3
$\begingroup$

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 $c(K^n)$?

$\endgroup$
0
$\begingroup$

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?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.