Chern classes of a blow-up at a point - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T01:47:33Z http://mathoverflow.net/feeds/question/92660 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/92660/chern-classes-of-a-blow-up-at-a-point Chern classes of a blow-up at a point gio 2012-03-30T08:44:24Z 2012-07-27T10:09:40Z <p>Let $X$ be a nonsingular projective variety over $\mathbb{C}$, and let $\widetilde{X}$ be the blow-up of X at a point $p\in X$. What relationships exist between the degrees of the Chern classes of $X$ (i.e. of the tangent bundle of $X$) and the degrees of the Chern classes of $\widetilde{X}$?</p> <p>Thanks.</p> http://mathoverflow.net/questions/92660/chern-classes-of-a-blow-up-at-a-point/92665#92665 Answer by Georges Elencwajg for Chern classes of a blow-up at a point Georges Elencwajg 2012-03-30T09:55:24Z 2012-03-30T09:55:24Z <p>For the first Chern class you get the simple formula<br> $$c_1(\tilde X)=p^*c_1(X)- (n-1)E$$ where $p:\tilde X \to X$ is the projection and $E$ the exceptional divisor. </p> <p>In general the formula is more complicated and I'll refer you to Fulton's <em>Intersection Theory</em>, where the formula you require is given in Theorem 15.4. </p> <p>In particular cases the relation may be quite simple: for example if $X$ is of dimension 3, it is just $c_2(\tilde X)=p^*(c_2(X)$ for the second Chern class, as proved in Griffiths-Harris's <em>Principles of Algebraic Geometry</em>, page 609. </p> http://mathoverflow.net/questions/92660/chern-classes-of-a-blow-up-at-a-point/92672#92672 Answer by Johannes NordstrÃ¶m for Chern classes of a blow-up at a point Johannes NordstrÃ¶m 2012-03-30T11:03:24Z 2012-03-30T11:03:24Z <p>Like Georges says, 15.4 of Fulton's Intersection Theory deals with the general theory. For this special case it's not too hard to work out the Chern classes by hand though.</p> <p>Let $f : \widetilde X \to X$ be the projection and $E \cong \mathbb{C}P^{n-1}$ the exceptional divisor. <code>$H^*(\widetilde X) \cong f^*H^*(X) \oplus \langle \textrm{Poincare duals of planes } P_k \textrm{ in } E$ $\textrm{of dimension }k = 1,\ldots, n-1\rangle$</code>. Note that $[P_{n-i}][P_{n-j}] = -[P_{n-i-j}]$, while <code>$(f^*\alpha) [P_k] = 0$</code> for any $\alpha \in H^i(X)$ ($i, k > 0$).</p> <p><code>$f^* c_i(X)$</code> and $c_i(\widetilde X)$ are equal outside the exceptional divisor, so their difference is Poincare dual to something in $E$. On the other hand the restriction of <code>$f^*c_i(X)$</code> to $E$ is 0 (for $i > 0$), while the restriction of $c_i(E)$ is $c_i(\mathcal{O}(1)^n \oplus \mathcal{O}(-1)) = \left({n\choose i} - {n \choose i-1}\right)H^i$, where $H \in H^2(E)$ is the hyperplane class. For $0 &lt; i &lt; n$ we deduce that <code>$c_i(\widetilde X) = f^*c_i(X) - \left({n\choose i} - {n \choose i-1}\right)[P_{n-i}]$</code> by comparing the evaluations on $P_i$.</p> http://mathoverflow.net/questions/92660/chern-classes-of-a-blow-up-at-a-point/92674#92674 Answer by Liviu Nicolaescu for Chern classes of a blow-up at a point Liviu Nicolaescu 2012-03-30T11:33:18Z 2012-07-27T10:09:40Z <p>Assume $X$ is smooth compact of dimension $n$ and $x_0\in X$ is the point where we perform the blowup. Set $X_* := X \setminus x_0$, $\tilde{X}_* := \tilde{X} \setminus E$. Denote by $N$ a tubular neighborhood of $E$ in $\tilde{X}_*$. By Mayer-Vietoris, the Chern classes of $\tilde{X}$ are determined once we know their restrictions to $X_*$ and $N$.</p> <p>We identify $\tilde{X}_*$ with $X_*$ via the blowdown map $p:\tilde{X}_* \to X_*$. The restriction of $c_k( \tilde{X})$ to $X_*$ is equal to the restriction of $c_k(X)$. The restriction of $c_k(\tilde{X})$ to $N$ is easy to determine since</p> <p>$$TN \cong \pi^* T\mathbb{CP}^{n-1} \oplus \pi^* H^*,$$</p> <p>where $\pi: N\to E= \mathbb{CP}^{n-1}$ is the natural projection and $H\to \mathbb{CP}^{n-1}$ is the hyperplane line bundle. Thus, </p> <p>$$c_k(\tilde{X})|_N = c_k( N ) = \pi^*c_k(\mathbb{CP}^{n-1} ) +\pi^* c_{k-1}(\mathbb{CP}^{n-1} ) \pi^* c_1(H^*)$$</p> <p>$$= \pi^*c_k(\mathbb{CP}^{n-1} ) - \pi^* c_{k-1}(\mathbb{CP}^{n-1} )\cup \pi^*[H].$$</p> <p>Things can be simplified a bit if we introduce the notation $h=\pi^*[H]\in H^2(N,\mathbb{Z})$ and we observe that for some integers $\nu_k$ and $\nu_{k-1}$</p> <p>$$\pi^*c_k(\mathbb{CP}^{n-1} ) =\nu_k h^k,$$</p> <p>$$\pi^* c_{k-1}(\mathbb{CP}^{n-1} )=\nu_{k-1} h^{k-1}.$$</p> <p>Then</p> <p>$$c_k(N) = ( \nu_k -\nu_{k-1} ) h^k.$$</p> <p>As for the integers $\nu_k$ they are determined from the equality</p> <p>$$1+c_1( \mathbb{CP}^{n-1} )+\cdots + c_{n-1}( \mathbb{CP}^{n-1} )= (1+H)^n - H^n$$</p> <p>$$= \sum_{k=0}^{n-1}\binom{n}{k} H^k.$$</p>