In 1961, Prof. Samuel I. Goldberg published his prescient conjecture that "Every compact, symplectic, Einstein manifold is Kähler." What do all these words mean ?
Kähler manifolds ( a class of complex manifolds which includes algebraic manifolds) possess a differential 2-form, w, which is non-degenerate ( i.e., wÙw is never 0 ), and closed ( i.e., dw = 0 ). Kähler manifolds are very special. Y-T. Siu recently showed that a (real) 4 - dimensional, locally complex manifold is Kähler if and only if all of its odd-degree Betti numbers, b2n+1, are even. Real manifolds which possess such a non-degenerate, closed differential 2-form, w, are called symplectic manifolds. They are necessarily even-dimensional. It used to be believed that closed, symplectic manifolds were always complex. However, in 1976, W. Thurston constructed a closed, symplectic 4-manifold which he showed could not be Kähler because its first Betti number, b1, is 3. In 1983, I generalized Thurston’s example to an infinite family of closed, non-Kähler, symplectic manifolds. Unexpectedly, in 1995, R. Gompf, using a procedure called "logarithmic transformation," constructed a plethora of closed, non-Kähler, symplectic 4-manifolds of almost any topology imaginable.
An n-dimensional manifold, Mn, is said to be Einstein if the its Ricci curvature 2-tensor, Ric(X,Y), is a constant multiple, (s/n).g(X,Y), of its Riemannian metric tensor, g. The factor, s, is called the scalar curvature of (M, g). If n > 3, then the scalar curvature, s, is a constant on an Einstein manifold. Examples of Einstein manifolds include n-spheres and complex projective spaces. The Einstein condition is fundamental to general relativity.
The classification by genus of 2-manifolds
is due to K. Kodiara. The dimension 3 and 4 cases are significantly harder.
At this moment (1997), a program developed by Thurston seems well on its way
to classifying 3-manifolds. In dimension 4, we have:
(1) In 1982, M. Freedman classified simply connected (no holes), closed topological
4-manifolds by relating them to unimodular, symmetric, bilinear forms.
For each such form, Q, there is either one or two topological 4-manifolds,
up to homeomorphism. If there are two, then only one has a differentiable structure.
(2) In 1983, S. K. Donaldson announced his instanton invariants on differentiable
4-manifolds based on Yang-Mills gauge field theory. In particular, he showed
that some unimodular symmetric bilinear forms (e.g., |E8ÅE8| )
cannot be associated to a differentiable 4-manifold. This leads to extremely
interesting "fake" calculuses on Euclidean 4-space. His theory also
showed that we have not completely distinguished all closed, differentiable
4-manifolds. The Donaldson invariants are extremely difficult
to calculate.
The connected sum, M1 # M2, of two n-manifolds is constructed by cutting a hole in each Mi , and the boundaries of the holes are joined by an n-tube( an (n -1) - dimensional manifold, Y, Cartesian producted with the interval (0,1) ) As recently as 1988, it was believed that every differentiable 4-manifold would be diffeomorphic to the connected sum of algebraic manifolds (with possibly the reversed orientation ) along with, possibly, the 4-sphere, S4, and/or S1´S3. This was very hopeful because all (real 4-dimensional) locally complex 2-manifolds, including the algebraic surfaces, were somewhat classified by K. Kodaira in the 1960’s using the Kodaira dimension, k. After many counterexamples to the connected sum of algebraic manifolds conjecture were constructed in the 1990’s, the current working hypothesis is ( was ! ) the
CONJECTURE: Every simply connected, closed, differentiable 4-manifold M can be decomposed as a connected sum M = M1 # . . . # Mn , where the Mi are symplectic 4-manifolds with, possibly, the reversed orientation, or are S4.
Sadly, this conjecture is false. Prof. Z. Szabo, using Gompf’s logarithmic transformations, found many counterexamples to this conjecture last year. Recently, Ron Fintushel and Ron Stern showed that there are as many examples of compact symplectic 4-manifolds as there are Alexander polynomials of knots; thereby relating the classification question to the unsolvable wrod problem. In late 1994, new hope of understanding all this arose in the:
SEIBERG-WITTEN INVARIANTS
The Seiberg-Witten equations are definable on any oriented, differentiable Riemannian 4-manifold, (M4, g). One chooses a Spinc structure P' (which is always possible) on the orthogonal frame bundle, P over M, and constructs the determinant line bundle, K , of P' . From K , one constructs the positive and negative spinors. You then choose a connection, A, on K with curvature FA and construct the associated Dirac operator, ¶A. The Seiberg-Witten equations are:
1) ¶A(FA+ ) = 0 and
2) F+A = yÄy* - ½|y|2Id
A solution is then a pair (A,y) where y is a positive spinor on M. The Seiberg-Witten equations are very easy to solve. The Seiberg-Witten invariant, SW(K), is the number of solutions, mod2, counted with signs. It is easy to see that a symplectic manifold admitting a metric of positive scalar curvature has SW(K) = 0. Using this theory, the past three years have seen the solution of the Thom Conjecture that an immersed 2-sphere in P2(C) minimizes the genus in a given 2-homology class and the proof that the Kodaira dimension, k(M4), is a differentiable invariant of the manifold. C. Taubes has shown that the Seiberg - Witten invariant is equal to the Gromov invariant defined in terms of embedded pseudo-holomorphic curves and that all symplectic manifolds with b2+ > 2 must have SW(K) = ± 1. A.-K. Liu has classified all rational and ruled complex surfaces. C. LeBrun has shown amazing results on symplectic, Einstein 4-manifolds. We fully expect the list of new, major results to go on and on.
We now see the importance of the Goldberg Conjecture ( "Every compact, symplectic, Einstein manifold is Kähler." ) in clarifying the Classification Conjecture for closed, differentiable 4-manifolds. This surprisingly difficult assertion has steadfastly resisted resolution. In 1987, K. Sekigawa used a very long, messy calculation to prove that Goldberg’s Conjecture is TRUE in all dimensions when the scalar curvature is non-negative ( s > 0). Independently, in 1984, I proved the same assertion in dimension four.
One may define the *-Ricci curvature tensor, Ric*, incorporating the almost Hermitian structure, J, on an almost Hermitian manifold in a manner analogous to that of the Ricci curvature tensor. An almost Hermitian manifold (M2m,g,J) which satisfies Ric* = (s*/2m)g for the possibly varying *-scalar curvature functional, s*, is called a (weakly) *-Einstein manifold. It is easy to verify that the norm squared, ||ÑJ||2, of the covariant differential of the almost complex structure, J, on an almost Kähler manifold is equal to 2(s* – s). Moreover, the almost Kähler manifold (M2m,g,J) is Kähler ifand only if ||ÑJ||2 = 0. We must pay particular attention to avoid assuming that s* is a global constant. A recent result of J. Armstrong implies that there is at least one point on a compact, four-dimensional Einstein almost Kähler manifold at which s = s*. Therefore, if s* is assumed to be constant on the Einstein almost Kähler 4-manifold, then s = s* everywhere and the manifold must be Kähler. Since 1987, I have been trying to prove Goldberg’s Conjecture in the negative scalar curvature case ( s < 0 ) using the Seiberg-Witten invariants. The analysis is very delicate, and involves a topological invariant, c1(M), called the first Chern class, which can be related to the total scalar curvature, i.e., the integral of s + s* over the compact manifold. I use the easily established fact that a closed, Einstein, symplectic manifold is Kähler if and only if its constant scalar curvature, s, is equal to its possibly varying *-scalar curvature, s*. (It is always true that s < s* ). In 1999, T. Oguro, Sekigawa and Yamada showed that a 4-dimensional compact almost Kähler manfold which is both Einstein and *-Einstein must be Kähler. However, it is very difficult to prove that a specific manifold is *-Einstein. Tedi Draghici, at Michigan State Univ., has also made great progress toward the Goldberg Conjecture and has proved many related results. I feel that the Goldberg Conjecture is TRUE in dimension four and is FALSE in higher dimensions. The recent work of Catanese and LeBrun, who constructed an 8-dimensional manifold which has two Kähler-Einstein structures of opposite sign, gives hope to the prospect of constructing a higher-dimensional counterexample to the Goldberg Conjecture.
I am also presently trying to prove a symplectic analogue of Siu’s Theorem:
CONJECTURE(Watson):
The odd-degree Betti numbers of a
compact, Einstein, symplectic 4-manifold, ( M4,
w ), are even.
The verification of this conjecture would imply that the dimension, b2+, of the positive eigenspace of the intersection form, Q, is odd, making an analysis of the 4-dimensional Goldberg Conjecture using Seiberg-Witten invariants much easier.
In 1976, I defined almost Hermitian submersions to be Riemannian submersions between almost Hermitian manifolds which commute with the two involved almost Hermitian structures. This is a natural mathematical object to study if one is seeking Riemannian submersions with minimally embedded fibres as I was. In my doctoral thesis and its sequel, I showed that a surjective manifold map commutes with the Laplacian, D, on 0-forms (resp. commutes with the codifferential, d, on 1-forms) if and only if the map is a Riemannian submersion with minimal fibres.
In 1998, I successfully proved that the four-dimensional total space of an almost Kähler submersion satisfies the Goldberg Conjecture without regard to the sign of the scalar curvature. This was accomplished by proving that the Einstein condition implies superminimal fibres which then implies that the almost complex structure is integrable. Compactness is not needed. This last remark is important, because John Armstrong has shown the possible existence of local counterexamples to the Goldberg Conjecture, while two Polish mathematicians produced a non-compact, Ricci-flat almost Kähler, non-Kähler manifold in February, 1998. Recently, Tedi Draghici, V. Apostolov and D. Kotschick proved that a compact, 4-dimensional almost Kähler manifold which satisfies the second curvature condition of Alfred Gray must be Kähler. Using this fact, I have recently extended my work on Almost Kähler submersions to the 6-dimensional total space case.
The Goldberg Conjecture is True
for the Four-dimensional Total Space of an Almost Kähler Submersion (to
appear), Journal of Geometry, 2000.
Superminimal Fibres in an Almost Hermitian Submersion, Boll. Unione
Mat. Ital., 8 (2000), 159-172.
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