Holomorphic Lefschetz fixed-point formula

<h2 id="statement">Statement</h2>
If f is an <a href="/facts/Automorphism/WnyxNFkQ">automorphism</a> of a compact complex manifold M with isolated fixed points, then

∑
          
            f
            (
            p
            )
            =
            p
          
        
        
          
            1
            
              det
              (
              1
              −
              
                A
                
                  p
                
              
              )
            
          
        
        =
        
          ∑
          
            q
          
        
        (
        −
        1
        
          )
          
            q
          
        
        trace
        ⁡
        (
        
          f
          
            ∗
          
        
        
          |
        
        
          H
          
            
              ∂
              ¯
            
          
          
            0
            ,
            q
          
        
        (
        M
        )
        )
      
    
    {\displaystyle \sum _{f(p)=p}{\frac {1}{\det(1-A_{p})}}=\sum _{q}(-1)^{q}\operatorname {trace} (f^{*}|H_{\overline {\partial }}^{0,q}(M))}

where

<ul><li>The sum is over the fixed points p of f</li>
<li>The linear transformation Ap is the action induced by f on the holomorphic <a href="/facts/Tangent_space/crvpbSeG">tangent space</a> at p</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Bott_residue_formula/ceoPVdqg">Bott residue formula</a></li></ul>

<ul><li><a href="/facts/Phillip_Griffiths/EHQJfBVI">Griffiths, Phillip</a>; <a href="/facts/Joe_Harris_(mathematician)/XPzjGwRf">Harris, Joseph</a> (1994), Principles of algebraic geometry, Wiley Classics Library, New York: <a href="/facts/John_Wiley_%2526_Sons/53g3LqJc">John Wiley & Sons</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-471-05059-9, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1288523">1288523</a></li></ul>

Holomorphic Lefschetz fixed-point formula open-in-new

Holomorphic Lefschetz fixed-point formula