The Gysin map for a singular hypersurface - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T06:38:07Z http://mathoverflow.net/feeds/question/104336 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/104336/the-gysin-map-for-a-singular-hypersurface The Gysin map for a singular hypersurface Alex Gavrilov 2012-08-09T10:05:26Z 2012-08-09T10:05:26Z <p>Let $X$ be a projective complex manifold and <code>$Y\subset X$</code> be an irreducible hypersurface. If $Y$ is <em>smooth</em>, there is a well known Gysin sequence. However, even if $Y$ is not smooth, a kind of Gysin map can still be devised. </p> <p>Consider a desingularization $f:\widetilde{Y}\to Y$ and the inclusion $i:Y\hookrightarrow X$. We have two maps </p> <p><code>$H^i(Y,\mathbb{Q})\to H^i(\widetilde{Y},\mathbb{Q})\to H^{i+2}(X,\mathbb{Q});$</code></p> <p>the first one is <code>$f^*$</code> and the second one is the Poincare dual to <code>$(i\circ f)^*$</code> (essentially the Gysin map). I am convinced that the composition does not depend on desingularization, though I do not know a rigorous proof of this. </p> <p><em>QUESTION</em>: Is the sequence </p> <p><code>$H^i(Y,\mathbb{Q})\to H^{i+2}(X,\mathbb{Q})\to H^{i+2}(X\setminus Y,\mathbb{Q})$</code></p> <p>exact (as it is in the smooth case)?</p> <p>All I know about it is a result of Deligne [Theorie de Hodge III. Publ. Math. IHES 44 (1974) pp. 5–77.; Corollary 8.2.8] that </p> <p><code>$H^i(\widetilde{Y},\mathbb{Q})\to H^{i+2}(X,\mathbb{Q})\to H^{i+2}(X\setminus Y,\mathbb{Q})$</code></p> <p>is exact, but this is much weaker than what I need.</p>