Gravitational instantons from rational elliptic surfaces

Hein, Hans-Joachim

doi:10.1090/s0894-0347-2011-00723-6

Cited by 73 publications

(152 citation statements)

References 47 publications

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“…X is a minimal elliptic surface, Theorem 2 is proved in our previous work joint with Z.L. Zhang [32], where we made crucial use of the result of Song and Tian [22,Lemma 3.4] and Hein [15,Section 3.3], i.e. surface case in Theorem 1.…”

Section: Remarkmentioning

confidence: 99%

“…Special case (2): minimal elliptic surfaces. When f : X → Σ in Theorem 8 is a minimal elliptic surface, the estimate (3.1) was proved by Song-Tian [22,Lemma 3.4] and Hein [15,Section 3.3]. Moreover, in [15,22], the constants β and N are obtained precisely according to the types of singular fibers in Kodaira's table [2, Section V.7].…”

Section: Limiting Singular Metrics On Riemann Surfacesmentioning

confidence: 99%

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Collapsing limits of the Kähler–Ricci flow and the continuity method

Zhang

2018

Math. Ann.

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We consider the Kähler-Ricci flow on certain Calabi-Yau fibration, which is a Calabi-Yau fibration with one dimensional base or a product of two Calabi-Yau fibrations with one dimensional bases. Assume the Kähler-Ricci flow on total space admits a uniform lower bound for Ricci curvature, then the flow converges in Gromov-Hausdorff topology to the metric completion of the regular part of generalized Kähler-Einstein current on the base, which is a compact length metric space homeomorphic to the base. The analogue results for the continuity method on such Calabi-Yau fibrations are also obtained. Moreover, we show the continuity method starting from a suitable Kähler metric on the total space of a Fano fibration with one dimensional base converges in Gromov-Hausdorff topology to a compact metric on the base. During the proof, we show the metric completion of the regular part of a generalized Kähler-Einstein current on a Riemann surface is compact.

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Section: Remarkmentioning

confidence: 99%

Section: Limiting Singular Metrics On Riemann Surfacesmentioning

confidence: 99%

Collapsing limits of the Kähler–Ricci flow and the continuity method

Zhang

2018

Math. Ann.

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“…Starting from a del Pezzo surface of degree nine and blowing up nine points gives a rational elliptic surface M [39]. It will be convenient to refer to this case as a Tian-Yau space of zero degree, so that we can extend the range of the degree to b = 0, 1, 2, .…”

Section: Single-sided Domain Walls and Tian-yau Spacesmentioning

confidence: 99%

Degenerations of K3, orientifolds and exotic branes

Chaemjumrus¹,

Hull

2019

J. High Energ. Phys.

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A recently constructed limit of K3 has a long neck consisting of segments, each of which is a nilfold fibred over a line, that are joined together with Kaluza-Klein monopoles. The neck is capped at either end by a Tian-Yau space, which is non-compact, hyperkähler and asymptotic to a nilfold fibred over a line. We show that the type IIA string on this degeneration of K3 is dual to the type I ′ string, with the Kaluza-Klein monopoles dual to the D8-branes and the Tian-Yau spaces providing a geometric dual to the O8 orientifold planes. At strong coupling, each O8-plane can emit a D8-brane to give an O8 * plane, so that there can be up to 18 D8-branes in the type I ′ string. In the IIA dual, this phenomenon occurs at weak coupling and there can be up to 18 Kaluza-Klein monopoles in the dual geometry. We consider further duals in which the Kaluza-Klein monopoles are dualised to NS5-branes or exotic branes. A 3-torus with H-flux can be realised in string theory as an NS5-brane wrapped on T 3 , with the 3-torus fibred over a line. T-dualising gives a 4-dimensional hyperkähler manifold which is a nilfold fibred over a line, which can be viewed as a Kaluza-Klein monopole wrapped on T 2 . Further T-dualities then give non-geometric spaces fibred over a line and can be regarded as wrapped exotic branes. These are all domain wall configurations, dual to the D8-brane. Type I ′ string theory is the natural home for D8-branes, and we dualise this to find string theory homes for each of these branes. The Kaluza-Klein monopoles arise in the IIA string on the degenerate K3. T-duals of this give exotic branes on non-geometric spaces.

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“…There are no hyperkähler metrics with growth between r 3 and r 4 [29]. However, Hein constructed an example of a hyperkähler metric with volume growth r 4/3 [20]. Chen-Chen call this an example of type ALG * since the growth rate 4 3 is the in ALG-like interval (1,2].…”

Section: Classification Of Noncompact Hyperkähler Manifolds Xmentioning

confidence: 99%

Perspectives on the Asymptotic Geometry of the Hitchin Moduli Space

Fredrickson

2019

SIGMA

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We survey some recent developments in the asymptotic geometry of the Hitchin moduli space, starting with an introduction to the Hitchin moduli space and hyperkähler geometry.Note: This is the model solution featured in [14,18,27]. The base curve is CP 1 with an irregular singularity at ∞ [15]. Nonabelian Hodge correspondenceThe Hitchin moduli space M is hyperkähler. As a consequence, it has a CP 1 -worth of complex structures, labeled by parameter ζ ∈ CP 1 . Two avatars of the Hitchin moduli space are• the Higgs bundle moduli space (ζ = 0), and• the moduli space of flat GL(n, C)-connections ζ ∈ C × .1 In a local holomorphic coordinate z and a local holomorphic frame for (E, ∂E), if ϕ = Φdz, then ϕ * h = h −1 Φ * hdz.

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Gravitational instantons from rational elliptic surfaces

Cited by 73 publications

References 47 publications

Collapsing limits of the Kähler–Ricci flow and the continuity method

Collapsing limits of the Kähler–Ricci flow and the continuity method

Degenerations of K3, orientifolds and exotic branes

Perspectives on the Asymptotic Geometry of the Hitchin Moduli Space

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