The Gysin map for a singular hypersurface

2012-08-09T10:05:26Z

Let $X$ be a projective complex manifold and $Y\subset X$ be an irreducible hypersurface. If $Y$ is smooth, there is a well known Gysin sequence. However, even if $Y$ is not smooth, a kind of Gysin map can still be devised.

Consider a desingularization $f:\widetilde{Y}\to Y$ and the inclusion $i:Y\hookrightarrow X$. We have two maps

$H^i(Y,\mathbb{Q})\to H^i(\widetilde{Y},\mathbb{Q})\to H^{i+2}(X,\mathbb{Q});$

the first one is $f^*$ and the second one is the Poincare dual to $(i\circ f)^*$ (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.

QUESTION: Is the sequence

$H^i(Y,\mathbb{Q})\to H^{i+2}(X,\mathbb{Q})\to H^{i+2}(X\setminus Y,\mathbb{Q})$

exact (as it is in the smooth case)?

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

$H^i(\widetilde{Y},\mathbb{Q})\to H^{i+2}(X,\mathbb{Q})\to H^{i+2}(X\setminus Y,\mathbb{Q})$

is exact, but this is much weaker than what I need.

The Gysin map for a singular hypersurface - MathOverflow

The Gysin map for a singular hypersurface