Calabi conjecture

In mathematics, the Calabi conjecture was a conjecture about the existence of certain "nice" Riemannian metrics on certain complex manifolds, made by Eugenio Calabi (1954, 1957) and proved by Shing-Tung Yau (1977, 1978). Yau received the Fields Medal in 1982 in part for this proof.

The Calabi conjecture states that a compact Kähler manifold has a unique Kähler metric in the same class whose Ricci form is any given 2-form representing the first Chern class. In particular if the first Chern class vanishes there is a unique Kähler metric in the same class with vanishing Ricci curvature; these are called Calabi–Yau manifolds.

More formally, the Calabi conjecture states:

If M is a compact Kähler manifold with Kähler metric ${\displaystyle g}$ and Kähler form ${\displaystyle \omega }$, and R is any (1,1)-form representing the manifold's first Chern class, then there exists a unique Kähler metric ${\displaystyle {\tilde {g}}}$ on M with Kähler form ${\displaystyle {\tilde {\omega }}}$ such that ${\displaystyle \omega }$ and ${\displaystyle {\tilde {\omega }}}$ represent the same class in cohomology ${\displaystyle H^{2}(M,\mathbb {R} )}$ and the Ricci form of ${\displaystyle {\tilde {\omega }}}$ is R.

The Calabi conjecture is closely related to the question of which Kähler manifolds have Kähler–Einstein metrics.

Kähler–Einstein metrics

A conjecture closely related to the Calabi conjecture states that if a compact Kähler variety has a negative, zero, or positive first Chern class then it has a Kähler–Einstein metric in the same class as its Kähler metric, unique up to rescaling. This was proved for negative first Chern classes independently by Thierry Aubin and Shing-Tung Yau in 1976. When the Chern class is zero it was proved by Yau as an easy consequence of the Calabi conjecture. These results were never explicitly conjectured by Calabi, but would have followed from results that he announced in his 1954 talk at the International Congress of Mathematicians.

When the first Chern class is positive, the above conjecture is actually false as a consequence of a result of Yozo Matsushima, which shows that the complex automorphism group of a Kähler–Einstein manifold of positive scalar curvature is necessarily reductive. For example, the complex projective plane blown up at 2 points has no Kähler–Einstein metric and so is a counterexample. Another problem arising from complex automorphisms is that they can lead to a lack of uniqueness for the Kähler–Einstein metric, even when it exists. However, complex automorphisms are not the only difficulty that arises in the positive case. Indeed, it was conjectured by Yau et al that when the first Chern class is positive, a Kähler manifold admits a Kähler–Einstein metric if and only if it is K-stable. A proof of this conjecture was published by Xiuxiong Chen, Simon Donaldson and Song Sun in January 2015,[1][2][3] and Tian gave a proof electronically published on September 16, 2015.[4][5]

On the other hand, in the special case of complex dimension two, a compact complex surface with positive first Chern class does admit a Kähler–Einstein metric if and only if its automorphism group if reductive. This important result is often attributed to Gang Tian. Since Tian’s proof, there have been some simplifications and refinements of arguments involved; cf. the paper by Odaka, Spotti, and Sun cited below. The complex surfaces that admit such Kähler–Einstein metrics are therefore exactly the complex projective plane, the product of two copies of a projective line, and blowups of the projective plane in 3 to 8 points in general position.

Outline of the proof of the Calabi conjecture

Calabi transformed the Calabi conjecture into a non-linear partial differential equation of complex Monge–Ampère type, and showed that this equation has at most one solution, thus establishing the uniqueness of the required Kähler metric.

Yau proved the Calabi conjecture by constructing a solution of this equation using the continuity method. This involves first solving an easier equation, and then showing that a solution to the easy equation can be continuously deformed to a solution of the hard equation. The hardest part of Yau's solution is proving certain a priori estimates for the derivatives of solutions.

Transformation of the Calabi conjecture to a differential equation

Suppose that M is a complex compact manifold with a Kähler form ω. Any other Kähler form in the same class is of the form

${\displaystyle \omega +dd'\phi }$

for some smooth function φ on M, unique up to addition of a constant. The Calabi conjecture is therefore equivalent to the following problem:

Let ${\displaystyle F=e^{f}}$ be a positive smooth function on M with average value 1. Then there is a smooth real function φ with

${\displaystyle (\omega +dd'\phi )^{m}=e^{f}\omega ^{m}}$

and φ is unique up to addition of a constant.

This is an equation of complex Monge–Ampère type for a single function φ. It is a particularly hard partial differential equation to solve, as it is non-linear in the terms of highest order. It is easy to solve it when f=0, as φ=0 is a solution. The idea of the continuity method is to show that it can be solved for all f by showing that the set of f for which it can be solved is both open and closed. Since the set of f for which it can be solved is non-empty, and the set of all f is connected, this shows that it can be solved for all f.

The map from smooth functions to smooth functions taking φ to F defined by

${\displaystyle F=(\omega +dd'\phi )^{m}/\omega ^{m}}$

is neither injective nor surjective. It is not injective because adding a constant to φ does not change F, and it is not surjective because F must be positive and have average value 1. So we consider the map restricted to functions φ that are normalized to have average value 0, and ask if this map is an isomorphism onto the set of positive ${\displaystyle F=e^{f}}$ with average value 1. Calabi and Yau proved that it is indeed an isomorphism. This is done in several steps, described below.

Uniqueness of the solution

Proving that the solution is unique involves showing that if

${\displaystyle (\omega +dd'\varphi _{1})^{m}=(\omega +dd'\varphi _{2})^{m}}$

then φ₁ and φ₂ differ by a constant (so must be the same if they are both normalized to have average value 0). Calabi proved this by showing that the average value of

${\displaystyle |d(\varphi _{1}-\varphi _{2})|^{2}}$

is given by an expression that is at most 0. As it is obviously at least 0, it must be 0, so

${\displaystyle d(\varphi _{1}-\varphi _{2})=0}$

which in turn forces φ₁ and φ₂ to differ by a constant.

The set of F is open

Proving that the set of possible F is open (in the set of smooth functions with average value 1) involves showing that if it is possible to solve the equation for some F, then it is possible to solve it for all sufficiently close F. Calabi proved this by using the implicit function theorem for Banach spaces: in order to apply this, the main step is to show that the linearization of the differential operator above is invertible.

The set of F is closed

This is the hardest part of the proof, and was the part done by Yau. Suppose that F is in the closure of the image of possible functions φ. This means that there is a sequence of functions φ₁, φ₂, ... such that the corresponding functions F₁, F₂,... converge to F, and the problem is to show that some subsequence of the φs converges to a solution φ. In order to do this, Yau finds some a priori bounds for the functions φ_i and their higher derivatives in terms of the higher derivatives of log(f_i). Finding these bounds requires a long sequence of hard estimates, each improving slightly on the previous estimate. The bounds Yau gets are enough to show that the functions φ_i all lie in a compact subset of a suitable Banach space of functions, so it is possible to find a convergent subsequence. This subsequence converges to a function φ with image F, which shows that the set of possible images F is closed.

References

1. Chen, Xiuxiong; Donaldson, Simon; Sun, Song Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities. J. Amer. Math. Soc. 28 (January 2015), no. 1, 183–197.
2. Chen, Xiuxiong; Donaldson, Simon; Sun, Song Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than 2π . J. Amer. Math. Soc. 28 (January 2015), no. 1, 199–234.
3. Chen, Xiuxiong; Donaldson, Simon; Sun, Song Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches 2π and completion of the main proof. J. Amer. Math. Soc. 28 (January 2015), no. 1, 235–278.
4. Gang Tian: K-Stability and Kähler-Einstein Metrics. Communications on Pure and Applied Mathematics, Volume 68, Issue 7, pages 1085–1156, July 2015 http://onlinelibrary.wiley.com/doi/10.1002/cpa.21578/abstract
5. Gang Tian: Corrigendum: K-stability and Kähler-Einstein metrics. Communications on Pure and Applied Mathematics, Volume 68, Issue 11, pages 2082–2083, September 2015 http://onlinelibrary.wiley.com/doi/10.1002/cpa.21612/full

• Thierry Aubin, Nonlinear Analysis on Manifolds, Monge–Ampère Equations ISBN 0-387-90704-1 This gives a proof of the Calabi conjecture and of Aubin's results on Kähler–Einstein metrics.
• Bourguignon, Jean-Pierre (1979), "Premières formes de Chern des variétés kählériennes compactes [d'après E. Calabi, T. Aubin et S. T. Yau]", Séminaire Bourbaki, 30e année (1977/78), Lecture Notes in Math., 710, Berlin, New York: Springer-Verlag, pp. 1–21, doi:10.1007/BFb0069970, ISBN 978-3-540-09243-8, MR 0554212 This gives a survey of the work of Aubin and Yau.
• Calabi, Eugenio (1954), "The space of Kähler metrics", Proc. Internat. Congress Math. Amsterdam, 2, pp. 206–207, archived from the original (PDF) on 2011-07-17, retrieved 2011-01-30
• Eugenio Calabi, The Space of Kähler Metrics, Proceedings of the International Congress of Mathematicians 1954, Volume II, pp. 206-7, E.P. Noordhoff, Groningen, 1956.
• Calabi, Eugenio (1957), "On Kähler manifolds with vanishing canonical class", in Fox, Ralph H.; Spencer, D. C.; Tucker, A. W. (eds.), Algebraic geometry and topology. A symposium in honor of S. Lefschetz, Princeton Mathematical Series, 12, Princeton University Press, pp. 78–89, MR 0085583
• Chen, Xiuxiong; Donaldson, Simon; Sun, Song (2013). "Kähler–Einstein Metrics and Stability". International Mathematics Research Notices. 2014 (8): 2119–2125. arXiv:1210.7494. doi:10.1093/imrn/rns279. MR 3194014.
• Dominic D. Joyce Compact Manifolds with Special Holonomy (Oxford Mathematical Monographs) ISBN 0-19-850601-5 This gives a simplified proof of the Calabi conjecture.
• Matsushima, Yozô (1957). "Sur la structure du groupe d'homéomorphismes analytiques d'une certaine variété kählérienne". Nagoya Mathematical Journal. 11: 145–150. doi:10.1017/s0027763000002026. MR 0094478.
• Y. Odaka, C. Spotti, and S. Sun, Compact moduli spaces of del Pezzo surfaces and Kähler-Einstein metrics. J. Differential Geom. 102 (2016), no. 1, 127–172.
• Tian, Gang (1990). "On Calabi's conjecture for complex surfaces with positive first Chern class". Inventiones Mathematicae. 101 (1): 101–172. Bibcode:1990InMat.101..101T. doi:10.1007/bf01231499. MR 1055713.
• Yau, Shing Tung (1977), "Calabi's conjecture and some new results in algebraic geometry", Proceedings of the National Academy of Sciences of the United States of America, 74 (5): 1798–1799, Bibcode:1977PNAS...74.1798Y, doi:10.1073/pnas.74.5.1798, ISSN 0027-8424, MR 0451180, PMC 431004, PMID 16592394
• Yau, Shing Tung (1978), "On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation. I", Communications on Pure and Applied Mathematics, 31 (3): 339–411, doi:10.1002/cpa.3160310304, MR 0480350

External links

• Yau, Shing Tung (2009), "Calabi-Yau manifold", Scholarpedia, 4 (8): 6524, Bibcode:2009SchpJ...4.6524Y, doi:10.4249/scholarpedia.6524