Do deformations of isolated hypersurface singularity naturally induce deformations of their divisors? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T11:56:26Z http://mathoverflow.net/feeds/question/93851 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/93851/do-deformations-of-isolated-hypersurface-singularity-naturally-induce-deformation Do deformations of isolated hypersurface singularity naturally induce deformations of their divisors? tarosano 2012-04-12T10:52:56Z 2012-04-12T17:44:37Z <p>Let $0 \in V =(f=0) \subset \mathbb{C}^{n+1}$ be an affine variety with an isolated hypersurface singularity at the origin for $n \ge 3$. Let $0 \in D=(x=f=0) \subset V$ be a divisor with only isolated singularity $0$ where $x$ is one of local coordinate.<br> Let $T^1_V, T^1_{(V,D)}$ be the set of 1st order deformations of $V$ and a pair $(V,D)$. We can consider the forgetting map $T^1_{(V,D)} \rightarrow T^1_V$. I want to consider the other way, that is, $T^1_V \rightarrow T^1_{(V,D)}$. </p> <p>For $\eta \in T^1_V$, we can consider a small deformation $\mathcal{V} \subset \mathbb{C}^4 \times \Delta^1 \rightarrow \Delta^1$ of $V$ over a small disc $\Delta^1 \subset \mathbb{C}$ induced by $\eta$. Set $\Gamma := (x=0)$. I can consider $\mathcal{D} := \Gamma \times \Delta^1 \cap \mathcal{V} \subset \mathcal{V}$ and this defines a deformation of $D$. </p> <p><strong>Question 1</strong> Does this construct $T^1_V \rightarrow T^1_{(V,D)}$? If yes, is it $\mathbb{C}$-linear? <strong>(Added)</strong> This construction seems to be not well-defined.</p> <p>Moreover, I want to know how this is induced by a sheaf homomorphism. By $n=\dim V \ge 3$, we have </p> <p>$T^1_V \simeq H^1(V', \Theta_{V'}) \simeq H^1(V', \Omega^2_{V'}(-K_V'))$, </p> <p>$T^1_{(V,D)} \simeq H^1(V', \Theta_{V'}(\log D')) \simeq H^1(V', \Omega_{V'}^2 (\log D') (-K_{V'} -D')).$</p> <p>We can consider an inclusion $\Omega^2_{V'} \hookrightarrow \Omega^2_{V'}(\log D'))$ and an isomorphism $O_{V'} \rightarrow O_{V'}(- D')$. Hence we have a homomorphism </p> <p>$\Omega_{V'}^2 (-K_{V'}) \rightarrow \Omega^2_{V'}(\log D')(-K_V' -D')$. </p> <p><strong>Question 2 (Changed)</strong> Is the above homomorphism induce the homomorphism zero?</p>