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This is Problem 16-D in Characteristic classes by John W. Milnor and James D. Stasheff.

Problem 16-D) If the complex manifold $K^n$ is complex analytically embedded in $K^{n+1}$ with dual cohomology class $u \in H^2(K^{n+1},\mathbb{Z})$, show that the total tangential Chern class $c(K^n)$ is equal to the restriction to $K^n$ of $c(K^{n+1})/(1+u)$. For any cohomology class $x \in H^{2n}(K^{n+1};\mathbb{Z})$ show that the Kronecker index $\langle x|K^n, \mu_{2n} \rangle$ is equal to $\langle xu, \mu_{2n+2} \rangle$. Using these constructions, compute $c(K^n)$ for a non-singular algebraic hypersurface $K^n$ of degree $d$ in $P^{n+1}(\mathbb{C})$, and prove that the characteristic number $s_n[K^n]$ is equal to $d(n+2-d^n)$. (An algebraic hypersurface of degree $d$ is the set of zeroes of a homogeneous polynomial of degree $d$.)

My approach)

Recall Theorem 11.3. If $M$ is embedded as a closed subset of $A$, then the composition of the two restriction homomorphisms $H^k(A,A-M) \rightarrow H^k(A) \rightarrow H^k(M)$, with $\mod 2$ coefficients, maps the fundamental class $u^{'}$ to the top Stiefel-Whitney class $w_k(v^k)$ of the normal bundle. Similarly, if $v^k$ is oriented, then the corresponding composition with integer coefficients maps the integral fundamental class $u^{'}$ to the Euler class $e(v^k)$.

Let $i : K^n \rightarrow K^{n+1}$ be an embedding and let $v^1$ be a complemental line bundle of $\tau_{K^n}$ of $K^n$ in $K^{n+1}$. $H^2(K^{n+1},K^{n+1}-K^n;\mathbb{Z}) \rightarrow H^2(K^{n+1};\mathbb{Z}) \rightarrow H^2(K^n;\mathbb{Z})$ Then the dual cohomology class $u \in H^2(K^{n+1})$ is sent to $e(v^1) = c_1(v^1) \in H^2(K^n;\mathbb{Z})$.Therefore, $i^{*}c(\tau_{K^{n+1}}) = c(i^{*}\tau_{K^{n+1}}) = c(\tau_{K^n} \oplus v^1) = c(\tau_{K^n})c(v^1) = c(\tau_{K^n})i^{*}(1+u).$

Recall the result from Problem 11-C, Let $M = M^n$ and $A = A^p$ be compact oriented manifolds with smooth embedding $i : M \rightarrow A$. Let $k = p-n$. Show that the Poincare duality isomorphism $\cap \mu_A : H^k(A) \rightarrow H_n(A)$ maps the cohomology class $u^{'}|A$ dual to $M$ to the homology class $(-1)^{nk}i_{*}(\mu_M)$. By using Problem 11-C, $\langle xu, \mu_{2n+2} \rangle = \langle x, u \cap \mu_{2n+2} \rangle = \langle x, i_{*}(\mu_{2n}) \rangle = \langle i^{*}(x), \mu_{2n} \rangle.$

Now, I want to compute a chern class $c(K^n)$ for a non-singular algebraic hypersurface $K^n$ of degree $d$ in $P^{n+1}(C)$. But I do not know a dual cohomology class when $K^n$ is embedded in $P^{n+1}(\mathbb{C})$. It may be integral multiples of $c(\gamma^1)$ where $\gamma^1$ is a canonical line bundle and this may be related with the degree of $K^n$. But I can not catch anything now. Can you explain how to compute $c(K^n)$?

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1 Answer 1

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Hint: You can see homogeneous polynomials of degree $k$ as sections of the bundle $(L^k)^*$ over $P^{n+1}(\mathbb{C})$. That the hypersurface is non-singular means that it intersects the zero section transversely. What is the normal bundle of this hypersurface?

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