# ag.algebraic geometry – Cohomology ring of a hypersurface in toric selection Answer

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ag.algebraic geometry – Cohomology ring of a hypersurface in toric selection

I cerebrate this will breathe finished, not less than for rational cohomology, utilizing the Lefschetz hyperplane theorem and the Hard Lefschetz theorem.

The pullback map $$H^i (X, mathbb Q) to H^i ( Z, mathbb Q)$$ is an isomorphism for $$i < n-1$$ and injective for $$i=n-1$$.

Thus Poincare duality provides an isomorphism $$H^i (Z, mathbb Q) = H^{2n-2-i}(X,mathbb Q)^vee$$ for $$i > n+1$$.

For $$i = n+1$$, the picture of $$H^i(X,mathbb Q)$$ inside $$H^i (Z,mathbb Q)$$ is a non-degenerate subspace for the Poincare duality pairing by the Hard Lefschetz theorem. So it has an orthogonal complement $$V$$, whose dimension you may compute.

We can thus narrate $$H^i (Z, mathbb Q)$$ as $$H^i(X,mathbb Q)$$ for $$i, $$H^i (X,mathbb Q) oplus V$$ for $$i=n$$, and $$H^{2n-2-i}(X,mathbb Q)^vee$$ for $$i>n$$.

Using this, we will secure the ring construction. Consider two courses $$alpha,beta$$, let’s compute the cup product $$alpha cup beta$$.

If $$alpha in H^i(X,mathbb Q)$$ and $$beta in H^j(X,mathbb Q)$$, then we will take $$alpha cup beta in H^{i+j}(X,mathbb Q)$$ as a result of the pullback map is appropriate with cup merchandise. If $$i+j>n$$, we necessity to know tips on how to map to $$H^{2n-2-i-j}(X,mathbb Q)^vee$$, which is equal to taking a category $$gamma$$ in $$H^{2n-2-i-j}(X,mathbb Q)$$ and integrating $$alpha cup beta cup gamma$$ over $$Z$$, which is the identical as integrating $$alpha cup beta cup gamma cup L$$ over $$X$$, which we will do by describing the ring construction of $$X$$.

If $$alpha in H^i (X,mathbb Q)$$ and $$beta in V$$, then $$alpha cup beta$$ has diploma $$>n-1$$ so it suffices to combine $$alpha cup beta cup gamma$$ over $$Z$$ for $$gamma in H^{n-1-i} (X,mathbb Q)$$. But $$(alpha cup beta cup gamma) = (alpha cup gamma) cup beta =0$$ since $$beta$$ is within the orthogonal complement of $$H^{n-1}(X,mathbb Q)$$.

If $$alpha in H^i (X,mathbb Q)$$ and $$beta in H^{2n-2-j}(X,mathbb Q)^vee$$, then it suffices to combine $$(alpha cup beta cup gamma)$$ for $$gamma in H^{2n-2-i-j}(X,mathbb Q)$$, however that is simply the linear figure $$beta$$ utilized to $$alpha cup gamma$$.

If $$alpha, beta in V$$, then $$alpha cup beta$$ has diploma $$2n-2$$ so it suffices to compute the Poincare duality pairing on $$V$$. This is a nondegenerate symmetric pairing if $$n-1$$ is plane or a nondegenerate symplectic pairing if $$n-1$$ is queer. There is a exclusive such as much as isomorphism, so we will at all times select that one.

Any different pairing may have diploma $$>2n-2$$ and thus vanish.

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