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Homotopy theory, homological algebra, algebraic treatments of manifolds.

5 votes

Cohomology of complete intersections

If $X$ is smooth then Lefschetz' hyperplane theorem and Poincaré duality yield that $H^i(X,\mathbb{Z})=H^i(\mathbb{P}^n,\mathbb{Z})$ for $i=0,\dots, 2\dim X$, $i\neq \dim X$. (This is proven in Dimca' …
LSpice's user avatar
  • 12.9k
2 votes

$b_2$ of the blow up of a complex $3$-fold in a curve

I am not sure whether $b_2(V')-b_2(V)=1$ always holds. Anyway, in the book of Peters and Steenbrink you can find "the Mayer-Vietoris sequence of the discriminant square". If $E$ is the exceptional div …
Remke Kloosterman's user avatar
3 votes

Cohomology of Hypersurface complement

Griffiths (On the periods of certain rational integrals. I, II. Ann. of Math. 90 (1969), 460-495 & 90 (1969), 496–541.) gave a procedure to calculate the Hodge numbers of Y. Let $S=\mathbf{C}[x_0,\dot …
Remke Kloosterman's user avatar