Local invariant cycle theorem

In mathematics, the local invariant cycle theorem was originally a conjecture of Griffiths which states that, given a surjective proper map p {\displaystyle p} from a Kähler manifold X {\displaystyle X} to the unit disk that has maximal rank everywhere except over 0, each cohomology class on p − 1 ( t ) , t ≠ 0 {\displaystyle p^{-1}(t),t\neq 0} is the restriction of some cohomology class on the entire X {\displaystyle X} if the cohomology class is invariant under a circle action (monodromy action); in short, H ∗ ⁡ ( X ) → H ∗ ⁡ ( p − 1 ( t ) ) S 1 {\displaystyle \operatorname {H} ^{*}(X)\to \operatorname {H} ^{*}(p^{-1}(t))^{S^{1}}} is surjective. The conjecture was first proved by Clemens.

Source: Wikipedia — Local invariant cycle theorem (CC BY-SA 4.0)

Local invariant cycle theorem

In mathematics, the local invariant cycle theorem was originally a conjecture of Griffiths which states that, given a surjective proper map p {\displaystyle p} from a Kähler manifold X {\displaystyle X} to the unit disk that has maximal rank everywhere except over 0, each cohomology class on p − 1 ( t ) , t ≠ 0 {\displaystyle p^{-1}(t),t\neq 0} is the restriction of some cohomology class on the entire X {\displaystyle X} if the cohomology class is invariant under a circle action (monodromy action); in short, H ∗ ⁡ ( X ) → H ∗ ⁡ ( p − 1 ( t ) ) S 1 {\displaystyle \operatorname {H} ^{*}(X)\to \operatorname {H} ^{*}(p^{-1}(t))^{S^{1}}} is surjective. The conjecture was first proved by Clemens.

Source: Wikipedia "Local invariant cycle theorem" · CC BY-SA 4.0

Share this article: X · Bluesky
Privacy Policy