Hermitian vs. Riemannian Geometry

Transcription

1 Hermitian vs. Riemannian Geometry Gabe Khan 1 1 Department of Mathematics The Ohio State University GSCAGT, May 2016

2 Outline of the talk Complex curves Background definitions What happens if the metric is flat? What happens if the metric is not flat?

3 Riemannian Surfaces A complex curve is a (real) 2-dimensional surface that is the solution set to an algebraic equation. For instance, one can obtain a double torus by the equation y 2 = x 5 3x 3 + 2x if you view the equation in CP 2. These surfaces are examples of Riemann surfaces and are the simplest examples of complex manifolds.

4 Riemannian Surfaces A complex curve is a (real) 2-dimensional surface that is the solution set to an algebraic equation. For instance, one can obtain a double torus by the equation y 2 = x 5 3x 3 + 2x if you view the equation in CP 2. These surfaces are examples of Riemann surfaces and are the simplest examples of complex manifolds.

5 Riemannian Surfaces A complex curve is a (real) 2-dimensional surface that is the solution set to an algebraic equation. For instance, one can obtain a double torus by the equation y 2 = x 5 3x 3 + 2x if you view the equation in CP 2. These surfaces are examples of Riemann surfaces and are the simplest examples of complex manifolds.

6 Riemannian Surfaces For surfaces, there is a nice parallel between the Riemannian geometry and the complex geometry. However, in higher dimensions things are a bit more complicated.

7 What makes a manifold complex? Given a Riemannian manifold M, an almost complex structure is a linear map J : T p M T p M such that J 2 = I and J varies smoothly in p. An almost complex structure is orthogonal to the metric if g(u, v) = g(ju, Jv) for all tangent vectors u and v. Essentially, J is multiplication by i. Almost complex manifolds are even-dimensional and oriented.

8 What makes a manifold complex? Given a Riemannian manifold M, an almost complex structure is a linear map J : T p M T p M such that J 2 = I and J varies smoothly in p. An almost complex structure is orthogonal to the metric if g(u, v) = g(ju, Jv) for all tangent vectors u and v. Essentially, J is multiplication by i. Almost complex manifolds are even-dimensional and oriented.

9 What makes a manifold complex? A complex structure is an almost complex structure that satisfies some extra integrability condition, which ensure that the Cauchy-Riemann equations hold. Equivalently, this implies that the tensor N J (X, Y ) = [X, Y ] + J([JX, Y ] + [X, JY ]) [JX, JY ] vanishes. This is the Newlander-Nirenberg theorem. There are quite a few restrictions that prevent a metric from admitting a complex structure.

10 What makes a manifold complex? A complex structure is an almost complex structure that satisfies some extra integrability condition, which ensure that the Cauchy-Riemann equations hold. Equivalently, this implies that the tensor N J (X, Y ) = [X, Y ] + J([JX, Y ] + [X, JY ]) [JX, JY ] vanishes. This is the Newlander-Nirenberg theorem. There are quite a few restrictions that prevent a metric from admitting a complex structure.

11 What makes a manifold complex? A complex structure is an almost complex structure that satisfies some extra integrability condition, which ensure that the Cauchy-Riemann equations hold. Equivalently, this implies that the tensor N J (X, Y ) = [X, Y ] + J([JX, Y ] + [X, JY ]) [JX, JY ] vanishes. This is the Newlander-Nirenberg theorem. There are quite a few restrictions that prevent a metric from admitting a complex structure.

12 Complex structures The nicest complex manifolds are Kähler manifolds. There are many ways to define the Kähler condition, but one of the simplest is that the complex structure J is parallel, i.e. J = 0. Equivalently, if we define ω as the 2-form g(j, ), we can define a Kähler metric as one that satisfies dω = 0. Kähler manifolds are a subclass of complex manifolds. The Kähler condition is very restrictive and there are a whole slew of obstructions from being a Kähler manifold. In some sense, most complex manifolds are not Kähler.

13 Complex structures The nicest complex manifolds are Kähler manifolds. There are many ways to define the Kähler condition, but one of the simplest is that the complex structure J is parallel, i.e. J = 0. Equivalently, if we define ω as the 2-form g(j, ), we can define a Kähler metric as one that satisfies dω = 0. Kähler manifolds are a subclass of complex manifolds. The Kähler condition is very restrictive and there are a whole slew of obstructions from being a Kähler manifold. In some sense, most complex manifolds are not Kähler.

14 Complex manifolds Being a Kähler manifold is very restrictive, but being a complex manifold is also restrictive so it takes some work to construct non-kähler complex manifolds. For instance, all smooth complex algebraic varieties are Kähler.

15 Complex manifolds Being a Kähler manifold is very restrictive, but being a complex manifold is also restrictive so it takes some work to construct non-kähler complex manifolds. For instance, all smooth complex algebraic varieties are Kähler.

16 Complex manifolds Kähler geometry has been studied extensively, so we generally consider the non-kähler case. Our work studies the interplay between the complex and Riemannian geometry. On Kähler manifolds, the Hermitian and Riemannian geometry coincide. A general question may be the following: Given a particular Riemannian manifold, what can we say about the moduli space of complex structures orthogonal to the metric?

17 Complex manifolds Kähler geometry has been studied extensively, so we generally consider the non-kähler case. Our work studies the interplay between the complex and Riemannian geometry. On Kähler manifolds, the Hermitian and Riemannian geometry coincide. A general question may be the following: Given a particular Riemannian manifold, what can we say about the moduli space of complex structures orthogonal to the metric?

18 Complex manifolds Kähler geometry has been studied extensively, so we generally consider the non-kähler case. Our work studies the interplay between the complex and Riemannian geometry. On Kähler manifolds, the Hermitian and Riemannian geometry coincide. A general question may be the following: Given a particular Riemannian manifold, what can we say about the moduli space of complex structures orthogonal to the metric?

19 Flat manifolds Let s get started with an example, in which the Riemannian geometry as simple as possible. We want to understand the possible complex structures when the Riemannian curvature is identically zero (i.e., flat manifolds). These all admit finite unbranched covers of tori, so we ll pass to a cover and just consider tori. In this case, all complex structures are balanced (dω n 1 = 0). Admitting a balanced metric has topological obstructions (Michelson) and is a birational invariant, so the Riemannian geometry strongly affects the complex geometry. However, we can say more.

20 Flat manifolds Let s get started with an example, in which the Riemannian geometry as simple as possible. We want to understand the possible complex structures when the Riemannian curvature is identically zero (i.e., flat manifolds). These all admit finite unbranched covers of tori, so we ll pass to a cover and just consider tori. In this case, all complex structures are balanced (dω n 1 = 0). Admitting a balanced metric has topological obstructions (Michelson) and is a birational invariant, so the Riemannian geometry strongly affects the complex geometry. However, we can say more.

21 2 and 4 dimensions In real dimension 2, the only flat orientable surface is the torus. The study of complex structures on tori is part of Teichmüller theory. However, on a given flat tori, there is a unique compatible complex structure. For surfaces, any complex structure is automatically Kähler. Moving on to real 4 dimensions, all of the complex structures on a flat 4 dimensional torus are Kähler. For a given flat 4-torus, the moduli space of complex structures orthogonal to the metric is SO(4)/U(2), which is isometric to CP 1.

22 2 and 4 dimensions In real dimension 2, the only flat orientable surface is the torus. The study of complex structures on tori is part of Teichmüller theory. However, on a given flat tori, there is a unique compatible complex structure. For surfaces, any complex structure is automatically Kähler. Moving on to real 4 dimensions, all of the complex structures on a flat 4 dimensional torus are Kähler. For a given flat 4-torus, the moduli space of complex structures orthogonal to the metric is SO(4)/U(2), which is isometric to CP 1.

23 6 dimensions In real 6 dimensions, the situation is more complicated, and more interesting. On a given flat 6-torus, there are some compatible Kähler structures. The moduli space of structures orthogonal to the metric is SO(6)/U(3). However, there are also non-kähler structures, known as BSV-tori. These spaces were originally constructed by Borisov, Salamon and Viaclovsky, which is why we refer to them as BSV-tori. Calabi discovered a related space in 1958, but he used a completely different method to produce it.

24 6 dimensions In real 6 dimensions, the situation is more complicated, and more interesting. On a given flat 6-torus, there are some compatible Kähler structures. The moduli space of structures orthogonal to the metric is SO(6)/U(3). However, there are also non-kähler structures, known as BSV-tori. These spaces were originally constructed by Borisov, Salamon and Viaclovsky, which is why we refer to them as BSV-tori. Calabi discovered a related space in 1958, but he used a completely different method to produce it.

25 What is a BSV-tori? Here s the idea: We start with a flat 6-dimensional manifold M = E T where E is an elliptic curve and T is a 4-dimensional torus. We view this as a trivial torus bundle over an elliptic curve. We know that the base admits a unique complex structure and that the fiber admits a CP 1 of complex structures. With that in mind, we vary the complex structure on the fiber depending on the base point.

26 BSV-tori Let J E be the complex structure determined by g E which makes E an elliptic curve. Given a map f : E CP 1, we get an associated almost complex manifold where the almost complex structure J is defined by J = J E + J f(x) at the point (x, y) in M = E T. This only produces an almost complex manifold. In order to produce a complex manifold, we must ensure that the integrability condition holds. This occurs if and only if the map f : E CP 1 is holomorphic. Note that holomorphic maps from an elliptic curve to CP 1 are just doubly-periodic meromorphic functions.

27 BSV-tori Let J E be the complex structure determined by g E which makes E an elliptic curve. Given a map f : E CP 1, we get an associated almost complex manifold where the almost complex structure J is defined by J = J E + J f(x) at the point (x, y) in M = E T. This only produces an almost complex manifold. In order to produce a complex manifold, we must ensure that the integrability condition holds. This occurs if and only if the map f : E CP 1 is holomorphic. Note that holomorphic maps from an elliptic curve to CP 1 are just doubly-periodic meromorphic functions.

28 BSV-tori Let J E be the complex structure determined by g E which makes E an elliptic curve. Given a map f : E CP 1, we get an associated almost complex manifold where the almost complex structure J is defined by J = J E + J f(x) at the point (x, y) in M = E T. This only produces an almost complex manifold. In order to produce a complex manifold, we must ensure that the integrability condition holds. This occurs if and only if the map f : E CP 1 is holomorphic. Note that holomorphic maps from an elliptic curve to CP 1 are just doubly-periodic meromorphic functions.

29 BSV-tori All of this motivates the following definition: Definition Let (E, g E ) be flat torus of real dimension 2, (T, g T ) be flat torus of real dimension 4 and let (M, g) be their metric product. Let J E be the complex structure determined by g E, which makes E an elliptic curve. Let f be a non-constant holomorphic map f : E CP 1. Since CP 1 = SO(4)/U(2) is the set of all complex structures on the flat 4-torus (T, g T ) compatible with the metric and the orientation, one may consider almost complex structures J on M defined by J = J E + J f(x) at the point (x, y) in M. Such a complex manifold is called a BSV-tori.

30 BSV-tori Theorem (K., Yang, Zheng, 2016) Let (M 6, g) be a compact Hermitian manifold whose Riemannian curvature tensor is identically zero. Then a finite unbranched cover of M is holomorphically isometric to either a flat complex torus or a BSV-torus. This provides a classification of complex structures on compact flat 6-dimensional spaces. Theorem Let M be a BSV-tori. Then M admits no pluri-closed Hermitian metrics. In particular, it admits no orthogonal metric which makes it a Kähler metric.

31 BSV-tori We are currently working to understand Riemannian-flat 8 dimensional Hermitian manifolds. A similar classification may be possible but there would be a lot more cases. In 10 or more dimensions, this may break down and we are not really sure what happens.

32 What happens if the metric is not flat? Some work of Salamon suggests that non-flat manifolds generically do not admit orthogonal complex structures and that if they admit one, it is unique. Therefore, the general case is very different from the flat case. In fact, if a metric admits 8 independent orthogonal complex structures, then it is locally conformally flat. The non-flat case can be a bit trickier. However, the situation is not completely hopeless.

33 What happens if the metric is not flat? Some work of Salamon suggests that non-flat manifolds generically do not admit orthogonal complex structures and that if they admit one, it is unique. Therefore, the general case is very different from the flat case. In fact, if a metric admits 8 independent orthogonal complex structures, then it is locally conformally flat. The non-flat case can be a bit trickier. However, the situation is not completely hopeless.

34 What happens if the metric is not flat? Some work of Salamon suggests that non-flat manifolds generically do not admit orthogonal complex structures and that if they admit one, it is unique. Therefore, the general case is very different from the flat case. In fact, if a metric admits 8 independent orthogonal complex structures, then it is locally conformally flat. The non-flat case can be a bit trickier. However, the situation is not completely hopeless.

35 Theorem (K., Lam, 2016) Let (M 2n, g) be a compact globally conformally Riemannian-flat Hermitian manifold and consider = n z i. Let K = inf x M Ric M, i=1 z i k = sup x M Ric M, R be the scalar curvature of M, d be the diameter of M and i be the injectivity radius of M. If λ 1 is an eigenvalue of, (i.e. u = λ 1 u) then we have the following estimate: λ C(d, K, k, n, R, R 2, i)

36 This looks complicated but all it is saying is that for some Riemannian manifolds, you can bound the lowest eigenvalue of the complex Laplacian in terms of the Riemannian geometry. We believe that this is true in much greater generality. I ll end the talk with the following conjecture. Conjecture Let (M 2n, g) be a compact Hermitian manifold and consider = n z i. If λ 1 is the principle eigenvalue of, i.e. i=1 z i u = λ 1 u and λ 1 is the smallest eigenvalue, then we can bound λ 1 from below solely using the Riemannian geometry of M.

37 Thank you for listening!