Algebraic geometry, categories and trace formula
                                 Bertrand To¨en
                               (CNRS, Toulouse)
                Clay Research Conference, Oxford, September 2017
                                           Algebraic geometry, categories and trace formula
                                                                         1 / 32
                                                                                                                                                                                                                                                                                                                                                                                                                   Homogeneous polynomials F ,...,F ∈ C[X ,...,X ]
                                                                                                                                                                                                                                                                                                                                                                                                                                              1       p       0      n
                                                                                                                                                                                                                                                                                                                                                                                                                                                                 n
                                                                                                                                                                                                                                                                                                                                                                                                                               X := {(x0,...,xn)/Fi(x) = 0} ⊂ P .
                                                                                                                                                                                                                                                                                                                                                                                                                                                                 C
                                                                                                                                                                                                                                                                                                                                                                                                                   Problem: read the topology of X in terms of the Fi’s.
             Topology of algebraic varieties
                                        Algebraic geometry, categories and trace formula
                                                                    2 / 32
             Topology of algebraic varieties
      Homogeneous polynomials F ,...,F ∈ C[X ,...,X ]
                                  1      p       0       n
                  X := {(x ,...,x )/F (x) = 0} ⊂ Pn.
                           0      n    i             C
      Problem: read the topology of X in terms of the Fi’s.
                                        Algebraic geometry, categories and trace formula
                                                                    2 / 32
              Topology of algebraic varieties
       Typical answers in low dimension
            (n = 1,p = 1): X finite set of cardinality deg(F1) counted
            with multiplicities.
            (n = 2 and p = 1): X is a compact Riemann surface and
                      g(X) = (d −1)(d −2)         d =deg(F )
                                     2                      1
            (g(X) is the arithmetic genus if X not smooth).
                                           Algebraic geometry, categories and trace formula
                                                                         3 / 32