Timeline for Cup Product with Ample Line Bundles

9 events

when toggle format	what		by	license	comment
Nov 21, 2019 at 7:49	vote	accept	Winnie_XP
Nov 21, 2019 at 6:15	review	Close votes
Nov 22, 2019 at 12:22
Nov 21, 2019 at 5:53	answer	added	abx		timeline score: 4
Nov 21, 2019 at 3:12	comment	added	Winnie_XP		@BalazsElek I see, thanks. Let me edit my post for more details.
Nov 21, 2019 at 3:12	history	edited	Winnie_XP	CC BY-SA 4.0	added 13 characters in body
Nov 21, 2019 at 3:08	comment	added	Balazs Elek		Yes, my comment only answers your question if $2k<n$.
Nov 21, 2019 at 3:03	comment	added	Winnie_XP		@BalazsElek Dear Balaza, thanks for your reply. I am a bit confused here. Say n=dim X, and k=n-1. HL says that $H^2\to H^{2n-2}$ is isomorphism, but I am actually consider $\cup c_1(L)\colon H^{2n-2}\to H^2n}$ here ?
Nov 21, 2019 at 2:58	comment	added	Balazs Elek		If $\alpha\in H^{2k}(X\mathbb{Q})$ and $2k<n$, where $n=\dim_\mathbb{C}(X)$ then this is a consequence of the Hard Lefschetz Theorem, which says that if $L$ is ample, then the map $(c_1(L))^{r} \cup -: H^{n-r}(X,\mathbb{Q}) \to H^{n+r}(X,\mathbb{Q})$ is an isomorphism.
Nov 21, 2019 at 2:04	history	asked	Winnie_XP	CC BY-SA 4.0