The Brauer group of analytic K3 surfaces

Abstract

We show that for complex analytic K3 surfaces any torsion class in H2(X,O∗ X) comes from an Azumaya algebra. In other words, the Brauer group equals the cohomological Brauer group. For algebraic surfaces, such results go back to Grothendieck. In our situation, we use twistor spaces to deform a given analytic K3 surface to suitable projective K3 surfaces, and then stable bundles and hyperholomorphy conditions to pass back and forth between the members of the twistor family. In analogy to the isomorphism Pic(X) ∼= H1(X,O∗ X), Grothendieck investigated in [8] the possibility of interpreting classes in H2(X,O∗ X) as geometric objects. He observed that the Brauer group Br(X), parameterizing equivalence classes of sheaves of Azumaya algebras on X, naturally injects into H2(X,O∗ X). It is not difficult to see that Br(X) ⊂ H2(X,O∗ X) is contained in the torsion part of H2(X,O∗ X) and Grothendieck asked: Is the natural injection Br(X ) ⊂ H (X ,O∗ X )tor an isomorphism? This question is of interest in various geometric categories, e.g. X might be a scheme, a complex space, a complex manifold, etc. It is also related to more recent developments in the application of complex algebraic geometry to conformal field theory. Certain elements in H2(X,O∗ X) have been interpreted as so-called B-fields, and those are used to construct super conformal field theories associated to Ricci-flat manifolds. Thus, understanding the geometric meaning of the cohomological Brauer group Br′(X) := H2(X,O∗ X)tor is also of interest for the mathematical interpretation of string theory and mirror symmetry. An affirmative answer to Grothendieck’s question has been given only in very few special cases: • If X is a complex curve, then H2(X,O∗ X) = 0. Hence, Br(X) = Br ′(X) = H2(X,O∗ X) = 0 (see [8, Cor.2.2] for the general case of a curve). • For smooth algebraic surfaces the surjectivity has been proved by Grothendieck [8, Cor.2.2] and for normal algebraic surfaces a proof was given more recently by Schroer [14]. • Hoobler [9] and Berkovich [3] gave an affirmative answer for abelian varieties of any dimension and Elencwajg and Narasimhan gave another proof for complex tori [6].