Local invariant cycle theorem

In mathematics, the local invariant cycle theorem was originally a conjecture of Griffiths which states that, given a surjective <a href="/facts/Proper_map/jJS4YK7o">proper map</a> 
 
 
 
 p
 
 
 {\displaystyle p}
 
 from a <a href="/facts/K%25C3%25A4hler_manifold/JFPM6Vxg">Kähler manifold</a> 
 
 
 
 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. The theorem is also a consequence of the <a href="/facts/BBD_decomposition/jUUyrrp1">BBD decomposition</a>.
Deligne also proved the following. Given a <a href="/facts/Proper_morphism/EJHIgjnr">proper morphism</a> 
 
 
 
 X
 →
 S
 
 
 {\displaystyle X\to S}
 
 over the spectrum 
 
 
 
 S
 
 
 {\displaystyle S}
 
 of the henselization of 
 
 
 
 k
 [
 T
 ]
 
 
 {\displaystyle k[T]}
 
, 
 
 
 
 k
 
 
 {\displaystyle k}
 
 an algebraically closed field, if 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is essentially smooth over 
 
 
 
 k
 
 
 {\displaystyle k}
 
 and 
 
 
 
 
 X
 
 
 η
 ¯
 
 
 
 
 
 {\displaystyle X_{\overline {\eta }}}
 
 smooth over 
 
 
 
 
 
 η
 ¯
 
 
 
 
 {\displaystyle {\overline {\eta }}}
 
, then the homomorphism on 
 
 
 
 
 Q
 
 
 
 {\displaystyle \mathbb {Q} }
 
-cohomology:

H
          
            ∗
          
        
        ⁡
        (
        
          X
          
            s
          
        
        )
        →
        
          H
          
            ∗
          
        
        ⁡
        (
        
          X
          
            
              η
              ¯
            
          
        
        
          )
          
            Gal
            ⁡
            (
            
              
                η
                ¯
              
            
            
              /
            
            η
            )
          
        
      
    
    {\displaystyle \operatorname {H} ^{*}(X_{s})\to \operatorname {H} ^{*}(X_{\overline {\eta }})^{\operatorname {Gal} ({\overline {\eta }}/\eta )}}

is surjective, where 
 
 
 
 s
 ,
 η
 
 
 {\displaystyle s,\eta }
 
 are the special and generic points and the homomorphism is the composition 
 
 
 
 
 H
 
 ∗
 
 
 ⁡
 (
 
 X
 
 s
 
 
 )
 ≃
 
 H
 
 ∗
 
 
 ⁡
 (
 X
 )
 →
 
 H
 
 ∗
 
 
 ⁡
 (
 
 X
 
 η
 
 
 )
 →
 
 H
 
 ∗
 
 
 ⁡
 (
 
 X
 
 
 η
 ¯
 
 
 
 )
 .
 
 
 {\displaystyle \operatorname {H} ^{*}(X_{s})\simeq \operatorname {H} ^{*}(X)\to \operatorname {H} ^{*}(X_{\eta })\to \operatorname {H} ^{*}(X_{\overline {\eta }}).}

Local invariant cycle theorem open-in-new

Local invariant cycle theorem