Complete non-compact G2{manifolds from asymptotically ...ucahlfo/Talks_files/ALC_G2_from_AC_CY3.pdfNon-compact complete examples of manifolds with special holonomy that collapse with

Complete non-compact G2–manifolds from

asymptotically conical Calabi–Yau 3-folds

Lorenzo Foscolo

Heriot-Watt University

joint with Mark Haskins and Johannes Nordstrom

EDGE Seminar, Edinburgh, September 21, 2017

September 20, 2017

Calabi–Yau 3-folds and G2–manifolds

smooth 6-manifold B endowed with an SU(3)–structure (ω,Ω) ω a non-degenerate 2-form

Ω a complex volume form almost complex structure J

compatibility: ω ∧ Ω = 0 and 4ω3 = 3Ω ∧ Ω Riemannian metric gω,Ω

(B, ω,Ω) is a Calabi–Yau 3-fold (CY) if dω = 0 = dΩ

smooth 7-manifold M endowed with a G2–structure φ φ a positive 3-form

16(uyφ) ∧ (vyφ) ∧ φ = gφ(u, v) volgφ

(M, φ) is a G2–manifold if dφ = 0 = d ∗ φ

(B, ω,Ω) CY 3-fold =⇒ M = B × S1, φ = dt ∧ ω + ReΩ G2–manifold

Key difference: whereas the Calabi–Yau Theorem yields examples ofCY 3-folds from algebraic geometry, nothing as powerful and general isknown about the existence of G2–manifolds

Calabi–Yau 3-folds and G2–manifolds

smooth 6-manifold B endowed with an SU(3)–structure (ω,Ω) ω a non-degenerate 2-form

Ω a complex volume form almost complex structure J

compatibility: ω ∧ Ω = 0 and 4ω3 = 3Ω ∧ Ω Riemannian metric gω,Ω

(B, ω,Ω) is a Calabi–Yau 3-fold (CY) if dω = 0 = dΩ

smooth 7-manifold M endowed with a G2–structure φ φ a positive 3-form

16(uyφ) ∧ (vyφ) ∧ φ = gφ(u, v) volgφ

(M, φ) is a G2–manifold if dφ = 0 = d ∗ φ

(B, ω,Ω) CY 3-fold =⇒ M = B × S1, φ = dt ∧ ω + ReΩ G2–manifold

Key difference: whereas the Calabi–Yau Theorem yields examples ofCY 3-folds from algebraic geometry, nothing as powerful and general isknown about the existence of G2–manifolds

Calabi–Yau 3-folds and G2–manifolds

smooth 6-manifold B endowed with an SU(3)–structure (ω,Ω) ω a non-degenerate 2-form

Ω a complex volume form almost complex structure J

compatibility: ω ∧ Ω = 0 and 4ω3 = 3Ω ∧ Ω Riemannian metric gω,Ω

(B, ω,Ω) is a Calabi–Yau 3-fold (CY) if dω = 0 = dΩ

smooth 7-manifold M endowed with a G2–structure φ φ a positive 3-form

16(uyφ) ∧ (vyφ) ∧ φ = gφ(u, v) volgφ

(M, φ) is a G2–manifold if dφ = 0 = d ∗ φ

(B, ω,Ω) CY 3-fold =⇒ M = B × S1, φ = dt ∧ ω + ReΩ G2–manifold

Key difference: whereas the Calabi–Yau Theorem yields examples ofCY 3-folds from algebraic geometry, nothing as powerful and general isknown about the existence of G2–manifolds

Main result

Theorem (F.–Haskins–Nordstrom, 2017)

Let (B, g0, ω0,Ω0) be an asymptotically conical Calabi–Yau 3-foldasymptotic to a Calabi–Yau cone (C, gC) and let M → B be a principalcircle bundle.

Assume that c1(M) = 0 but c1(M) ∪ [ω0] = 0.

Then for every ϵ > 0 sufficiently small there exists an S1–invariantG2–holonomy metric gϵ on M with the following properties.

(M, gϵ) is an ALC manifold: as r → ∞, gϵ = gC + ϵ2θ2∞ + O(r−ν).

(M, gϵ) collapses to (B, g0) with bounded curvature as ϵ → 0:gϵ ∼C k,α g0 + ϵ2θ2 as ϵ → 0.

Main result: comments Only 4 non-trivial examples of simply connected complete non-compact

G2–manifolds are currently known: three asymptotically conical examples due to Bryant–Salamon (1989); an explicit example due to Brandhuber–Gomis–Gubser–Gukov (2001) moving

in a 1-parameter family whose existence was rigorously established byBogoyavlenskaya (2013).

We produce infinitely many new examples.

Non-compact complete examples of manifolds with special holonomy thatcollapse with globally bounded curvature are a new higher-dimensionalphenomenon: the only hyperkahler 4-manifold with a tri-holomorphiccircle action without fixed points is R3 × S1.

Connections to physics: Type IIA String theory compactified on AC CY3-fold (B, ω0,Ω0) with RR 2-form flux dθ satisfying [dθ]∪ [ω0] = 0 as theweak-coupling limit of M theory compactified on an ALC G2–manifold.

Large N duality: duality between Type IIA String theory on thedeformed/resolved conifold with D6-branes/RR fluxes more transparentby passing to M theory (Atiyah–Maldacena–Vafa).

Main result: comments Only 4 non-trivial examples of simply connected complete non-compact

G2–manifolds are currently known: three asymptotically conical examples due to Bryant–Salamon (1989); an explicit example due to Brandhuber–Gomis–Gubser–Gukov (2001) moving

in a 1-parameter family whose existence was rigorously established byBogoyavlenskaya (2013).

We produce infinitely many new examples. Non-compact complete examples of manifolds with special holonomy that

collapse with globally bounded curvature are a new higher-dimensionalphenomenon: the only hyperkahler 4-manifold with a tri-holomorphiccircle action without fixed points is R3 × S1.

Connections to physics: Type IIA String theory compactified on AC CY3-fold (B, ω0,Ω0) with RR 2-form flux dθ satisfying [dθ]∪ [ω0] = 0 as theweak-coupling limit of M theory compactified on an ALC G2–manifold.

Large N duality: duality between Type IIA String theory on thedeformed/resolved conifold with D6-branes/RR fluxes more transparentby passing to M theory (Atiyah–Maldacena–Vafa).

Main result: comments Only 4 non-trivial examples of simply connected complete non-compact

G2–manifolds are currently known: three asymptotically conical examples due to Bryant–Salamon (1989); an explicit example due to Brandhuber–Gomis–Gubser–Gukov (2001) moving

in a 1-parameter family whose existence was rigorously established byBogoyavlenskaya (2013).

We produce infinitely many new examples. Non-compact complete examples of manifolds with special holonomy that

collapse with globally bounded curvature are a new higher-dimensionalphenomenon: the only hyperkahler 4-manifold with a tri-holomorphiccircle action without fixed points is R3 × S1.

Connections to physics: Type IIA String theory compactified on AC CY3-fold (B, ω0,Ω0) with RR 2-form flux dθ satisfying [dθ]∪ [ω0] = 0 as theweak-coupling limit of M theory compactified on an ALC G2–manifold.

Large N duality: duality between Type IIA String theory on thedeformed/resolved conifold with D6-branes/RR fluxes more transparentby passing to M theory (Atiyah–Maldacena–Vafa).

Main result: comments Only 4 non-trivial examples of simply connected complete non-compact

G2–manifolds are currently known: three asymptotically conical examples due to Bryant–Salamon (1989); an explicit example due to Brandhuber–Gomis–Gubser–Gukov (2001) moving

in a 1-parameter family whose existence was rigorously established byBogoyavlenskaya (2013).

We produce infinitely many new examples. Non-compact complete examples of manifolds with special holonomy that

collapse with globally bounded curvature are a new higher-dimensionalphenomenon: the only hyperkahler 4-manifold with a tri-holomorphiccircle action without fixed points is R3 × S1.

Connections to physics: Type IIA String theory compactified on AC CY3-fold (B, ω0,Ω0) with RR 2-form flux dθ satisfying [dθ]∪ [ω0] = 0 as theweak-coupling limit of M theory compactified on an ALC G2–manifold.

Large N duality: duality between Type IIA String theory on thedeformed/resolved conifold with D6-branes/RR fluxes more transparentby passing to M theory (Atiyah–Maldacena–Vafa).

The Gibbons–Hawking Ansatz

Recall the Gibbons–Hawking Ansatz (1978): local form of hyperkahlermetrics in dimension 4 with a triholomorphic circle action

U open subset of R3

h positive function on U

M → U a principal U(1)–bundle with a connection θ

g = h gR3 + h−1θ2 is a hyperkahler metric on M

(h, θ) satisfies the monopole equation ∗dh = dθ

Goal: a higher-dimensional analogue for G2–manifolds

Cvetic–Gibbons–Lu–Pope (2002), Kaste–Minasian–Petrini–Tomasiello (2003),

Apostolov–Salamon (2004)

The Gibbons–Hawking Ansatz

Recall the Gibbons–Hawking Ansatz (1978): local form of hyperkahlermetrics in dimension 4 with a triholomorphic circle action

U open subset of R3

h positive function on U

M → U a principal U(1)–bundle with a connection θ

g = h gR3 + h−1θ2 is a hyperkahler metric on M

(h, θ) satisfies the monopole equation ∗dh = dθ

Goal: a higher-dimensional analogue for G2–manifolds

Cvetic–Gibbons–Lu–Pope (2002), Kaste–Minasian–Petrini–Tomasiello (2003),

Apostolov–Salamon (2004)

The Apostolov–Salamon equations

M7 → B6 a principal circle bundle with connection θ

h : B → R+

(ω,Ω) an SU(3)–structure on B

φ = θ ∧ ω + h34ReΩ, ∗φφ = −θ ∧ h

14 ImΩ+ 1

2hω2,

gφ =√h gω,Ω + h−1θ2

Torsion-free G2–structure on M if and only if

(

43h

34 , θ

)satisfies the Calabi–Yau monopole equations

∗dh = h14 dθ ∧ ReΩ, dθ ∧ ω2 = 0

the SU(3)–structure (ω,Ω) has constrained torsion

dω = 0, d(h

34ReΩ

)+ dθ ∧ ω = 0, d

(h

14 ImΩ

)= 0

The Apostolov–Salamon equations

M7 → B6 a principal circle bundle with connection θ

h : B → R+

(ω,Ω) an SU(3)–structure on B

φ = θ ∧ ω + h34ReΩ, ∗φφ = −θ ∧ h

14 ImΩ+ 1

2hω2,

gφ =√h gω,Ω + h−1θ2

Torsion-free G2–structure on M if and only if

(

43h

34 , θ

)satisfies the Calabi–Yau monopole equations

∗dh = h14 dθ ∧ ReΩ, dθ ∧ ω2 = 0

the SU(3)–structure (ω,Ω) has constrained torsion

dω = 0, d(h

34ReΩ

)+ dθ ∧ ω = 0, d

(h

14 ImΩ

)= 0

Adiabatic limit of the AS equations

Introduce a small parameter ϵ > 0:

φ = ϵ θ ∧ ω + h34ReΩ, gφ =

√h gω,Ω + ϵ2h−1θ2

The ϵ–dependent Apostolov–Salamon equations:

∗dh = ϵ h14 dθ ∧ ReΩ, dθ ∧ ω2 = 0,

dω = 0, d(h

34ReΩ

)+ ϵ dθ ∧ ω = 0, d

(h

14 ImΩ

)= 0.

Formal limit as ϵ → 0: h0 ≡ 1 and (ω0,Ω0) is a CY structure on B.

Linearisation over the collapsed limit: Calabi–Yau monopole

∗dh = dθ ∧ ImΩ0, dθ ∧ ω20 = 0

infinitesimal deformation of the SU(3)–structure

dω = 0, dRe Ω+ 34dh∧ReΩ0+dθ∧ω0 = 0, dIm Ω+ 1

4dh∧ImΩ0 = 0

Adiabatic limit of the AS equations

Introduce a small parameter ϵ > 0:

φ = ϵ θ ∧ ω + h34ReΩ, gφ =

√h gω,Ω + ϵ2h−1θ2

The ϵ–dependent Apostolov–Salamon equations:

∗dh = ϵ h14 dθ ∧ ReΩ, dθ ∧ ω2 = 0,

dω = 0, d(h

34ReΩ

)+ ϵ dθ ∧ ω = 0, d

(h

14 ImΩ

)= 0.

Formal limit as ϵ → 0: h0 ≡ 1 and (ω0,Ω0) is a CY structure on B.

Linearisation over the collapsed limit: Calabi–Yau monopole

∗dh = dθ ∧ ImΩ0, dθ ∧ ω20 = 0

infinitesimal deformation of the SU(3)–structure

dω = 0, dRe Ω+ 34dh∧ReΩ0+dθ∧ω0 = 0, dIm Ω+ 1

4dh∧ImΩ0 = 0

Adiabatic limit of the AS equations

Introduce a small parameter ϵ > 0:

φ = ϵ θ ∧ ω + h34ReΩ, gφ =

√h gω,Ω + ϵ2h−1θ2

The ϵ–dependent Apostolov–Salamon equations:

∗dh = ϵ h14 dθ ∧ ReΩ, dθ ∧ ω2 = 0,

dω = 0, d(h

34ReΩ

)+ ϵ dθ ∧ ω = 0, d

(h

14 ImΩ

)= 0.

Formal limit as ϵ → 0: h0 ≡ 1 and (ω0,Ω0) is a CY structure on B.

Linearisation over the collapsed limit: Calabi–Yau monopole

∗dh = dθ ∧ ImΩ0, dθ ∧ ω20 = 0

infinitesimal deformation of the SU(3)–structure

dω = 0, dRe Ω+ 34dh∧ReΩ0+dθ∧ω0 = 0, dIm Ω+ 1

4dh∧ImΩ0 = 0

Adiabatic limit of the AS equations

Introduce a small parameter ϵ > 0:

φ = ϵ θ ∧ ω + h34ReΩ, gφ =

√h gω,Ω + ϵ2h−1θ2

The ϵ–dependent Apostolov–Salamon equations:

∗dh = ϵ h14 dθ ∧ ReΩ, dθ ∧ ω2 = 0,

dω = 0, d(h

34ReΩ

)+ ϵ dθ ∧ ω = 0, d

(h

14 ImΩ

)= 0.

Formal limit as ϵ → 0: h0 ≡ 1 and (ω0,Ω0) is a CY structure on B.

Linearisation over the collapsed limit: Calabi–Yau monopole

∗dh = dθ ∧ ImΩ0, dθ ∧ ω20 = 0

infinitesimal deformation of the SU(3)–structure

d ω = 0, dRe Ω+ 34dh∧ReΩ0+dθ∧ω0 = 0, dIm Ω+ 1

4dh∧ImΩ0 = 0

Abelian Hermitian Yang–Mills connections

Start with an AC Calabi–Yau 3-fold (B, ω0,Ω0)

We look for Calabi–Yau monopole on B:

∗dh = dθ ∧ ImΩ0, dθ ∧ ω20 = 0

There is a basic dichotomy: h ≡ 0 and θ is a Hermitian Yang–Mills (HYM) connection (h, θ) has singularities (e.g. Dirac-type singularities along a special

Lagrangian submanifold L ⊂ B)

In this talk we only consider the former case

Moreover, a necessary condition to solve

dRe Ω + 34dh ∧ ReΩ0 + dθ ∧ ω0 = 0

is [dθ] ∪ [ω0] = 0 ∈ H4(B).

Abelian Hermitian Yang–Mills connections

Start with an AC Calabi–Yau 3-fold (B, ω0,Ω0)

We look for Calabi–Yau monopole on B:

∗dh = dθ ∧ ImΩ0, dθ ∧ ω20 = 0

There is a basic dichotomy: h ≡ 0 and θ is a Hermitian Yang–Mills (HYM) connection (h, θ) has singularities (e.g. Dirac-type singularities along a special

Lagrangian submanifold L ⊂ B)

In this talk we only consider the former case

Moreover, a necessary condition to solve

dRe Ω + 34dh ∧ ReΩ0 + dθ ∧ ω0 = 0

is [dθ] ∪ [ω0] = 0 ∈ H4(B).

Abelian Hermitian Yang–Mills connections

Start with an AC Calabi–Yau 3-fold (B, ω0,Ω0)

We look for Calabi–Yau monopole on B:

∗dh = dθ ∧ ImΩ0, dθ ∧ ω20 = 0

There is a basic dichotomy: h ≡ 0 and θ is a Hermitian Yang–Mills (HYM) connection (h, θ) has singularities (e.g. Dirac-type singularities along a special

Lagrangian submanifold L ⊂ B)

In this talk we only consider the former case

Moreover, a necessary condition to solve

dRe Ω + 34dh ∧ ReΩ0 + dθ ∧ ω0 = 0

is [dθ] ∪ [ω0] = 0 ∈ H4(B).

Abelian Hermitian Yang–Mills connections

Start with an AC Calabi–Yau 3-fold (B, ω0,Ω0)

We look for Calabi–Yau monopole on B:

∗dh = dθ ∧ ImΩ0, dθ ∧ ω20 = 0

There is a basic dichotomy: h ≡ 0 and θ is a Hermitian Yang–Mills (HYM) connection (h, θ) has singularities (e.g. Dirac-type singularities along a special

Lagrangian submanifold L ⊂ B)

In this talk we only consider the former case

Moreover, a necessary condition to solve

dRe Ω + 34dh ∧ ReΩ0 + dθ ∧ ω0 = 0

is [dθ] ∪ [ω0] = 0 ∈ H4(B).

Solving the AS equations for small ϵ

(B, ω0,Ω0) AC Calabi–Yau manifold

principal U(1)–bundle M → B with c1(M) = 0 & c1(M) ∪ [ω0] = 0

HYM connection θ on M with coexact curvature: dθ = d∗ρ, dρ = 0

Solutionh = 0, θ, ω = 0, Ω = −(∗ρ+ iρ)

of the linearised AS equations

closed ALC S1–invariant G2–structure on M with torsion O(ϵ2)

φ(1)ϵ = ϵ θ ∧ ω0 + ReΩ0 − ϵ ∗ρ

Construct formal solution of the non-linear AS equations as a formalpower series in ϵ

Prove the series has a positive radius of convergence (in weighted

Holder spaces)

Solving the AS equations for small ϵ

(B, ω0,Ω0) AC Calabi–Yau manifold

principal U(1)–bundle M → B with c1(M) = 0 & c1(M) ∪ [ω0] = 0

HYM connection θ on M with coexact curvature: dθ = d∗ρ, dρ = 0

Solutionh = 0, θ, ω = 0, Ω = −(∗ρ+ iρ)

of the linearised AS equations

closed ALC S1–invariant G2–structure on M with torsion O(ϵ2)

φ(1)ϵ = ϵ θ ∧ ω0 + ReΩ0 − ϵ ∗ρ

Construct formal solution of the non-linear AS equations as a formalpower series in ϵ

Prove the series has a positive radius of convergence (in weighted

Holder spaces)

Solving the AS equations for small ϵ

(B, ω0,Ω0) AC Calabi–Yau manifold

principal U(1)–bundle M → B with c1(M) = 0 & c1(M) ∪ [ω0] = 0

HYM connection θ on M with coexact curvature: dθ = d∗ρ, dρ = 0

Solutionh = 0, θ, ω = 0, Ω = −(∗ρ+ iρ)

of the linearised AS equations

closed ALC S1–invariant G2–structure on M with torsion O(ϵ2)

φ(1)ϵ = ϵ θ ∧ ω0 + ReΩ0 − ϵ ∗ρ

Construct formal solution of the non-linear AS equations as a formalpower series in ϵ

Prove the series has a positive radius of convergence (in weighted

Holder spaces)

Solving the AS equations for small ϵ

(B, ω0,Ω0) AC Calabi–Yau manifold

principal U(1)–bundle M → B with c1(M) = 0 & c1(M) ∪ [ω0] = 0

HYM connection θ on M with coexact curvature: dθ = d∗ρ, dρ = 0

Solutionh = 0, θ, ω = 0, Ω = −(∗ρ+ iρ)

of the linearised AS equations

closed ALC S1–invariant G2–structure on M with torsion O(ϵ2)

φ(1)ϵ = ϵ θ ∧ ω0 + ReΩ0 − ϵ ∗ρ

Construct formal solution of the non-linear AS equations as a formalpower series in ϵ

Prove the series has a positive radius of convergence (in weighted

Holder spaces)

Examples from crepant resolutions

Calabi–Yau cones “Kodaira Embedding” for Kahler cones (Ornea–Verbitsky, van Coevering):

C(Σ) Kahler cone with H1(Σ;R) = 0 ⇐⇒ X = C(Σ) ∪ 0 affine variety inCN with an isolated singularity invariant under a torus T ⊂ U(N)

conical Kahler metric ⇐⇒ “polarization”: ξ ∈ t acting on coordinatefunctions with positive weights

Collins–Szekelyhidi, 2015: CY cone structure ⇐⇒ (X , ξ) is K-stable

π : B → X crepant resolution: Ω0 = π∗ΩC

Existence of AC CY metric in each Kahler class containing AC metrics(Joyce 2001, van Coevering 2010, Goto 2012, Conlon–Hein 2013)

In order to apply our Theorem we need a circle bundle M → B withc1(M) ∪ [ω0] = 0 ∈ H4(B). There are no solutions for generic [ω0], but a rank–b2(Σ) sublattice of

H2(B;Z) when [ω0] is integral

Examples from crepant resolutions

Calabi–Yau cones “Kodaira Embedding” for Kahler cones (Ornea–Verbitsky, van Coevering):

C(Σ) Kahler cone with H1(Σ;R) = 0 ⇐⇒ X = C(Σ) ∪ 0 affine variety inCN with an isolated singularity invariant under a torus T ⊂ U(N)

conical Kahler metric ⇐⇒ “polarization”: ξ ∈ t acting on coordinatefunctions with positive weights

Collins–Szekelyhidi, 2015: CY cone structure ⇐⇒ (X , ξ) is K-stable

π : B → X crepant resolution: Ω0 = π∗ΩC

Existence of AC CY metric in each Kahler class containing AC metrics(Joyce 2001, van Coevering 2010, Goto 2012, Conlon–Hein 2013)

In order to apply our Theorem we need a circle bundle M → B withc1(M) ∪ [ω0] = 0 ∈ H4(B). There are no solutions for generic [ω0], but a rank–b2(Σ) sublattice of

H2(B;Z) when [ω0] is integral

Examples from crepant resolutions

Calabi–Yau cones “Kodaira Embedding” for Kahler cones (Ornea–Verbitsky, van Coevering):

C(Σ) Kahler cone with H1(Σ;R) = 0 ⇐⇒ X = C(Σ) ∪ 0 affine variety inCN with an isolated singularity invariant under a torus T ⊂ U(N)

conical Kahler metric ⇐⇒ “polarization”: ξ ∈ t acting on coordinatefunctions with positive weights

Collins–Szekelyhidi, 2015: CY cone structure ⇐⇒ (X , ξ) is K-stable

π : B → X crepant resolution: Ω0 = π∗ΩC

Existence of AC CY metric in each Kahler class containing AC metrics(Joyce 2001, van Coevering 2010, Goto 2012, Conlon–Hein 2013)

In order to apply our Theorem we need a circle bundle M → B withc1(M) ∪ [ω0] = 0 ∈ H4(B). There are no solutions for generic [ω0], but a rank–b2(Σ) sublattice of

H2(B;Z) when [ω0] is integral

Examples from small resolutions of CY cones

Consider the isolated hypersurface singularity Xp ⊂ C4 defined by

xy + zp+1 − wp+1 = 0

Collins–Szekelyhidi (2015): Xp carries a Calabi–Yau cone metric

(this uses K-stability)

Brieskorn (1968): Xp has a small resolution B → Xp.

b4(B) = 0b2(B) = p (chain of p rational curves exceptional set of resolution)

Goto (2012): B carries a p–parameter family of AC CY structures

circle bundle M → B has b2(M) = p − 1 and b3(M) = p

infinitely many new simply connected complete G2–manifoldsand families of complete non-compact G2–metrics of arbitrarily highdimension

Examples from small resolutions of CY cones

Consider the isolated hypersurface singularity Xp ⊂ C4 defined by

xy + zp+1 − wp+1 = 0

Collins–Szekelyhidi (2015): Xp carries a Calabi–Yau cone metric

(this uses K-stability)

Brieskorn (1968): Xp has a small resolution B → Xp.

b4(B) = 0b2(B) = p (chain of p rational curves exceptional set of resolution)

Goto (2012): B carries a p–parameter family of AC CY structures

circle bundle M → B has b2(M) = p − 1 and b3(M) = p

infinitely many new simply connected complete G2–manifoldsand families of complete non-compact G2–metrics of arbitrarily highdimension

Examples from small resolutions of CY cones

Consider the isolated hypersurface singularity Xp ⊂ C4 defined by

xy + zp+1 − wp+1 = 0

Collins–Szekelyhidi (2015): Xp carries a Calabi–Yau cone metric

(this uses K-stability)

Brieskorn (1968): Xp has a small resolution B → Xp.

b4(B) = 0b2(B) = p (chain of p rational curves exceptional set of resolution)

Goto (2012): B carries a p–parameter family of AC CY structures

circle bundle M → B has b2(M) = p − 1 and b3(M) = p

infinitely many new simply connected complete G2–manifoldsand families of complete non-compact G2–metrics of arbitrarily highdimension

Examples from small resolutions of CY cones

Consider the isolated hypersurface singularity Xp ⊂ C4 defined by

xy + zp+1 − wp+1 = 0

Collins–Szekelyhidi (2015): Xp carries a Calabi–Yau cone metric

(this uses K-stability)

Brieskorn (1968): Xp has a small resolution B → Xp.

b4(B) = 0b2(B) = p (chain of p rational curves exceptional set of resolution)

Goto (2012): B carries a p–parameter family of AC CY structures

circle bundle M → B has b2(M) = p − 1 and b3(M) = p

infinitely many new simply connected complete G2–manifoldsand families of complete non-compact G2–metrics of arbitrarily highdimension