A gluing construction of K3 surfaces

Abstract

We develop a new method for constructing K3 surfaces. We construct such a K3 surface X by patching two open complex surfaces obtained as the complements of tubular neighborhoods of elliptic curves embedded in blow-ups of the projective planes at general nine points. Our construction has 19 complex dimensional degrees of freedom. For general parameters, the K3 surface X is neither Kummer nor projective. By the argument based on the concrete computation of the period map, we also investigate which points in the period domain correspond to K3 surfaces obtained by such construction.

A gluing construction of projective K3 surfaces

“…Moreover, for any s ∈ ∆ \ {0} with sufficiently small |s| << 1, the K3 surface X s admits an ample line bundle L s = L + ∨ L − by Theorem 1.2 (ii). Hence we have an 18 dimensional family of projective K3 surfaces, whose Kodaira-Spencer map is injective by [KU,Theorem 1.1]. Moreover it follows from [KU] that there exists a holomorphic immersion F b : C → X b mentioned in Theorem 1.1 (see also Remark 2.3).…”

On the complement of a hypersurface with flat normal bundle which corresponds to a semipositive line bundle

“…Indeed, as it follows from [CLPT,Proposition 2.7] that the fibration as in the theorem exists if the natural map H 1 (X, O X ) → H 1 (Y, O Y ) is not injective. Therefore it is sufficient to consider the case where Pic As N m Y /X is not holomorphically trivial for any positive integer m, one has that, for any neighborhood Ω of Y , there exists a compact Levi-flat hypersurface H in Ω \ Y such that each leaf of the Levi foliation of H is dense in H. Then it holds that any holomorphic function on Ω \ Y is constant (see the proof of [KU,Lemma 2.2] for example). Therefore it is clear that the Hartogs type extension theorem holds on X \ Y .…”

Hermitian metrics on the anti-canonical bundle of the blow-up of the projective plane at nine points

“…Note that we can naturally regard L as a holomorphic line bundle, since each line bundle L y is isomorphic (on Y 0 × {y} via the first projection as a holomorphic line bundle) to the C * -flat line bundle on Y 0 which corresponds to the C *representation π 1 (Y 0 , * ) → C * defined by γ 1 → 1 and γ 2 → exp(2π √ −1(p(y) + q(y) • τ )). It follows from the fact that Pic 0 (Y 0 ) has a property as the coarse moduli (see [KU,§A.1] for example), there exists an isomorphism i :…”