Einstein Metrics, Harmonic Forms, & Symplectic Four-Manifolds Claude

Einstein Metrics,

Harmonic Forms, &

Symplectic Four-Manifolds

Claude LeBrun Stony Brook University

Sardegna, 2014/9/20

1 Deﬁnition. A Riemannian metric h is said to be Einstein if it has constant Ricci curvature— i.e.

2 Deﬁnition. A Riemannian metric h is said to be Einstein if it has constant Ricci curvature— i.e. r = λh for some constant λ ∈ R.

3 Deﬁnition. A Riemannian metric h is said to be Einstein if it has constant Ricci curvature— i.e. r = λh for some constant λ ∈ R.

“. . . the greatest blunder of my life!” — A. Einstein, to G. Gamow

4 Deﬁnition. A Riemannian metric h is said to be Einstein if it has constant Ricci curvature— i.e. r = λh for some constant λ ∈ R.

5 Deﬁnition. A Riemannian metric h is said to be Einstein if it has constant Ricci curvature— i.e. r = λh for some constant λ ∈ R.

As punishment . . .

6 Deﬁnition. A Riemannian metric h is said to be Einstein if it has constant Ricci curvature— i.e. r = λh for some constant λ ∈ R.

λ called Einstein constant.

7 Deﬁnition. A Riemannian metric h is said to be Einstein if it has constant Ricci curvature— i.e. r = λh for some constant λ ∈ R.

λ called Einstein constant.

Has same sign as the scalar curvature j ij s = rj = R ij.

8 Deﬁnition. A Riemannian metric h is said to be Einstein if it has constant Ricci curvature— i.e. r = λh for some constant λ ∈ R.

λ called Einstein constant.

Has same sign as the scalar curvature j ij s = rj = R ij.

volg(Bε(p)) ε2 4 n = 1 − s + O(ε ) cnε 6(n+2)

...... ε...... r ......

9 Geometrization Problem:

10 Geometrization Problem:

Given a smooth compact manifold Mn, can one

11 Geometrization Problem:

Given a smooth compact manifold Mn, can one decompose M in Einstein and collapsed pieces?

12 Geometrization Problem:

Given a smooth compact manifold Mn, can one decompose M in Einstein and collapsed pieces?

When n =3, answer is “Yes.”

13 Geometrization Problem:

Given a smooth compact manifold Mn, can one decompose M in Einstein and collapsed pieces?

When n =3, answer is “Yes.”

Proof by Ricci ﬂow. Perelman et al.

14 Geometrization Problem:

Given a smooth compact manifold Mn, can one decompose M in Einstein and collapsed pieces?

When n =3, answer is “Yes.”

Proof by Ricci ﬂow. Perelman et al.

Perhaps reasonable in other dimensions?

15 Recognition Problem:

16 Recognition Problem:

Suppose Mn admits Einstein metric h.

17 Recognition Problem:

Suppose Mn admits Einstein metric h.

What, if anything, does h then tell us about M?

18 Recognition Problem:

Suppose Mn admits Einstein metric h.

What, if anything, does h then tell us about M?

Can we recognize M by looking at h?

19 Recognition Problem:

Suppose Mn admits Einstein metric h.

What, if anything, does h then tell us about M?

Can we recognize M by looking at h?

When n =3, h has constant sectional curvature!

20 Recognition Problem:

Suppose Mn admits Einstein metric h.

What, if anything, does h then tell us about M?

Can we recognize M by looking at h?

When n =3, h has constant sectional curvature!

3 3 3 So M has universal cover S , R , H ...

21 Recognition Problem:

Suppose Mn admits Einstein metric h.

What, if anything, does h then tell us about M?

Can we recognize M by looking at h?

When n =3, h has constant sectional curvature!

3 3 3 So M has universal cover S , R , H ...

But when n ≥ 5, situation seems hopeless.

22 Recognition Problem:

Suppose Mn admits Einstein metric h.

What, if anything, does h then tell us about M?

Can we recognize M by looking at h?

When n =3, h has constant sectional curvature!

3 3 3 So M has universal cover S , R , H ...

But when n ≥ 5, situation seems hopeless.

{Einstein metrics on Sn}/∼ is highly disconnected.

23 Recognition Problem:

Suppose Mn admits Einstein metric h.

What, if anything, does h then tell us about M?

Can we recognize M by looking at h?

When n =3, h has constant sectional curvature!

3 3 3 So M has universal cover S , R , H ...

But when n ≥ 5, situation seems hopeless.

{Einstein metrics on Sn}/∼ is highly disconnected.

When n ≥ 4, situation is more encouraging. . .

24 Moduli Spaces of Einstein metrics

25 Moduli Spaces of Einstein metrics

+ E (M) = {Einstein h}/(Diﬀeos × R )

26 Moduli Spaces of Einstein metrics

+ E (M) = {Einstein h}/(Diﬀeos × R )

Known to be connected for certain4-manifolds:

27 Moduli Spaces of Einstein metrics

+ E (M) = {Einstein h}/(Diﬀeos × R )

Known to be connected for certain4-manifolds:

4 4 M = T ,K 3, H /Γ, CH2/Γ.

28 Moduli Spaces of Einstein metrics

+ E (M) = {Einstein h}/(Diﬀeos × R )

Known to be connected for certain4-manifolds:

4 4 M = T ,K 3, H /Γ, CH2/Γ.

Berger, Hitchin, Besson-Courtois-Gallot, L.

29 Moduli Spaces of Einstein metrics

+ E (M) = {Einstein h}/(Diﬀeos × R )

Known to be connected for certain4-manifolds:

4 4 M = T ,K 3, H /Γ, CH2/Γ.

Berger, Hitchin, Besson-Courtois-Gallot, L.

30 Moduli Spaces of Einstein metrics

+ E (M) = {Einstein h}/(Diﬀeos × R )

Known to be connected for certain4-manifolds:

4 4 M = T ,K 3, H /Γ, CH2/Γ.

Berger, Hitchin, Besson-Courtois-Gallot,L.

31 Moduli Spaces of Einstein metrics

+ E (M) = {Einstein h}/(Diﬀeos × R )

Known to be connected for certain4-manifolds:

4 4 M = T ,K 3, H /Γ, CH2/Γ.

Berger, Hitchin, Besson-Courtois-Gallot, L.

32 Four Dimensions is Exceptional

33 Four Dimensions is Exceptional

When n =4, Einstein metrics are genuinely non- trivial: not typically spaces of constant curvature.

34 Four Dimensions is Exceptional

When n =4, Einstein metrics are genuinely non- trivial: not typically spaces of constant curvature.

There are beautiful and subtle global obstructions to the existence of Einstein metrics on 4-manifolds.

35 Four Dimensions is Exceptional

When n =4, Einstein metrics are genuinely non- trivial: not typically spaces of constant curvature.

There are beautiful and subtle global obstructions to the existence of Einstein metrics on 4-manifolds.

But might allow for geometrization of4-manifolds by decomposition into Einstein and collapsed pieces.

36 Four Dimensions is Exceptional

When n =4, Einstein metrics are genuinely non- trivial: not typically spaces of constant curvature.

There are beautiful and subtle global obstructions to the existence of Einstein metrics on 4-manifolds.

But might allow for geometrization of4-manifolds by decomposition into Einstein and collapsed pieces.

One key question:

37 Four Dimensions is Exceptional

When n =4, Einstein metrics are genuinely non- trivial: not typically spaces of constant curvature.

There are beautiful and subtle global obstructions to the existence of Einstein metrics on 4-manifolds.

But might allow for geometrization of4-manifolds by decomposition into Einstein and collapsed pieces.

One key question:

Does enough rigidity really hold in dimension four to make this a genuine geometrization?

38 Symplectic 4-manifolds:

39 Symplectic 4-manifolds:

A laboratory for exploring Einstein metrics.

40 Symplectic 4-manifolds:

A laboratory for exploring Einstein metrics.

K¨ahlergeometry is a rich source of examples.

41 Symplectic 4-manifolds:

A laboratory for exploring Einstein metrics.

K¨ahlergeometry is a rich source of examples.

If M admits a K¨ahlermetric, it of course admits a symplectic form ω.

42 Symplectic 4-manifolds:

A laboratory for exploring Einstein metrics.

K¨ahlergeometry is a rich source of examples.

If M admits a K¨ahlermetric, it of course admits a symplectic form ω.

On such manifolds, Seiberg-Witten theory mimics K¨ahlergeometry when treating non-K¨ahlermetrics.

43 Symplectic 4-manifolds:

A laboratory for exploring Einstein metrics.

K¨ahlergeometry is a rich source of examples.

If M admits a K¨ahlermetric, it of course admits a symplectic form ω.

On such manifolds, Seiberg-Witten theory mimics K¨ahlergeometry when treating non-K¨ahlermetrics.

Some Suggestive Questions. If (M4, ω) is a symplectic 4-manifold, when does M4 admit an Einstein metric h (unrelated to ω)? What if we also require λ ≥ 0?

44 Symplectic 4-manifolds:

A laboratory for exploring Einstein metrics.

K¨ahlergeometry is a rich source of examples.

If M admits a K¨ahlermetric, it of course admits a symplectic form ω.

On such manifolds, Seiberg-Witten theory mimics K¨ahlergeometry when treating non-K¨ahlermetrics.

Some Suggestive Questions. If (M4,ω ) is a symplectic 4-manifold, when does M4 admit an Einstein metric h (unrelated to ω)? What if we also require λ ≥ 0?

45 Symplectic 4-manifolds:

A laboratory for exploring Einstein metrics.

K¨ahlergeometry is a rich source of examples.

If M admits a K¨ahlermetric, it of course admits a symplectic form ω.

On such manifolds, Seiberg-Witten theory mimics K¨ahlergeometry when treating non-K¨ahlermetrics.

Some Suggestive Questions. If (M4,ω ) is a symplectic 4-manifold, when does M4 admit an Einstein metric h (unrelated to ω)? What if we also require λ ≥ 0?

46 Symplectic 4-manifolds:

A laboratory for exploring Einstein metrics.

K¨ahlergeometry is a rich source of examples.

If M admits a K¨ahlermetric, it of course admits a symplectic form ω.

On such manifolds, Seiberg-Witten theory mimics K¨ahlergeometry when treating non-K¨ahlermetrics.

Some Suggestive Questions. If (M4,ω ) is a symplectic 4-manifold, when does M4 admit an Einstein metric h (unrelated to ω)? What if we also require λ ≥ 0?

47 Symplectic 4-manifolds:

A laboratory for exploring Einstein metrics.

K¨ahlergeometry is a rich source of examples.

If M admits a K¨ahlermetric, it of course admits a symplectic form ω.

On such manifolds, Seiberg-Witten theory mimics K¨ahlergeometry when treating non-K¨ahlermetrics.

Some Suggestive Questions. If (M4,ω ) is a symplectic 4-manifold, when does M4 admit an Einstein metric h (unrelated to ω)? What if we also require λ ≥ 0?

48 Theorem (L ’09). Suppose that M is a smooth compact oriented 4-manifold which admits a symplectic structure ω. Then M also admits an Einstein metric h with λ ≥ 0 if and only if

49 Theorem (L ’09). Suppose that M is a smooth compact oriented 4-manifold which admits a symplectic structure ω. Then M also admits an Einstein metric h with λ ≥ 0 if and only if

50 Theorem (L ’09). Suppose that M is a smooth compact oriented 4-manifold which admits a symplectic structure ω. Then M also admits an Einstein metric h with λ ≥ 0 if and only if

51 Theorem (L ’09). Suppose that M is a smooth compact oriented 4-manifold which admits a symplectic structure ω. Then M also admits an Einstein metric h with λ ≥ 0 if and only if

52 Theorem (L ’09). Suppose that M is a smooth compact oriented 4-manifold which admits a symplectic structure ω. Then M also admits an Einstein metric h with λ ≥ 0 if and only if

53 Theorem (L ’09). Suppose that M is a smooth compact oriented 4-manifold which admits a symplectic structure ω. Then M also admits an Einstein metric h with λ ≥ 0 if and only if

54 Theorem (L ’09). Suppose that M is a smooth compact oriented 4-manifold which admits a symplectic structure ω. Then M also admits an Einstein metric h with λ ≥ 0 if and only if  CP #k , 0 ≤ k ≤ 8,  2 CP2  2 2 S × S ,  K3, diﬀ  M ≈ K3/Z2,  T 4,  T 4/ ,T 4/ ,T 4/ ,T 4/ ,  Z2 Z3 Z4 Z6  4 4 4 T /(Z2 ⊕ Z2),T /(Z3 ⊕ Z3), or T /(Z2 ⊕ Z4).

55 Theorem (L ’09). Suppose that M is a smooth compact oriented 4-manifold which admits a symplectic structure ω. Then M also admits an Einstein metric h with λ ≥ 0 if and only if  #k , 0 ≤ k ≤ 8, CP2 CP2  2 2 S × S ,  K3, diﬀ  M ≈ K3/Z2,  T 4,  T 4/ ,T 4/ ,T 4/ ,T 4/ ,  Z2 Z3 Z4 Z6  4 4 4 T /(Z2 ⊕ Z2),T /(Z3 ⊕ Z3), or T /(Z2 ⊕ Z4).

56 Theorem (L ’09). Suppose that M is a smooth compact oriented 4-manifold which admits a symplectic structure ω. Then M also admits an Einstein metric h with λ ≥ 0 if and only if  #k , 0 ≤ k ≤ 8, CP2 CP2  2 2 S × S  K3 diﬀ  M ≈ K3/Z2,  T 4,  T 4/ ,T 4/ ,T 4/ ,T 4/ ,  Z2 Z3 Z4 Z6  4 4 4 T /(Z2 ⊕ Z2),T /(Z3 ⊕ Z3), or T /(Z2 ⊕ Z4).

57 Theorem (L ’09). Suppose that M is a smooth compact oriented 4-manifold which admits a symplectic structure ω. Then M also admits an Einstein metric h with λ ≥ 0 if and only if  #k , 0 ≤ k ≤ 8, CP2 CP2  2 2 S × S ,  K3 diﬀ  M ≈ K3/Z2,  T 4,  T 4/ ,T 4/ ,T 4/ ,T 4/ ,  Z2 Z3 Z4 Z6  4 4 4 T /(Z2 ⊕ Z2),T /(Z3 ⊕ Z3), or T /(Z2 ⊕ Z4).

58 Theorem (L ’09). Suppose that M is a smooth compact oriented 4-manifold which admits a symplectic structure ω. Then M also admits an Einstein metric h with λ ≥ 0 if and only if  #k , 0 ≤ k ≤ 8, CP2 CP2  2 2 S × S ,  K3, diﬀ  M ≈ K3/Z2,  T 4,  T 4/ ,T 4/ ,T 4/ ,T 4/ ,  Z2 Z3 Z4 Z6  4 4 4 T /(Z2 ⊕ Z2),T /(Z3 ⊕ Z3), or T /(Z2 ⊕ Z4).

59 Theorem (L ’09). Suppose that M is a smooth compact oriented 4-manifold which admits a symplectic structure ω. Then M also admits an Einstein metric h with λ ≥ 0 if and only if  #k , 0 ≤ k ≤ 8, CP2 CP2  2 2 S × S ,  K3, diﬀ  M ≈ K3/Z2,  T 4,  T 4/ ,T 4/ ,T 4/ ,T 4/ ,  Z2 Z3 Z4 Z6  4 4 4 T /(Z2 ⊕ Z2),T /(Z3 ⊕ Z3), or T /(Z2 ⊕ Z4).

60 Theorem (L ’09). Suppose that M is a smooth compact oriented 4-manifold which admits a symplectic structure ω. Then M also admits an Einstein metric h with λ ≥ 0 if and only if  #k , 0 ≤ k ≤ 8, CP2 CP2  2 2 S × S ,  K3, diﬀ  M ≈ K3/Z2,  T 4,  T 4/ ,T 4/ ,T 4/ ,T 4/ ,  Z2 Z3 Z4 Z6  4 4 4 T /(Z2 ⊕ Z2),T /(Z3 ⊕ Z3), or T /(Z2 ⊕ Z4).

61 Theorem (L ’09). Suppose that M is a smooth compact oriented 4-manifold which admits a symplectic structure ω. Then M also admits an Einstein metric h with λ ≥ 0 if and only if  #k , 0 ≤ k ≤ 8, CP2 CP2  2 2 S × S ,  K3, diﬀ  M ≈ K3/Z2,  T 4,  T 4/ ,T 4/ ,T 4/ ,T 4/ ,  Z2 Z3 Z4 Z6  4 4 4 T /(Z2 ⊕ Z2),T /(Z3 ⊕ Z3), or T /(Z2 ⊕ Z4).

62 Theorem (L ’09). Suppose that M is a smooth compact oriented 4-manifold which admits a symplectic structure ω. Then M also admits an Einstein metric h with λ ≥ 0 if and only if  #k , 0 ≤ k ≤ 8, CP2 CP2  2 2 S × S ,  K3, diﬀ  M ≈ K3/Z2,  T 4,  T 4/ ,T 4/ ,T 4/ ,T 4/ ,  Z2 Z3 Z4 Z6  4 4 4 T /(Z2 ⊕ Z2),T /(Z3 ⊕ Z3), or T /(Z2 ⊕ Z4).

63 Conventions:

CP2 = reverse oriented CP2.

64 Conventions:

CP2 = reverse oriented CP2.

Connected sum#:

......

65 Conventions:

CP2 = reverse oriented CP2.

Connected sum#:

......

66 Conventions:

CP2 = reverse oriented CP2.

Connected sum#:

......

67 Conventions:

CP2 = reverse oriented CP2.

Connected sum#:

......

68 Conventions:

CP2 = reverse oriented CP2.

Connected sum#:

......

69 Conventions:

CP2 = reverse oriented CP2.

Connected sum#:

......

70 Conventions:

CP2 = reverse oriented CP2.

Connected sum#:

......

71 Theorem (L ’09). Suppose that M is a smooth compact oriented 4-manifold which admits a symplectic structure ω. Then M also admits an Einstein metric h with λ ≥ 0 if and only if  #k , 0 ≤ k ≤ 8, CP2 CP2  2 2 S × S ,  K3, diﬀ  M ≈ K3/Z2,  T 4,  T 4/ ,T 4/ ,T 4/ ,T 4/ ,  Z2 Z3 Z4 Z6  4 4 4 T /(Z2 ⊕ Z2),T /(Z3 ⊕ Z3), or T /(Z2 ⊕ Z4).

72 Theorem (L ’09). Suppose that M is a smooth compact oriented 4-manifold which admits a symplectic structure ω. Then M also admits an Einstein metric h with λ ≥ 0 if and only if  #k , 0 ≤ k ≤ 8, CP2 CP2  2 2 S × S ,  K3, diﬀ  M ≈ K3/Z2,  T 4,  T 4/ ,T 4/ ,T 4/ ,T 4/ ,  Z2 Z3 Z4 Z6  4 4 4 T /(Z2 ⊕ Z2),T /(Z3 ⊕ Z3), or T /(Z2 ⊕ Z4). Del Pezzo surfaces, K3 surface, Enriques surface, Abelian surface, Hyper-elliptic surfaces.

73 But we understand some cases better than others!

CP2#kCP2, 0 ≤ k ≤ 8, S2 × S2, K3, M === K3/Z2, T 4, 4 4 4 4 T /Z2,T /Z3,T /Z4,T /Z6, 4 4 4 T /(Z2 ⊕ Z2),T /(Z3 ⊕ Z3), or T /(Z2 ⊕ Z4).

74 Deﬁnitive list . . .

CP2#kCP2, 0 ≤ k ≤ 8, S2 × S2, K3, M === K3/Z2, T 4, 4 4 4 4 T /Z2,T /Z3,T /Z4,T /Z6, 4 4 4 T /(Z2 ⊕ Z2),T /(Z3 ⊕ Z3), or T /(Z2 ⊕ Z4).

75 But we understand some cases better than others!

CP2#kCP2, 0 ≤ k ≤ 8, S2 × S2, K3, M === K3/Z2, T 4, 4 4 4 4 T /Z2,T /Z3,T /Z4,T /Z6, 4 4 4 T /(Z2 ⊕ Z2),T /(Z3 ⊕ Z3), or T /(Z2 ⊕ Z4).

76 But we understand some cases better than others!

CP2#kCP2, 0 ≤ k ≤ 8, S2 × S2, K3, M === K3/Z2, T 4, 4 4 4 4 T /Z2,T /Z3,T /Z4,T /Z6, 4 4 4 T /(Z2 ⊕ Z2),T /(Z3 ⊕ Z3), or T /(Z2 ⊕ Z4).

77 But we understand some cases better than others!

CP2#kCP2, 0 ≤ k ≤ 8, S2 × S2, K3, M === K3/Z2, T 4, 4 4 4 4 T /Z2,T /Z3,T /Z4,T /Z6, 4 4 4 T /(Z2 ⊕ Z2),T /(Z3 ⊕ Z3), or T /(Z2 ⊕ Z4).

78 But we understand some cases better than others!

CP2#kCP2, 0 ≤ k ≤ 8, S2 × S2, K3, M === K3/Z2, T 4, 4 4 4 4 T /Z2,T /Z3,T /Z4,T /Z6, 4 4 4 T /(Z2 ⊕ Z2),T /(Z3 ⊕ Z3), or T /(Z2 ⊕ Z4).

Below the line:

79 But we understand some cases better than others!

CP2#kCP2, 0 ≤ k ≤ 8, S2 × S2, K3, M === K3/Z2, T 4, 4 4 4 4 T /Z2,T /Z3,T /Z4,T /Z6, 4 4 4 T /(Z2 ⊕ Z2),T /(Z3 ⊕ Z3), or T /(Z2 ⊕ Z4).

Below the line: Every Einstein metric is Ricci-ﬂat K¨ahler.

80 But we understand some cases better than others!

CP2#kCP2, 0 ≤ k ≤ 8, S2 × S2, K3, M === K3/Z2, T 4, 4 4 4 4 T /Z2,T /Z3,T /Z4,T /Z6, 4 4 4 T /(Z2 ⊕ Z2),T /(Z3 ⊕ Z3), or T /(Z2 ⊕ Z4).

Below the line: Every Einstein metric is Ricci-flat Kähler. Moduli space E (M)= {Einstein metrics}/Diffeomorphisms

81 But we understand some cases better than others!

CP2#kCP2, 0 ≤ k ≤ 8, S2 × S2, K3, M === K3/Z2, T 4, 4 4 4 4 T /Z2,T /Z3,T /Z4,T /Z6, 4 4 4 T /(Z2 ⊕ Z2),T /(Z3 ⊕ Z3), or T /(Z2 ⊕ Z4).

Below the line: Every Einstein metric is Ricci-flat Kähler. + Moduli space E (M) = {Einstein h}/(Diffeos×R )

82 But we understand some cases better than others!

CP2#kCP2, 0 ≤ k ≤ 8, S2 × S2, K3, M === K3/Z2, T 4, 4 4 4 4 T /Z2,T /Z3,T /Z4,T /Z6, 4 4 4 T /(Z2 ⊕ Z2),T /(Z3 ⊕ Z3), or T /(Z2 ⊕ Z4).

Below the line: Every Einstein metric is Ricci-ﬂat K¨ahler. Moduli space E (M) completely understood.

83 But we understand some cases better than others!

CP2#kCP2, 0 ≤ k ≤ 8, S2 × S2, K3, M === K3/Z2, T 4, 4 4 4 4 T /Z2,T /Z3,T /Z4,T /Z6, 4 4 4 T /(Z2 ⊕ Z2),T /(Z3 ⊕ Z3), or T /(Z2 ⊕ Z4).

Below the line: Every Einstein metric is Ricci-ﬂat K¨ahler. Moduli space E (M) connected!

84 But we understand some cases better than others!

CP2#kCP2, 0 ≤ k ≤ 8, S2 × S2, K3, M === K3/Z2, T 4, 4 4 4 4 T /Z2,T /Z3,T /Z4,T /Z6, 4 4 4 T /(Z2 ⊕ Z2),T /(Z3 ⊕ Z3), or T /(Z2 ⊕ Z4).

Below the line: Every Einstein metric is Ricci-ﬂat K¨ahler. Moduli space E (M) connected!

85 Above the line:

CP2#kCP2, 0 ≤ k ≤ 8, S2 × S2, K3, M === K3/Z2, T 4, 4 4 4 4 T /Z2,T /Z3,T /Z4,T /Z6, 4 4 4 T /(Z2 ⊕ Z2),T /(Z3 ⊕ Z3), or T /(Z2 ⊕ Z4).

Below the line: Every Einstein metric is Ricci-ﬂat K¨ahler. Moduli space E (M) connected!

86 Above the line: Know an Einstein metric on each manifold.

CP2#kCP2, 0 ≤ k ≤ 8, S2 × S2, K3, M === K3/Z2, T 4, 4 4 4 4 T /Z2,T /Z3,T /Z4,T /Z6, 4 4 4 T /(Z2 ⊕ Z2),T /(Z3 ⊕ Z3), or T /(Z2 ⊕ Z4).

Below the line: Every Einstein metric is Ricci-ﬂat K¨ahler. Moduli space E (M) connected!

87 Above the line: Moduli space E (M) =6 ∅. Is it connected?

CP2#kCP2, 0 ≤ k ≤ 8, S2 × S2, K3, M === K3/Z2, T 4, 4 4 4 4 T /Z2,T /Z3,T /Z4,T /Z6, 4 4 4 T /(Z2 ⊕ Z2),T /(Z3 ⊕ Z3), or T /(Z2 ⊕ Z4).

Below the line: Every Einstein metric is Ricci-ﬂat K¨ahler. Moduli space E (M) connected!

88 Above the line: Moduli space E (M) =6 ∅. But is it connected?

CP2#kCP2, 0 ≤ k ≤ 8, S2 × S2, K3, M === K3/Z2, T 4, 4 4 4 4 T /Z2,T /Z3,T /Z4,T /Z6, 4 4 4 T /(Z2 ⊕ Z2),T /(Z3 ⊕ Z3), or T /(Z2 ⊕ Z4).

Below the line: Every Einstein metric is Ricci-ﬂat K¨ahler. Moduli space E (M) connected!

89 Objective of this lecture:

90 Objective of this lecture:

Describe modest recent progress on this issue.

91 Objective of this lecture:

Describe modest recent progress on this issue.

For M a Del Pezzo surface, will deﬁne explicit open

92 Objective of this lecture:

Describe modest recent progress on this issue.

For M a Del Pezzo surface, will deﬁne explicit open

U ⊂ {Riemannian metrics on M}

93 Objective of this lecture:

Describe modest recent progress on this issue.

For M a Del Pezzo surface, will deﬁne explicit open

U ⊂ {Riemannian metrics on M} such that

94 Objective of this lecture:

Describe modest recent progress on this issue.

For M a Del Pezzo surface, will deﬁne explicit open

U ⊂ {Riemannian metrics on M} such that

• Every known Einstein metric belongs to U ;

95 Objective of this lecture:

Describe modest recent progress on this issue.

For M a Del Pezzo surface, will deﬁne explicit open

U ⊂ {Riemannian metrics on M} such that

• Every known Einstein metric belongs to U ;

• These form a connected family, mod diﬀeos; and

96 Objective of this lecture:

Describe modest recent progress on this issue.

For M a Del Pezzo surface, will deﬁne explicit open

U ⊂ {Riemannian metrics on M} such that

• Every known Einstein metric belongs to U ;

• These form a connected family, mod diﬀeos; and

• No other Einstein metrics belong to U !

97 Formulation will depend on. . .

98 Special character of dimension 4:

99 Special character of dimension 4:

The Lie group SO(4) is not simple:

100 Special character of dimension 4:

The Lie group SO(4) is not simple:

so(4) ∼= so(3) ⊕ so(3).

101 Special character of dimension 4:

The Lie group SO(4) is not simple:

so(4) ∼= so(3) ⊕ so(3). On oriented (M4,g ),= ⇒

102 Special character of dimension 4:

The Lie group SO(4) is not simple:

so(4) ∼= so(3) ⊕ so(3). On oriented (M4,g ),= ⇒ Λ2 =Λ + ⊕ Λ−

103 Special character of dimension 4:

The Lie group SO(4) is not simple:

so(4) ∼= so(3) ⊕ so(3). On oriented (M4,g ),= ⇒ Λ2 =Λ + ⊕ Λ− whereΛ ± are (±1)-eigenspaces of ? :Λ 2 → Λ2,

104 Special character of dimension 4:

The Lie group SO(4) is not simple:

so(4) ∼= so(3) ⊕ so(3). On oriented (M4,g ),= ⇒ Λ2 =Λ + ⊕ Λ− whereΛ ± are (±1)-eigenspaces of ? :Λ 2 → Λ2, ?2 = 1.

Λ+ self-dual 2-forms. Λ− anti-self-dual 2-forms.

105 Riemann curvature of g R :Λ 2 → Λ2

106 Riemann curvature of g R :Λ 2 → Λ2 splits into 4 irreducible pieces:

107 Riemann curvature of g R :Λ 2 → Λ2 splits into 4 irreducible pieces: Λ+∗ Λ−∗

+ s Λ W + + 12 ˚r

− s Λ ˚r W − + 12

108 Riemann curvature of g R :Λ 2 → Λ2 splits into 4 irreducible pieces: Λ+∗ Λ−∗

+ s Λ W + + 12 ˚r

− s Λ ˚r W − + 12 where s = scalar curvature ˚r = trace-free Ricci curvature W + = self-dual Weyl curvature (conformally invariant) W − = anti-self-dual Weyl curvature

109 Riemann curvature of g R :Λ 2 → Λ2 splits into 4 irreducible pieces: Λ+∗ Λ−∗

+ s Λ W + + 12 ˚r

− s Λ ˚r W − + 12 where s = scalar curvature ˚r = trace-free Ricci curvature W + = self-dual Weyl curvature (conformally invariant) 00 W − = anti-self-dual Weyl curvature

110 4 Smooth compact M has invariants b±(M), deﬁned in terms of intersection pairing

111 4 Smooth compact M has invariants b±(M), deﬁned in terms of intersection pairing

2 2 H (M, R) × H (M, R) −→ R Z ([ϕ] , [ψ]) 7−→ ϕ ∧ ψ M

112 4 Smooth compact M has invariants b±(M), deﬁned in terms of intersection pairing

2 2 H (M, R) × H (M, R) −→ R Z ([ϕ] , [ψ]) 7−→ ϕ ∧ ψ M Diagonalize:

113 4 Smooth compact M has invariants b±(M), deﬁned in terms of intersection pairing

2 2 H (M, R) × H (M, R) −→ R Z ([ϕ] , [ψ]) 7−→ ϕ ∧ ψ M Diagonalize:

  +1  ...     +1     | {z }  .  b+(M)       −1   ..   b−(M) .   −1

114 4 Smooth compact M has invariants b±(M), deﬁned in terms of intersection pairing

2 2 H (M, R) × H (M, R) −→ R Z ([ϕ] , [ψ]) 7−→ ϕ ∧ ψ M Diagonalize:

  +1  ...     +1     | {z }  .  b+(M)       −1   ..   b−(M) .   −1

115 Hodge theory:

2 2 H (M, R) = {ϕ ∈ Γ(Λ ) | dϕ = 0, d ?ϕ = 0}.

116 Hodge theory:

2 2 H (M, R) = {ϕ ∈ Γ(Λ ) | dϕ = 0, d ?ϕ = 0}. Since ? is involution of RHS,= ⇒

2 + − H (M, R) = Hg ⊕H g ,

117 Hodge theory:

2 2 H (M, R) = {ϕ ∈ Γ(Λ ) | dϕ = 0, d ?ϕ = 0}. Since ? is involution of RHS,= ⇒

2 + − H (M, R) = Hg ⊕H g , where

± ± Hg = {ϕ ∈ Γ(Λ ) | dϕ = 0} self-dual & anti-self-dual harmonic forms.

118 Hodge theory:

2 2 H (M, R) = {ϕ ∈ Γ(Λ ) | dϕ = 0, d ?ϕ = 0}. Since ? is involution of RHS,= ⇒

2 + − H (M, R) = Hg ⊕H g , where

± ± Hg = {ϕ ∈ Γ(Λ ) | dϕ = 0} self-dual & anti-self-dual harmonic forms. Then

± b±(M) = dimHg .

119 ...... + ...... H ...... g ...... − ...... H . . . . g ......

2 {a | a · a = 0} ⊂H (M, R)

120 ...... + ...... H ...... g ...... − ...... H ...... g ......

2 {a | a · a = 0} ⊂ H (M, R)

121 ...... + ...... H ...... 0 ...... g ...... − ...... H ...... 0 ...... g ......

2 {a | a · a = 0} ⊂ H (M, R)

122 ...... + ...... H ...... g ...... − ...... H ...... g ......

± b±(M) = dimHg .

123 ...... + ...... H ...... 0 ...... g ...... − ...... H ...... 0 ...... g ......

± b±(M) = dimHg .

124 ...... + ...... H ...... g ...... − ...... H ...... g ......

± b±(M) = dimHg .

125 ...... + ...... H ...... g ...... − ...... H . . . . g ......

± b±(M) = dimHg .

126 More Technical Question. When does a compact complex surface (M4,J ) admit an Einstein metric h which is Hermitian, in the sense that

127 More Technical Question. When does a compact complex surface (M4,J ) admit an Einstein metric h which is Hermitian, in the sense that h(·, ·) = h(J·,J ·)?

128 More Technical Question. When does a compact complex surface (M4,J ) admit an Einstein metric h which is Hermitian, in the sense that h(·, ·) = h(J·,J ·)?

K¨ahler if the 2-form ω = h(J·, ·) is closed: dω = 0. But we do not assume this!

129 More Technical Question. When does a compact complex surface (M4,J ) admit an Einstein metric h which is Hermitian, in the sense that h(·, ·) = h(J·,J ·)?

130 Theorem. A compact complex surface (M4,J ) admits an Einstein metric h which is Hermitian 4 with respect to J ⇐⇒ c1(M ,J ) “has a sign.”

131 Theorem. A compact complex surface (M4,J ) admits an Einstein metric h which is Hermitian 4 with respect to J ⇐⇒ c1(M ,J ) “has a sign.” More precisely, ∃ such h with Einstein constant λ ⇐⇒ there is a K¨ahlerform ω such that 4 c1(M ,J ) = λ[ω].

132 Theorem. A compact complex surface (M4,J ) admits an Einstein metric h which is Hermitian 4 with respect to J ⇐⇒ c1(M ,J ) “has a sign.” More precisely, ∃ such h with Einstein constant λ ⇐⇒ there is a K¨ahlerform ω such that 4 c1(M ,J ) = λ[ω]. Moreover, this metric is unique, up to isometry, if λ =6 0 .

133 Theorem. A compact complex surface (M4,J ) admits an Einstein metric h which is Hermitian 4 with respect to J ⇐⇒ c1(M ,J ) “has a sign.” More precisely, ∃ such h with Einstein constant λ ⇐⇒ there is a K¨ahlerform ω such that 4 c1(M ,J ) = λ[ω]. Moreover, this metric is unique, up to isometry, if λ =6 0 .

Aubin, Yau, Siu, Tian... K¨ahlercase.

134 Theorem. A compact complex surface (M4,J ) admits an Einstein metric h which is Hermitian 4 with respect to J ⇐⇒ c1(M ,J ) “has a sign.” More precisely, ∃ such h with Einstein constant λ ⇐⇒ there is a K¨ahlerform ω such that 4 c1(M ,J ) = λ[ω]. Moreover, this metric is unique, up to isometry, if λ =6 0 .

Aubin, Yau, Siu, Tian... K¨ahlercase.

Chen-L-Weber (2008),L (2012): non-K¨ahlercase.

135 Theorem. A compact complex surface (M4,J ) admits an Einstein metric h which is Hermitian 4 with respect to J ⇐⇒ c1(M ,J ) “has a sign.” More precisely, ∃ such h with Einstein constant λ ⇐⇒ there is a K¨ahlerform ω such that 4 c1(M ,J ) = λ[ω]. Moreover, this metric is unique, up to isometry, if λ =6 0 .

Aubin, Yau, Siu, Tian... K¨ahlercase.

Chen-L-Weber (2008),L (2012): non-K¨ahlercase. Only two metrics arise in non-K¨ahlercase!

136 Theorem. A compact complex surface (M4,J ) admits an Einstein metric h which is Hermitian 4 with respect to J ⇐⇒ c1(M ,J ) “has a sign.” More precisely, ∃ such h with Einstein constant λ ⇐⇒ there is a K¨ahlerform ω such that 4 c1(M ,J ) = λ[ω]. Moreover, this metric is unique, up to isometry, if λ =6 0 .

Key Point. Metrics are actually conformally K¨ahler.

137 Theorem. A compact complex surface (M4,J ) admits an Einstein metric h which is Hermitian 4 with respect to J ⇐⇒ c1(M ,J ) “has a sign.” More precisely, ∃ such h with Einstein constant λ ⇐⇒ there is a K¨ahlerform ω such that 4 c1(M ,J ) = λ[ω]. Moreover, this metric is unique, up to isometry, if λ =6 0 .

Key Point. Metrics are actually conformally K¨ahler.

138 Theorem. A compact complex surface (M4,J ) admits an Einstein metric h which is Hermitian 4 with respect to J ⇐⇒ c1(M ,J ) “has a sign.” More precisely, ∃ such h with Einstein constant λ ⇐⇒ there is a K¨ahlerform ω such that 4 c1(M ,J ) = λ[ω]. Moreover, this metric is unique, up to isometry, if λ =6 0 .

Key Point. Metrics are actually conformally K¨ahler.

h = s−2g

139 Theorem. A compact complex surface (M4,J ) admits an Einstein metric h which is Hermitian 4 with respect to J ⇐⇒ c1(M ,J ) “has a sign.” More precisely, ∃ such h with Einstein constant λ ⇐⇒ there is a K¨ahlerform ω such that 4 c1(M ,J ) = λ[ω]. Moreover, this metric is unique, up to isometry, if λ =6 0 .

Key Point. Metrics are actually conformally K¨ahler.

Strictly 4-dimensional phenomenon!

140 Theorem. A compact complex surface (M4,J ) admits an Einstein metric h which is Hermitian 4 with respect to J ⇐⇒ c1(M ,J ) “has a sign.” More precisely, ∃ such h with Einstein constant λ ⇐⇒ there is a K¨ahlerform ω such that 4 c1(M ,J ) = λ[ω]. Moreover, this metric is unique, up to isometry, if λ =6 0 .

Key Point. Metrics are actually conformally K¨ahler.

Strictly 4-dimensional phenomenon!

“Riemannian Goldberg-Sachs Theorem.”

141 So, which complex surfaces admit

142 So, which complex surfaces admit

Einstein Hermitian metrics with λ > 0?

143 Del Pezzo surfaces:

144 Del Pezzo surfaces:

4 (M ,J ) for which c1 is a K¨ahlerclass [ω].

145 Del Pezzo surfaces:

4 (M ,J ) for which c1 is a K¨ahlerclass [ω]. Shorthand: “c1 > 0.”

146 Del Pezzo surfaces:

4 (M ,J ) for which c1 is a K¨ahlerclass [ω]. Shorthand: “c1 > 0.”

Blow-up of CP2 at k distinct points,0 ≤ k ≤ 8, in general position, or CP1 × CP1.

147 Del Pezzo surfaces:

4 (M ,J ) for which c1 is a K¨ahlerclass [ω]. Shorthand: “c1 > 0.”

Blow-up of CP2 at k distinct points,0 ≤ k ≤ 8, in general position, or CP1 × CP1.

148 Del Pezzo surfaces:

4 (M ,J ) for which c1 is a K¨ahlerclass [ω]. Shorthand: “c1 > 0.”

Blow-up of CP2 at k distinct points,0 ≤ k ≤ 8, in general position, or CP1 × CP1.

149 Del Pezzo surfaces:

4 (M ,J ) for which c1 is a K¨ahlerclass [ω]. Shorthand: “c1 > 0.”

Blow-up of CP2 at k distinct points,0 ≤ k ≤ 8, in general position, or CP1 × CP1.

...... •...... •• ...... 2 ...... CP ......

150 Blowing up:

151 Blowing up: If N is a complex surface, may replace p ∈ N with CP1 to obtain blow-up

...... M ......

...... N . r ......

152 Blowing up: If N is a complex surface, may replace p ∈ N with CP1 to obtain blow-up

...... M ......

...... N . r ......

153 Blowing up: If N is a complex surface, may replace p ∈ N with CP1 to obtain blow-up

...... M ......

...... N . r ......

154 Blowing up: If N is a complex surface, may replace p ∈ N with CP1 to obtain blow-up M ≈ N#CP2 in which added CP1 has normal bundle O(−1).

...... M ......

...... N . r ......

155 156 Blowing up: If N is a complex surface, may replace p ∈ N with CP1 to obtain blow-up M ≈ N#CP2 in which added CP1 has normal bundle O(−1).

...... M ......

...... N . r ......

157 158 Blowing up: If N is a complex surface, may replace p ∈ N with CP1 to obtain blow-up M ≈ N#CP2 in which added CP1 has normal bundle O(−1).

...... M ......

...... N . r ......

159 160 Blowing up: If N is a complex surface, may replace p ∈ N with CP1 to obtain blow-up M ≈ N#CP2 in which added CP1 has normal bundle O(−1).

...... M ......

...... N . r ......

161 Del Pezzo surfaces:

4 (M ,J ) for which c1 is a K¨ahlerclass [ω]. Shorthand: “c1 > 0.”

Blow-up of CP2 at k distinct points,0 ≤ k ≤ 8, in general position, or CP1 × CP1.

...... •...... •• ...... 2 ...... CP ......

162 Del Pezzo surfaces:

4 (M ,J ) for which c1 is a K¨ahlerclass [ω]. Shorthand: “c1 > 0.”

Blow-up of CP2 at k distinct points,0 ≤ k ≤ 8, in general position, or CP1 × CP1.

...... •...... •• ...... 2 ...... CP ......

No3 on a line, no 5 on conic, no 8 on nodal cubic.

163 Del Pezzo surfaces:

4 (M ,J ) for which c1 is a K¨ahlerclass [ω]. Shorthand: “c1 > 0.”

Blow-up of CP2 at k distinct points,0 ≤ k ≤ 8, in general position, or CP1 × CP1.

...... •• ...... •...... •• ...... 2 ...... CP ...... • ......

No3 on a line, no6 on conic, no 8 on nodal cubic.

164 Del Pezzo surfaces:

4 (M ,J ) for which c1 is a K¨ahlerclass [ω]. Shorthand: “c1 > 0.”

Blow-up of CP2 at k distinct points,0 ≤ k ≤ 8, in general position, or CP1 × CP1.

...... •...... •...... • ...... • ...... • ...... • ...... • ...... • ...... 2 ...... CP ......

No3 on a line, no6 on conic, no8 on nodal cubic.

165 Del Pezzo surfaces:

4 (M ,J ) for which c1 is a K¨ahlerclass [ω]. Shorthand: “c1 > 0.”

Blow-up of CP2 at k distinct points,0 ≤ k ≤ 8, in general position, or CP1 × CP1.

166 Del Pezzo surfaces:

4 (M ,J ) for which c1 is a K¨ahlerclass [ω]. Shorthand: “c1 > 0.”

Blow-up of CP2 at k distinct points,0 ≤ k ≤ 8, in general position, or CP1 × CP1.

For each topological type:

167 Del Pezzo surfaces:

4 (M ,J ) for which c1 is a K¨ahlerclass [ω]. Shorthand: “c1 > 0.”

Blow-up of CP2 at k distinct points,0 ≤ k ≤ 8, in general position, or CP1 × CP1.

For each topological type:

Moduli space of such (M4,J ) is connected.

168 Del Pezzo surfaces:

4 (M ,J ) for which c1 is a K¨ahlerclass [ω]. Shorthand: “c1 > 0.”

Blow-up of CP2 at k distinct points,0 ≤ k ≤ 8, in general position, or CP1 × CP1.

For each topological type:

Moduli space of such (M4,J ) is connected.

Just a point if b2(M) ≤ 5.

169 Every del Pezzo surface has b+ = 1. ⇐⇒

170 Every del Pezzo surface has b+ = 1. ⇐⇒

Up to scale, ∀ h, ∃! self-dual harmonic 2-form ω:

171 Every del Pezzo surface has b+ = 1. ⇐⇒

Up to scale, ∀ h, ∃! self-dual harmonic 2-form ω:

dω = 0,?ω = ω.

172 Every del Pezzo surface has b+ = 1. ⇐⇒

Up to scale, ∀ h, ∃! self-dual harmonic 2-form ω:

dω = 0,?ω = ω.

173 Every del Pezzo surface has b+ = 1. ⇐⇒

Up to scale, ∀ h, ∃! self-dual harmonic 2-form ω:

dω = 0,?ω = ω.

Such a form deﬁnes a symplectic structure, except at points where ω = 0.

174 Every del Pezzo surface has b+ = 1. ⇐⇒

Up to scale, ∀ h, ∃! self-dual harmonic 2-form ω:

dω = 0,?ω = ω.

Such a form deﬁnes a symplectic structure, except at points where ω = 0.

175 Every del Pezzo surface has b+ = 1. ⇐⇒

Up to scale, ∀ h, ∃! self-dual harmonic 2-form ω:

dω = 0,?ω = ω.

Such a form deﬁnes a symplectic structure, except at points where ω = 0.

Deﬁnition. Let M be smooth 4-manifold with b+(M) =1 . We will say that a Riemannian metric h on M is of symplectic type if associated SD harmonic ω is nowhere zero.

176 Every del Pezzo surface has b+ = 1. ⇐⇒

Up to scale, ∀ h, ∃! self-dual harmonic 2-form ω:

dω = 0,?ω = ω.

Such a form deﬁnes a symplectic structure, except at points where ω = 0.

Deﬁnition. Let M be smooth 4-manifold with b+(M) =1 . We will say that a Riemannian metric h on M is of symplectic type if associated SD harmonic ω is nowhere zero.

177 Every del Pezzo surface has b+ = 1. ⇐⇒

Up to scale, ∀ h, ∃! self-dual harmonic 2-form ω:

dω = 0,?ω = ω.

Such a form deﬁnes a symplectic structure, except at points where ω = 0.

Deﬁnition. Let M be smooth 4-manifold with b+(M) =1 . We will say that a Riemannian metric h on M is of symplectic type if associated SD harmonic ω is nowhere zero.

178 Every del Pezzo surface has b+ = 1. ⇐⇒

Up to scale, ∀ h, ∃! self-dual harmonic 2-form ω:

dω = 0,?ω = ω.

Such a form deﬁnes a symplectic structure, except at points where ω = 0.

Deﬁnition. Let M be smooth 4-manifold with b+(M) =1 . We will say that a Riemannian metric h on M is of symplectic type if associated SD harmonic ω is nowhere zero.

179 Every del Pezzo surface has b+ = 1. ⇐⇒

Up to scale, ∀ h, ∃! self-dual harmonic 2-form ω:

dω = 0,?ω = ω.

Such a form deﬁnes a symplectic structure, except at points where ω = 0.

Deﬁnition. Let M be smooth 4-manifold with b+(M) =1 . We will say that a Riemannian metric h on M is of symplectic type if associated SD harmonic ω is nowhere zero.

180 Every del Pezzo surface has b+ = 1. ⇐⇒

Up to scale, ∀ h, ∃! self-dual harmonic 2-form ω:

dω = 0,?ω = ω.

Such a form deﬁnes a symplectic structure, except at points where ω = 0.

Deﬁnition. Let M be smooth 4-manifold with b+(M) =1 . We will say that a Riemannian metric h on M is of symplectic type if associated SD harmonic ω is nowhere zero.

• open condition;

181 Every del Pezzo surface has b+ = 1. ⇐⇒

Up to scale, ∀ h, ∃! self-dual harmonic 2-form ω:

dω = 0,?ω = ω.

Such a form deﬁnes a symplectic structure, except at points where ω = 0.

Deﬁnition. Let M be smooth 4-manifold with b+(M) =1 . We will say that a Riemannian metric h on M is of symplectic type if associated SD harmonic ω is nowhere zero.

• open condition; • conformally invariant; and

182 Every del Pezzo surface has b+ = 1. ⇐⇒

Up to scale, ∀ h, ∃! self-dual harmonic 2-form ω:

dω = 0,?ω = ω.

Such a form deﬁnes a symplectic structure, except at points where ω = 0.

Deﬁnition. Let M be smooth 4-manifold with b+(M) =1 . We will say that a Riemannian metric h on M is of symplectic type if associated SD harmonic ω is nowhere zero.

• open condition; • conformally invariant; and • holds in K¨ahler case.

183 Deﬁnition. Let M be smooth compact 4-manifold with b+(M) =1 . We will say that a Riemannian metric h is of positive symplectic type if Weyl curvature and SD harmonic form ω satisfy

184 Deﬁnition. Let M be smooth compact 4-manifold with b+(M) =1 . We will say that a Riemannian metric h is of positive symplectic type if Weyl curvature and SD harmonic form ω satisfy

185 Deﬁnition. Let M be smooth compact 4-manifold with b+(M) =1 . We will say that a Riemannian metric h is of positive symplectic type if Weyl curvature and SD harmonic form ω satisfy

186 Deﬁnition. Let M be smooth compact 4-manifold with b+(M) =1 . We will say that a Riemannian metric h is of positive symplectic type if Weyl curvature and SD harmonic form ω satisfy

187 Deﬁnition. Let M be smooth compact 4-manifold with b+(M) =1 . We will say that a Riemannian metric h is of positive symplectic type if Weyl curvature and SD harmonic form ω satisfy

W+(ω,ω ) > 0

188 Deﬁnition. Let M be smooth compact 4-manifold with b+(M) =1 . We will say that a Riemannian metric h is of positive symplectic type if Weyl curvature and SD harmonic form ω satisfy