Timeline for Hypersurfaces without variable cohomology

Current License: CC BY-SA 3.0

10 events

when toggle format	what		by	license	comment
Jan 29, 2016 at 5:04	history	edited	Honglu	CC BY-SA 3.0	added 53 characters in body
Jan 28, 2016 at 19:31	history	edited	Honglu	CC BY-SA 3.0	deleted 77 characters in body
Jan 28, 2016 at 16:56	answer	added	Will Sawin		timeline score: 2
Jan 28, 2016 at 16:29	history	edited	Honglu	CC BY-SA 3.0	correct mistakes in the motivation
Jan 28, 2016 at 16:27	comment	added	Daniel Litt		This will be pretty rare--the morphism in question is a map of Hodge structures, so it is surjective (hence an isomorphism by Lefschetz) iff it is an isomorphism on every Hodge component. The obstruction to surjectivity on $H^n(X, \mathcal{O}_X)\to H^n(Y, \mathcal{O}_Y)$ is the kernel of the map $H^{n+1}(X, \mathcal{O}_X(-Y))\to H^{n+1}(X, \mathcal{O}_X)$. But for $Y$ sufficiently ample, this kernel will be huge.
Jan 28, 2016 at 16:27	history	edited	Honglu	CC BY-SA 3.0	Add my motivation and stuff like that.
Jan 28, 2016 at 16:04	history	edited	Honglu	CC BY-SA 3.0	added 26 characters in body
Jan 28, 2016 at 16:01	comment	added	Honglu		@JasonStarr Right. But I only need the restriction to be surjective. I think in your example it's still ok. Probably it's better for me to change boundary divisors into torus invariant divisors.
Jan 28, 2016 at 10:25	comment	added	Jason Starr		What do you mean by "boundary divisors"? For the smooth projective toric variety $X=\mathbb{P}^1\times \mathbb{P}^2$ with the standard action of $\mathbb{G}_m\times (\mathbb{G}_m)^2$, one of the irreducible components of the complement of the open orbit is $Y=\{0\}\times \mathbb{P}^2$. The restriction map $H^2(X;\mathbb{Q})\to H^2(Y;\mathbb{Q})$ is not injective.
Jan 28, 2016 at 1:24	history	asked	Honglu	CC BY-SA 3.0