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 .

A NOTE ON SPRAYS OF ISOTROPIC CURVATURE

A basic goal of this paper is to calculate, Weyl curvature of R-flat (Ricci-flat) spray of isotropic curvature and a locally projectively R-flat (Ricci-flat) spray, which is a projective invariance. Besides, the equivalents of E ̅-curvature and H-curvature that are closely related to the mean Berwald curvature have been found for a locally projectively R-flat spray of isotropic curvature.

Four-dimensional Einstein manifolds with Heisenberg symmetry

We classify Einstein metrics on $$\mathbb {R}^4$$ R 4 invariant under a four-dimensional group of isometries including a principal action of the Heisenberg group. We consider metrics which are either Ricci-flat or of negative Ricci curvature. We show that all of the Ricci-flat metrics, including the simplest ones which are hyper-Kähler, are incomplete. By contrast, those of negative Ricci curvature contain precisely two complete examples: the complex hyperbolic metric and a metric of cohomogeneity one known as the one-loop deformed universal hypermultiplet.

Investigation of Pseudo-Ricci Symmetric Spacetimes in Gray’s Subspaces

In the present paper, we focused our attention to study pseudo-Ricci symmetric spacetimes in Gray’s decomposition subspaces. It is proved that PRS n spacetimes are Ricci flat in trivial, A , and B subspaces, whereas perfect fluid in subspaces I , I ⊕ A , and I ⊕ B , and have zero scalar curvature in subspace A ⊕ B . Finally, it is proved that pseudo-Ricci symmetric GRW spacetimes are vacuum, and as a consequence of this result, we address several corollaries.

Resolution à la Kronheimer of $$\mathbb {C}^3/\Gamma $$ singularities and the Monge–Ampère equation for Ricci-flat Kähler metrics in view of D3-brane solutions of supergravity

In this paper, we analyze the relevance of the generalized Kronheimer construction for the gauge/gravity correspondence. We begin with the general structure of D3-brane solutions of type IIB supergravity on smooth manifolds $$Y^\Gamma $$ Y Γ that are supposed to be the crepant resolution of quotient singularities $$\mathbb {C}^3/\Gamma $$ C 3 / Γ with $$\Gamma $$ Γ a finite subgroup of SU(3). We emphasize that nontrivial 3-form fluxes require the existence of imaginary self-dual harmonic forms $$\omega ^{2,1}$$ ω 2 , 1 . Although excluded in the classical Kronheimer construction, they may be reintroduced by means of mass deformations. Next we concentrate on the other essential item for the D3-brane construction, namely, the existence of a Ricci-flat metric on $$Y^\Gamma $$ Y Γ . We study the issue of Ricci-flat Kähler metrics on such resolutions $$Y^\Gamma $$ Y Γ , with particular attention to the case $$\Gamma =\mathbb {Z}_4$$ Γ = Z 4 . We advance the conjecture that on the exceptional divisor of $$Y^\Gamma $$ Y Γ the Kronheimer Kähler metric and the Ricci-flat one, that is locally flat at infinity, coincide. The conjecture is shown to be true in the case of the Ricci-flat metric on $$\mathrm{tot} K_{{\mathbb {W}P}[112]}$$ tot K W P [ 112 ] that we construct, i.e., the total space of the canonical bundle of the weighted projective space $${\mathbb {W}P}[112]$$ W P [ 112 ] , which is a partial resolution of $$\mathbb {C}^3/\mathbb {Z}_4$$ C 3 / Z 4 . For the full resolution, we have $$Y^{\mathbb {Z}_4}={\text {tot}} K_{\mathbb {F}_{2}}$$ Y Z 4 = tot K F 2 , where $$\mathbb {F}_2$$ F 2 is the second Hirzebruch surface. We try to extend the proof of the conjecture to this case using the one-parameter Kähler metric on $$\mathbb {F}_2$$ F 2 produced by the Kronheimer construction as initial datum in a Monge–Ampère (MA) equation. We exhibit three formulations of this MA equation, one in terms of the Kähler potential, the other two in terms of the symplectic potential but with two different choices of the variables. In both cases, one can establish a series solution in powers of the variable along the fibers of the canonical bundle. The main property of the MA equation is that it does not impose any condition on the initial geometry of the exceptional divisor, rather it uniquely determines all the subsequent terms as local functionals of this initial datum. Although a formal proof is still missing, numerical and analytical results support the conjecture. As a by-product of our investigation, we have identified some new properties of this type of MA equations that we believe to be so far unknown.