Introduction Existence Deformations Other applications Scalar-flat K¨ ahler ALE metrics on minimal resolutions Jeff Viaclovsky University of Wisconsin May 19, 2015 Vanderbilt University Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions
Introduction Existence Deformations Other applications ALE metrics Definition A complete Riemannian manifold ( X 4 , g ) is called asymptotically locally Euclidean or ALE of order τ if there exists a finite subgroup Γ ⊂ SO(4) acting freely on S 3 and a diffeomorphism ψ : X \ K → ( R 4 \ B (0 , R )) / Γ where K is a compact subset of X , and such that under this identification, ( ψ ∗ g ) ij = δ ij + O ( ρ − τ ) , ∂ | k | ( ψ ∗ g ) ij = O ( ρ − τ − k ) , for any partial derivative of order k , as r → ∞ , where ρ is the distance to some fixed basepoint. Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions
Introduction Existence Deformations Other applications Minimal resolutions Definition Let Γ ⊂ U(2) be a finite subgroup which acts freely on S 3 . Then, a smooth complex surface ˜ X is called a minimal resolution of C 2 / Γ if there is a mapping π : ˜ X → C 2 / Γ such that 1 The restriction π : ˜ X \ π − 1 (0) → C 2 / Γ \ { 0 } is a biholomorphism; 2 π − 1 (0) is a divisor in ˜ X containing no − 1 curves. Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions
Introduction Existence Deformations Other applications Minimal resolutions Definition Let Γ ⊂ U(2) be a finite subgroup which acts freely on S 3 . Then, a smooth complex surface ˜ X is called a minimal resolution of C 2 / Γ if there is a mapping π : ˜ X → C 2 / Γ such that 1 The restriction π : ˜ X \ π − 1 (0) → C 2 / Γ \ { 0 } is a biholomorphism; 2 π − 1 (0) is a divisor in ˜ X containing no − 1 curves. • These resolutions are minimal in the sense that, given any other resolution π Y : Y → C 2 / Γ , there is a proper analytic map p : Y → ˜ X such that π Y = π ◦ p . Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions
Introduction Existence Deformations Other applications Minimal resolutions Definition Let Γ ⊂ U(2) be a finite subgroup which acts freely on S 3 . Then, a smooth complex surface ˜ X is called a minimal resolution of C 2 / Γ if there is a mapping π : ˜ X → C 2 / Γ such that 1 The restriction π : ˜ X \ π − 1 (0) → C 2 / Γ \ { 0 } is a biholomorphism; 2 π − 1 (0) is a divisor in ˜ X containing no − 1 curves. • These resolutions are minimal in the sense that, given any other resolution π Y : Y → C 2 / Γ , there is a proper analytic map p : Y → ˜ X such that π Y = π ◦ p . • For each such Γ , up to isomorphism there exists a unique such minimal resolution, and in 1968, Brieskorn completely described these resolutions complex-analytically. Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions
Introduction Existence Deformations Other applications Existence of scalar-flat K¨ ahler ALE metrics Theorem (Lock-V) Let Γ ⊂ U(2) be a finite subgroup which acts freely on S 3 . Then the minimal resolution ˜ X of X = C 2 / Γ admits scalar-flat K¨ ahler ALE metrics. Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions
Introduction Existence Deformations Other applications Small deformations of complex structure Theorem (Lock-V) Let Γ ⊂ U(2) be a finite subgroup finite subgroup which acts freely on S 3 . Then some small deformations of the minimal resolution ˜ X of X = C 2 / Γ admit scalar-flat K¨ ahler ALE metrics. Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions
Introduction Existence Deformations Other applications Existence of extremal K¨ ahler metrics Theorem (Lock-V) Let Γ ⊂ U(2) be a finite subgroup which acts freely on S 3 . Then there exist extremal K¨ ahler metrics on certain K¨ ahler classes on the rational surfaces which arise as complex analytic compactifications of the minimal resolution ˜ X of X = C 2 / Γ . Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions
Introduction Existence Deformations Other applications A non-existence result for Ricci-flat metrics Conjecture of Bando-Kasue-Nakajima: the only simply connected Ricci-flat ALE metrics in dimension four are hyperk¨ ahler. Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions
Introduction Existence Deformations Other applications A non-existence result for Ricci-flat metrics Conjecture of Bando-Kasue-Nakajima: the only simply connected Ricci-flat ALE metrics in dimension four are hyperk¨ ahler. Related to this, Theorem (Lock-V) Let Γ ⊂ U(2) be a finite subgroup which acts freely on S 3 , and let X be diffeomorphic to the minimal resolution of C 2 / Γ or any iterated blow-up thereof. If g is a Ricci-flat ALE metric on X , then Γ ⊂ SU(2) and g is hyperk¨ ahler. Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions
Introduction Existence Deformations Other applications A non-existence result for Ricci-flat metrics Conjecture of Bando-Kasue-Nakajima: the only simply connected Ricci-flat ALE metrics in dimension four are hyperk¨ ahler. Related to this, Theorem (Lock-V) Let Γ ⊂ U(2) be a finite subgroup which acts freely on S 3 , and let X be diffeomorphic to the minimal resolution of C 2 / Γ or any iterated blow-up thereof. If g is a Ricci-flat ALE metric on X , then Γ ⊂ SU(2) and g is hyperk¨ ahler. • We do not assume that g is K¨ ahler, only assumption is about the diffeomorphism type. Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions
Introduction Existence Deformations Other applications Applications to self-dual examples Using similar ideas, we can obtain some new examples of self-dual metrics on N # CP 2 (the connected sum of N copies of the complex projective plane). Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions
Introduction Existence Deformations Other applications Applications to self-dual examples Using similar ideas, we can obtain some new examples of self-dual metrics on N # CP 2 (the connected sum of N copies of the complex projective plane). Theorem (Lock-V) There exist sequences of self-dual metrics on N # CP 2 limiting to orbifolds with singularities which are not cyclic. These metrics admit a conformally isometric S 1 -action. Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions
Introduction Existence Deformations Other applications Applications to self-dual examples Using similar ideas, we can obtain some new examples of self-dual metrics on N # CP 2 (the connected sum of N copies of the complex projective plane). Theorem (Lock-V) There exist sequences of self-dual metrics on N # CP 2 limiting to orbifolds with singularities which are not cyclic. These metrics admit a conformally isometric S 1 -action. • To describe the orbifold groups at the singular points, we need some background on finite subgroups of SO(4) acting freely on S 3 , which I will describe next. Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions
Introduction Existence Deformations Other applications SO(4) • Identification of C 2 and H = { x 0 + x 1 ˆ i + x 2 ˆ j + x 3 ˆ k } : ( z 1 , z 2 ) ∈ C 2 ← → z 1 + z 2 ˆ j ∈ H Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions
Introduction Existence Deformations Other applications SO(4) • Identification of C 2 and H = { x 0 + x 1 ˆ i + x 2 ˆ j + x 3 ˆ k } : ( z 1 , z 2 ) ∈ C 2 ← → z 1 + z 2 ˆ j ∈ H • Double cover of SO(4) : φ : S 3 × S 3 → SO(4) φ ( q 1 , q 2 )( h ) = q 1 ∗ h ∗ ¯ q 2 Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions
Introduction Existence Deformations Other applications SO(4) • Identification of C 2 and H = { x 0 + x 1 ˆ i + x 2 ˆ j + x 3 ˆ k } : ( z 1 , z 2 ) ∈ C 2 ← → z 1 + z 2 ˆ j ∈ H • Double cover of SO(4) : φ : S 3 × S 3 → SO(4) φ ( q 1 , q 2 )( h ) = q 1 ∗ h ∗ ¯ q 2 • Double cover of U(2) : φ : S 1 × S 3 → U(2) Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions
Introduction Existence Deformations Other applications U(2) actions ( φ : S 1 × S 3 → U(2) ) • L ( q, p ) ⊂ U(2) : The subgroup generated by � exp(2 πi/p ) � 0 . 0 exp(2 πiq/p ) Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions
Introduction Existence Deformations Other applications U(2) actions ( φ : S 1 × S 3 → U(2) ) • L ( q, p ) ⊂ U(2) : The subgroup generated by � exp(2 πi/p ) � 0 . 0 exp(2 πiq/p ) • S 3 ∼ = SU(2) : Let h 1 + h 2 ˆ j ∈ S 3 , then � h 1 � � z 1 � − ¯ h 2 ( z 1 + z 2 ˆ j ) ∗ ( h 1 + h 2 ˆ j ) ← → . ¯ h 2 h 1 z 2 Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions
Introduction Existence Deformations Other applications U(2) actions ( φ : S 1 × S 3 → U(2) ) • L ( q, p ) ⊂ U(2) : The subgroup generated by � exp(2 πi/p ) � 0 . 0 exp(2 πiq/p ) • S 3 ∼ = SU(2) : Let h 1 + h 2 ˆ j ∈ S 3 , then � h 1 � � z 1 � − ¯ h 2 ( z 1 + z 2 ˆ j ) ∗ ( h 1 + h 2 ˆ j ) ← → . ¯ h 2 h 1 z 2 • Left quaternionic multiplication: e iθ ∗ ( z 1 + z 2 ˆ j ) ← → L (1 , p ) . Jeff Viaclovsky Scalar-flat K¨ ahler ALE metrics on minimal resolutions
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