

Degree as coefficient in the linear equivalence class of the generic projection to a divisor in $\mathbb{CP}^{\dim X+1}$, given by the possibly reducible hypersurface $\pi_{L_{n-k-2}}(X_k)=:\bar{X}_k$: $$[\bar{X}_k]\sim^{lin.}(\deg X_k)\cdot [L_k],$$ where linear equivalence is considering every divisor mod a rational divisor, i.e. $[D]\in \operatorname{Cl}\,(\mathbb{CP}^{k+1}) :=\operatorname{Div}(\mathbb{CP}^{k+1})/\sim^{lin.}$ with $D_1\sim^{lin.}D_2 :\Leftrightarrow \exists f\in\mathbb{C}(X_k)$ such that $D_1-D_2=\operatorname{div}(f)$ and Div is the free abelian group generated by the hypersurfaces. Equivalently, degree as the coefficient in the rational equivalence class of the Chow group of order $\dim X$, i.e. $[X_k]\in A_k(\mathbb{CP}^n)$: $$[X_k]\sim^{rat.}(\deg X_k)\cdot [L_k],$$

• J.

Degree as intersection number of self-intersecting $\dim X$ times the hyperplane twisting sheaf $\mathcal{O}_{\mathbb{P}^n_k}(1)\vert_{X}$, where the intersection of invertible sheaves (line bundles) is defined by$$(\mathcal{L_1}\cdot ... \cdot\mathcal{L_m})_{X}:=\chi_{X}-\sum_i\chi_{X}(\mathcal{L}_i^{-1})+\sum_{i<j}\chi_{X}(\mathcal{L}_i^{-1}\otimes\mathcal{L}_j^{-1})-\cdots+(-1)^n\chi_{X}(\mathcal{L}_1^{-1}\otimes\cdots\otimes\mathcal{L}_m^{-1})$$ for $m\geq k$ line bundles on $X_k$, and $\chi_X$ the Euler characteristics of the bundle. That is to say: $$\deg X:= (\mathcal{O}_{\mathbb{P}^n_k}(1)\vert_{X})^{\dim X}.$$