Every Other Hodge Isometry of the Cohomology of a K3 Surface is Induced by an Autoequivalence of the Derived Category

Using a result of Orlov on Fourier-Mukai transforms we construct a homomorphism from the automorphism group of the derived category to the automorphism group of the Mukai lattice on a K3 surface. Applying Mukai's moduli space construction we show that this homomorphism is surjective up to index 2. Let X be a projective K3 surface over the eld C. Its singular cohomology H∗(X,Z) is free of rank 24. The only nonvanishing components appear in degrees 0, 2, 4 and have ranks of 1, 22, 1, respectively. The intersection pairing (cup product) on H(X,Z) has signature (3,19) and is even and unimodular. In the Hodge decomposition of the complex cohomology H(X,C) = H ⊕H ⊕H the outer summands give a plane P := H⊕H which is de ned over the reals. P is a positive plane and is called the period of the surface. By the Torelli theorem for K3 surfaces P determines X up to isomorphy. A reference for generalities on K3 surfaces and a proof of the Torelli theorem is the series of lectures in [Be]. Furthermore, H(X,R) contains the lattice H(X,Z). We consider the following automorphisms of the vector space H(X,R): Aut(H(X)) := {φ̃ ∈ GL(H(X,Z)) isometry with φ̃C(H) = H} Thus we deal only with those automorphisms which respect all three structures: pairing, Hodge structure, and lattice. We enhance these structures to H∗(X,R). A new pairing is de ned on the whole lattice H∗(X,Z) by setting (v, v′) := −rs′ − r′s+ α.α′ for v = (r, α, s), v′ = (r′, α′, s′) This could also be written as (v, v′) := − ∫ X v∨ · v′ where multiplication takes place in the ring H∗(X,Z) (i.e. is the usual intersection product) and the dual is de ned to be v∨ := (r,−α, s).