Hermitian–Yang–Mills Connections on Collapsing Elliptically Fibered K3 Surfaces

Let $$X\rightarrow {{\mathbb {P}}}^1$$ X → P 1 be an elliptically fibered K3 surface, admitting a sequence $$\omega _{i}$$ ω i of Ricci-flat metrics collapsing the fibers. Let V be a holomorphic SU(n) bundle over X, stable with respect to $$\omega _i$$ ω i . Given the corresponding sequence $$\Xi _i$$ Ξ i of Hermitian–Yang–Mills connections on V, we prove that, if E is a generic fiber, the restricted sequence $$\Xi _i|_{E}$$ Ξ i | E converges to a flat connection $$A_0$$ A 0 . Furthermore, if the restriction $$V|_E$$ V | E is of the form $$\oplus _{j=1}^n{\mathcal {O}}_E(q_j-0)$$ ⊕ j = 1 n O E ( q j - 0 ) for n distinct points $$q_j\in E$$ q j ∈ E , then these points uniquely determine $$A_0$$ A 0 .

Moduli spaces of stable sheaves over quasi-polarized surfaces, and the relative Strange Duality morphism

The main result of the present paper is a construction of relative moduli spaces of stable sheaves over the stack of quasipolarized projective surfaces. For this, we use the theory of good moduli spaces, whose study was initiated by Alper. As a corollary, we extend the relative Strange Duality morphism to the locus of quasipolarized K3 surfaces.

Stability of some vector bundles on Hilbert schemes of points on K3 surfaces

Let X be a projective K3 surfaces. In two examples where there exists a fine moduli space M of stable vector bundles on X, isomorphic to a Hilbert scheme of points, we prove that the universal family $${\mathcal {E}}$$ E on $$X\times M$$ X × M can be understood as a complete flat family of stable vector bundles on M parametrized by X, which identifies X with a smooth connected component of some moduli space of stable sheaves on M.

Modular curves and the refined distance conjecture

We test the refined distance conjecture in the vector multiplet moduli space of 4D $$ \mathcal{N} $$ N = 2 compactifications of the type IIA string that admit a dual heterotic description. In the weakly coupled regime of the heterotic string, the moduli space geometry is governed by the perturbative heterotic dualities, which allows for exact computations. This is reflected in the type IIA frame through the existence of a K3 fibration. We identify the degree d = 2N of the K3 fiber as a parameter that could potentially lead to large distances, which is substantiated by studying several explicit models. The moduli space geometry degenerates into the modular curve for the congruence subgroup Γ0(N)+. In order to probe the large N regime, we initiate the study of Calabi-Yau threefolds fibered by general degree d > 8 K3 surfaces by suggesting a construction as complete intersections in Grassmann bundles.

Solutions to the Hull–Strominger System with Torus Symmetry

We construct new smooth solutions to the Hull–Strominger system, showing that the Fu–Yau solution on torus bundles over K3 surfaces can be generalized to torus bundles over K3 orbifolds. In particular, we prove that, for $$13 \le k \le 22$$ 13 ≤ k ≤ 22 and $$14\le r\le 22$$ 14 ≤ r ≤ 22 , the smooth manifolds $$S^1\times \sharp _k(S^2\times S^3)$$ S 1 × ♯ k ( S 2 × S 3 ) and $$\sharp _r (S^2 \times S^4) \sharp _{r+1} (S^3 \times S^3)$$ ♯ r ( S 2 × S 4 ) ♯ r + 1 ( S 3 × S 3 ) , have a complex structure with trivial canonical bundle and admit a solution to the Hull–Strominger system.