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Kähler-Einstein Metrics on Fano Manifolds

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Kähler-Einstein Metrics on Fano Manifolds K¨ahler-Einstein metrics on Fano manifolds Ved Datar UC Berkeley Jan 02, 2018 Ved Datar (UC Berkeley) K¨ahler-Einstein metrics on Fano manifolds Jan 02, 2018 1 / 36 Outline 1 Introduction: Some history and the main theorem 2 K-stability 3 Outline of the proof 4 What next? Ved Datar (UC Berkeley) K¨ahler-Einstein metrics on Fano manifolds Jan 02, 2018 2 / 36 Outline 1 Introduction: Some history and the main theorem 2 K-stability 3 Outline of the proof 4 What next? Ved Datar (UC Berkeley) K¨ahler-Einstein metrics on Fano manifolds Jan 02, 2018 2 / 36 Outline 1 Introduction: Some history and the main theorem 2 K-stability 3 Outline of the proof 4 What next? Ved Datar (UC Berkeley) K¨ahler-Einstein metrics on Fano manifolds Jan 02, 2018 2 / 36 Outline 1 Introduction: Some history and the main theorem 2 K-stability 3 Outline of the proof 4 What next? Ved Datar (UC Berkeley) K¨ahler-Einstein metrics on Fano manifolds Jan 02, 2018 2 / 36 Introduction: Some history and the main theorem Uniformization Theorem. Theorem (Uniformization theorem) Any compact Riemann surface admits a metric of constant Gauss curvature. Given any oriented compact 2-d Riemannian manifold (M, g0),thereexistsametric 2' g = e g0 with constant Gauss curvature. Ved Datar (UC Berkeley) K¨ahler-Einstein metrics on Fano manifolds Jan 02, 2018 3 / 36 Introduction: Some history and the main theorem Uniformization Theorem. Theorem (Uniformization theorem) Any compact Riemann surface admits a metric of constant Gauss curvature. Given any oriented compact 2-d Riemannian manifold (M, g0),thereexistsametric 2' g = e g0 with constant Gauss curvature. Ved Datar (UC Berkeley) K¨ahler-Einstein metrics on Fano manifolds Jan 02, 2018 3 / 36 Introduction: Some history and the main theorem For a surface with metric (in isothermal coordinates) ds2 = h(dx 2 + dy 2), the Gauss curvature form is given by @2 log h KdA = p 1 dz dz¯, − − @z@z¯ ^ where z = x + p 1y. By Gauss-Bonnet, − KdA=2⇡(M). ZM So depending on the sign of χ(M), a compact Riemann surface admits a metric of constant Gauss curvature K = 1orK =0. ± Remark A purely PDE proof of the case K =1(ie. M = S 2)isthehardest.Thisisaharbingerof things to come! Ved Datar (UC Berkeley) K¨ahler-Einstein metrics on Fano manifolds Jan 02, 2018 4 / 36 Introduction: Some history and the main theorem For a surface with metric (in isothermal coordinates) ds2 = h(dx 2 + dy 2), the Gauss curvature form is given by @2 log h KdA = p 1 dz dz¯, − − @z@z¯ ^ where z = x + p 1y. By Gauss-Bonnet, − KdA=2⇡(M). ZM So depending on the sign of χ(M), a compact Riemann surface admits a metric of constant Gauss curvature K = 1orK =0. ± Remark A purely PDE proof of the case K =1(ie. M = S 2)isthehardest.Thisisaharbingerof things to come! Ved Datar (UC Berkeley) K¨ahler-Einstein metrics on Fano manifolds Jan 02, 2018 4 / 36 Introduction: Some history and the main theorem For a surface with metric (in isothermal coordinates) ds2 = h(dx 2 + dy 2), the Gauss curvature form is given by @2 log h KdA = p 1 dz dz¯, − − @z@z¯ ^ where z = x + p 1y. By Gauss-Bonnet, − KdA=2⇡(M). ZM So depending on the sign of χ(M), a compact Riemann surface admits a metric of constant Gauss curvature K = 1orK =0. ± Remark A purely PDE proof of the case K =1(ie. M = S 2)isthehardest.Thisisaharbingerof things to come! Ved Datar (UC Berkeley) K¨ahler-Einstein metrics on Fano manifolds Jan 02, 2018 4 / 36 Introduction: Some history and the main theorem For a surface with metric (in isothermal coordinates) ds2 = h(dx 2 + dy 2), the Gauss curvature form is given by @2 log h KdA = p 1 dz dz¯, − − @z@z¯ ^ where z = x + p 1y. By Gauss-Bonnet, − KdA=2⇡(M). ZM So depending on the sign of χ(M), a compact Riemann surface admits a metric of constant Gauss curvature K = 1orK =0. ± Remark A purely PDE proof of the case K =1(ie. M = S 2)isthehardest.Thisisaharbingerof things to come! Ved Datar (UC Berkeley) K¨ahler-Einstein metrics on Fano manifolds Jan 02, 2018 4 / 36 Introduction: Some history and the main theorem For a surface with metric (in isothermal coordinates) ds2 = h(dx 2 + dy 2), the Gauss curvature form is given by @2 log h KdA = p 1 dz dz¯, − − @z@z¯ ^ where z = x + p 1y. By Gauss-Bonnet, − KdA=2⇡(M). ZM So depending on the sign of χ(M), a compact Riemann surface admits a metric of constant Gauss curvature K = 1orK =0. ± Remark A purely PDE proof of the case K =1(ie. M = S 2)isthehardest.Thisisaharbingerof things to come! Ved Datar (UC Berkeley) K¨ahler-Einstein metrics on Fano manifolds Jan 02, 2018 4 / 36 Introduction: Some history and the main theorem K¨ahler manifolds n Let M be a compact complex manifold with dimCM = n. ARiemannianmetricg is called K¨ahler if there are local coordinates (x 1, , x 2n)in ··· which 2 gjk = δjk + O( x ), | | and such that for j =1, , n, ··· z j = x j + p 1x n+j − are local holomorphic coordinates. One can associate a (1, 1) form in the following way - If J denotes the canonical complex structure @ @ J = , J2 = id, @x j @x n+j − then we define the K¨ahler form⇣ ! by⌘ !( , )=g(J , ). · · · · g K¨ahler d! =0. () Ved Datar (UC Berkeley) K¨ahler-Einstein metrics on Fano manifolds Jan 02, 2018 5 / 36 Introduction: Some history and the main theorem K¨ahler manifolds n Let M be a compact complex manifold with dimCM = n. ARiemannianmetricg is called K¨ahler if there are local coordinates (x 1, , x 2n)in ··· which 2 gjk = δjk + O( x ), | | and such that for j =1, , n, ··· z j = x j + p 1x n+j − are local holomorphic coordinates. One can associate a (1, 1) form in the following way - If J denotes the canonical complex structure @ @ J = , J2 = id, @x j @x n+j − then we define the K¨ahler form⇣ ! by⌘ !( , )=g(J , ). · · · · g K¨ahler d! =0. () Ved Datar (UC Berkeley) K¨ahler-Einstein metrics on Fano manifolds Jan 02, 2018 5 / 36 Introduction: Some history and the main theorem K¨ahler manifolds n Let M be a compact complex manifold with dimCM = n. ARiemannianmetricg is called K¨ahler if there are local coordinates (x 1, , x 2n)in ··· which 2 gjk = δjk + O( x ), | | and such that for j =1, , n, ··· z j = x j + p 1x n+j − are local holomorphic coordinates. One can associate a (1, 1) form in the following way - If J denotes the canonical complex structure @ @ J = , J2 = id, @x j @x n+j − then we define the K¨ahler form⇣ ! by⌘ !( , )=g(J , ). · · · · g K¨ahler d! =0. () Ved Datar (UC Berkeley) K¨ahler-Einstein metrics on Fano manifolds Jan 02, 2018 5 / 36 Introduction: Some history and the main theorem K¨ahler manifolds n Let M be a compact complex manifold with dimCM = n. ARiemannianmetricg is called K¨ahler if there are local coordinates (x 1, , x 2n)in ··· which 2 gjk = δjk + O( x ), | | and such that for j =1, , n, ··· z j = x j + p 1x n+j − are local holomorphic coordinates. One can associate a (1, 1) form in the following way - If J denotes the canonical complex structure @ @ J = , J2 = id, @x j @x n+j − then we define the K¨ahler form⇣ ! by⌘ !( , )=g(J , ). · · · · g K¨ahler d! =0. () Ved Datar (UC Berkeley) K¨ahler-Einstein metrics on Fano manifolds Jan 02, 2018 5 / 36 Introduction: Some history and the main theorem Properties of the K¨ahler form ! is a closed, real form (ie.! ¯ = !), and so represents a cohomology class in 2 H (M, R). Locally we can write p 1 ↵ β ! = − g ¯dz dz¯ , 2 ↵β ^ @ @ where g ¯ = g , . Then the matrix g ¯ is a positive definite Hermitian ↵β @z↵ @z¯β { ↵β } matrix. ⇣ ⌘ Conversely, given such a form !, g( , )=!( , J )definesaRiemannianmetric. · · · · 2 Any class in H (M, R)whichcontainsaK¨ahlermetric,iscalledaK¨ahlerclass.The 2 set of K¨ahler classes H (M, R)isanopenconvexcone. K⇢ ¯ (@@ Lemma) If [!1]=[!2], then there exists a ' C 1(M, R)suchthat 2 !2 = !1 + p 1@@'. − Note: Henceforth we will abuse notation and refer to ! as simply the K¨ahlermetric. Ved Datar (UC Berkeley) K¨ahler-Einstein metrics on Fano manifolds Jan 02, 2018 6 / 36 Introduction: Some history and the main theorem Properties of the K¨ahler form ! is a closed, real form (ie.! ¯ = !), and so represents a cohomology class in 2 H (M, R). Locally we can write p 1 ↵ β ! = − g ¯dz dz¯ , 2 ↵β ^ @ @ where g ¯ = g , . Then the matrix g ¯ is a positive definite Hermitian ↵β @z↵ @z¯β { ↵β } matrix. ⇣ ⌘ Conversely, given such a form !, g( , )=!( , J )definesaRiemannianmetric. · · · · 2 Any class in H (M, R)whichcontainsaK¨ahlermetric,iscalledaK¨ahlerclass.The 2 set of K¨ahler classes H (M, R)isanopenconvexcone. K⇢ ¯ (@@ Lemma) If [!1]=[!2], then there exists a ' C 1(M, R)suchthat 2 !2 = !1 + p 1@@'. − Note: Henceforth we will abuse notation and refer to ! as simply the K¨ahlermetric. Ved Datar (UC Berkeley) K¨ahler-Einstein metrics on Fano manifolds Jan 02, 2018 6 / 36 Introduction: Some history and the main theorem Properties of the K¨ahler form ! is a closed, real form (ie.! ¯ = !), and so represents a cohomology class in 2 H (M, R).
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