Surfaces Moduli K3 surfaces Algebraic geometry Details Collapsing Ricci-flat metrics on K3 surfaces Jeff Viaclovsky University of California, Irvine June 27, 2018 Tokyo Institute of Technology Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces
Surfaces Moduli K3 surfaces Algebraic geometry Details Surfaces Compact orientable surfaces are classified by their genus γ : γ = 0 γ = 1 γ ≥ 2 . Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces
Surfaces Moduli K3 surfaces Algebraic geometry Details Metrics and distance Definition A Riemannian metric g on a manifold M is a smooth choice of positive definite inner product on each tangent space T p M . This gives us a way to compute distances, since the length of a path α : [ a, b ] → M is given by � b � L ( α ) = g ( ˙ α, ˙ α ) dt. a The distance between points p 1 and p 2 in the infimum L over all smooth paths connecting p 1 and p 2 . Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces
Surfaces Moduli K3 surfaces Algebraic geometry Details Volumes Let e 1 and e 2 be an ONB of the tangent space T p M , and let e 1 and e 2 denote the dual basis of T ∗ p M . That is, e 1 ( e 1 ) = 1 , e 1 ( e 2 ) = 0 e 2 ( e 1 ) = 0 , e 2 ( e 2 ) = 1 . The form e 1 ∧ e 2 is a well-defined 2 -form, independent of the choice of basis, which is called the volume form, and we let dV g ≡ e 1 ∧ e 2 . The volume of a region U ⊂ M is defined by � V ol ( R ) = dV g . U Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces
Surfaces Moduli K3 surfaces Algebraic geometry Details Gaussian Curvature Definition The Gaussian curvature K : M → R is defined by 1 + K ( p ) V ol ( B ( p, r )) = πr 2 � 12 r 2 + O ( r 4 ) � , as r → 0 , where B ( p, r ) is a ball of radius r centered at p . Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces
Surfaces Moduli K3 surfaces Algebraic geometry Details Constant Gauss curvature A “best” metric on a surface is one which has constant curvature 1 spherical K = 0 flat − 1 hyperbolic. It turns out that such a metric always exists on a compact surface, this is part of the uniformization theorem. Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces
Surfaces Moduli K3 surfaces Algebraic geometry Details The sphere The flat metric in R 3 restricts to a very nice metric on the unit sphere in R 3 , called the round metric . The round metric is the unique “best” metric on the 2 -sphere: Theorem If ( S 2 , g ) is any Riemannian metric on the 2 -sphere with constant curvature 1 , then there exists a diffeomorphism ϕ : S 2 → S 2 so that ϕ ∗ g is the standard round metric. Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces
Surfaces Moduli K3 surfaces Algebraic geometry Details The torus The usual picture of a torus (as a surface of revolution in R 3 ) does not represent a “best” metric. Instead, we define it as � � ( T 2 , g ) = R 2 / Z ⊕ Z , g 0 where the action is by integer translations in either coordinate direction, and g 0 is the flat metric, so K ≡ 0 . Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces
Surfaces Moduli K3 surfaces Algebraic geometry Details Lattices More generally, we can consider a lattice in R 2 generated by { 1 , τ } where τ is a complex number in the upper half plane. Consider the quotient � � ( T 2 , g ) = R 2 / Z ⊕ Z · τ, g 0 . For any τ in the upper half plane, we get a flat metric on the torus. Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces
Surfaces Moduli K3 surfaces Algebraic geometry Details Moduli of flat metrics It turns out that the lattices determined by τ and τ ′ determine the same flat metric on T 2 if and only if τ ′ = aτ + b cτ + d, where � a � b ∈ SL(2 , Z ) / {± 1 } . c d Consequently, there is a 2 -dimensional family of flat metrics on a torus. Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces
Surfaces Moduli K3 surfaces Algebraic geometry Details Moduli of flat metrics The above family contains all flat metrics on a torus (up to diffeomorphism): Theorem If g is any flat metric on the torus T 2 , then there exists a diffeomorphism ϕ : T 2 → T 2 such that ϕ ∗ g is the quotient of the flat metric on R 2 by a lattice. So, up to scaling, there is a 2 -dimensional moduli space of solutions to the equation K = 0 . Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces
Surfaces Moduli K3 surfaces Algebraic geometry Details Compactification There are 2 ways to compactify this moduli space. • Scaling to unit diameter, a torus limits to a circle, with only one of the S 1 directions shrinking. (We say that the torus limits to the circle in the Gromov-Hausdorff sense. • Algebraically, we can add a singular elliptic curve, a nodal cubic curve Figure: y 2 = x 3 − x 2 Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces
Surfaces Moduli K3 surfaces Algebraic geometry Details Higher genus The Gauss-Bonnet theorem states that � KdV g = 2 πχ ( M ) , M where χ ( M ) = V − E + F is the Euler characteristic. If K = − 1 , then χ ( M ) = 2 − 2 γ < 0 , so the genus is ≥ 2 . Riemann showed that there is a moduli space of solutions to the equation K = − 1 of dimension dim R ( M γ ) = 6 γ − 6 Compactification is more involved, involving limits with hyperbolic cusps. Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces
Surfaces Moduli K3 surfaces Algebraic geometry Details Scalar curvature In higher dimensions, the expansion of volume of a ball is R ( p ) V ol ( B ( p, r )) = ω n r n � 6( n + 2) r 2 + O ( r 4 ) � 1 − , as r → 0 . This defines a curvature quantity R called the scalar curvature . Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces
Surfaces Moduli K3 surfaces Algebraic geometry Details Ricci Curvature The expansion of the volume element in normal coordinates is 1 − 1 dx 1 ∧ · · · ∧ dx n , � 6 R kl x k x l + O ( r 3 ) � dV g = as r → 0 . This defines a tensor Ric = R kl dx k ⊗ dx l , called the Ricci tensor . Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces
Surfaces Moduli K3 surfaces Algebraic geometry Details Einstein metrics The condition that Ric ( g ) = λ · g, for some constant λ , is a generalization of the constant Gaussian curvature case to higher dimensions, and solutions are known as Einstein metrics . Einstein was interested in solutions of this equation on a Lorentzian manifold (with metric g of signature (1 , 3) ), in which case the equations are hyperbolic. But in recent years mathematicians have become interested in solutions on a Riemannian manifold (with a positive definite metric), and such solutions are also very important in physics. Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces
Surfaces Moduli K3 surfaces Algebraic geometry Details Moduli of Einstein metrics For a manifold M of dimension n , the moduli space of solutions is defined by M λ = { g ∈ Met ( M ) | Ric ( g ) = λ · g } / Diff ( M ) We must mod out by the infinite dimensional group Diff ( M ) of diffeomorphisms of M , which makes things tricky. The trick is to show that there is a “slice” for the diffeomorphism group action on the space of metrics (Ebin). Then Einstein’s equations become elliptic. The only failure of ellipticity comes from the diffeomorphism action directions. (Details omitted). Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces
Surfaces Moduli K3 surfaces Algebraic geometry Details Ellipticity The equation Ric ( g ) = λ · g, is nonlinear, and, modulo the diffeomorphism directions, very roughly looks like ∆ g = g ∗ ∇ g so is “quasilinear elliptic”. Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces
Surfaces Moduli K3 surfaces Algebraic geometry Details The linearized operator The moduli space of Einstein metrics near a given Einstein metric can be studied using the linearized operator. Indeed, if we write F ( θ ) = Ric ( g + θ ) − λ · ( g + θ ) , then F ′ ( h ) = d � dtF ( th ) � � t =0 After getting rid of the diffeomorphism directions, the linearized operator is of the form F ′ ( h ) = ∆ h + lower order terms , where the lower order terms involve the curvature of g , so is an elliptic operator. Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces
Surfaces Moduli K3 surfaces Algebraic geometry Details Lyapunov-Schmidt reduction One would like to say that the moduli space of Einstein metrics looks like the kernel of the linearized operator. However, since the operator is nonlinear, this may or may not be true. What is true is the following: Theorem There exists a mapping Ψ : Ker ( F ′ ) → Coker ( F ′ ) such that the moduli space of Einstein metrics near g looks like Ψ − 1 (0) . That is, the zero space of the operator F mapping between between infinite-dimensional spaces looks exactly like the zero set of a smooth map between finite-dimensional spaces. Jeff Viaclovsky Collapsing Ricci-flat metrics on K3 surfaces
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