ag.algebraic geometry - Decomposition theorem and blow-ups

In this example we have $p : X to Y$ and we may assume, wlog, that $X$ is isomorphic to the total space of the normal bundle to the surface, and $p$ is the contraction of the zero section.

Then, by the Deligne construction, $IC(Y) = tau_{le -1} j_* mathbb{Q}[3]$, where $j : Y^0 hookrightarrow Y$ is the inclusion of the smooth locus (which is isomorphic to $X^0$ the complement of the zero section in $X$).

In order to work this out, we can use the Leray-Hirsch spectral sequence

$E_2^{p,q} = H^p(S) otimes H^q(mathbb{C}^*) Rightarrow H^{p+q}(X^0)$

this converges at $E_3$ and we get that the degree 0, 1 and 2 parts of the cohomology of $X^0$ is given by the primitive classes in $H^i(S)$ for $i = 0, 1, 2$. Note that this is everything in degrees 0 and 1, but in degree two the primitive classes form a codimension one subspace $P_2 subset H^2(S)$.

The Deligne construction above, gives us that $IC(Y)_0 = H^0(S)[3] oplus H^1(S)[2] oplus P_2[1]$.

(This is a general fact: whenever you take a cone over a smooth projective variety, the stalk of the intersection cohomology complex at 0 is given by the primitive classes with respect to the ample bundle used to embed the variety. This follows by exactly the same arguments given above.)

Then the decomposition theorem gives

$p_* mathbb{Q} = mathbb{Q}_0[1] oplus ( IC(Y) oplus H^3(S) ) oplus H^4(S)[-1]$.

EDIT: fixed typos pointed out by Chris.

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