In this section, O×hol(Z) stands for the multiplicative group of the algebra Ohol(Z) of all holomorphic functions on a complex space Z, and Ohol,+(Z) for its additive group.

The following proposition is Assertion (1) of Theorem 0.3:

In this section we explain the precise complexity problem studied by the GCT program, namely ¯¯¯¯¯¯¯¯¯¯¯¯VPws≠VNP, and place it in the context of Valiant’s algebraic model of NP-completeness [63, 64]. In particular, we compare this to the conjecture…

The authors are indebted to Pierre Liardet for valuable comments, and to the developpers of the PARI/GP system on which numerical experiments were performed.

Let W=Mn(C), the space of complex matrices of size n. Its dimension is N=n2. The hypersurface in PW defined by the determinant is dual to the variety of rank one matrices, the Segre product Pn−1×Pn−1⊂PN−1.

On montre facilement qu’un diviseur E sur une variété (lisse) X est exceptionnel si par un point général de E passe une courbe C⊂E avec E⋅C<0. On sait démontrer que cette propriété caractérise les diviseurs exceptionnels si dim(X)=2. On démontre ici…