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Calabi-Yau Manifolds at the Interface of Math and Physics by Shiu-Yuen Cheng*, Lizhen Ji†, Liping Wang‡, and Hao Xu§

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Calabi-Yau Manifolds at the Interface of Math and Physics by Shiu-Yuen Cheng*, Lizhen Ji†, Liping Wang‡, and Hao Xu§ Calabi-Yau Manifolds at the Interface of Math and Physics by Shiu-Yuen Cheng*, Lizhen Ji†, Liping Wang‡, and Hao Xu§ There were once a young man and an older man. could be built with them? Maybe there was not Their mathematics paths crossed and their names be- enough structure. How about adding complex struc- came merged into the long-lasting notion of Calabi- tures? Would they admit canonical metrics? Or per- Yau manifolds. haps there was still not enough structure. How about How did this happen? Why did Calabi make his Kähler manifolds—structures between complex ge- famous conjecture? How did Yau prove it? Why are ometry and algebraic geometry? This was what Calabi Calabi-Yau manifolds so important? And how are asked 60 years ago, and what Yau answered 24 years they used? All these and other questions take time to later. He cracked a geometric problem by unraveling answer and real experts to explain. They can probably hard nonlinear differential equations. be best understood through the interactions between The Calabi-Yau metric has been an invaluable tool math and physics, and that’s where their true value in differential and algebraic geometry. Here is an in- lies. Although these questions have a long history, complete list of long-standing conjectures and prob- the future will undoubtedly bring more questions and lems solved by using the Calabi-Yau metric and Yau’s surprises regarding Calabi-Yau manifolds. technique of the a priori estimate. Indeed, one must go back to more than 150 years 1. The Severi conjecture, that every complex sur- ago, when Riemann introduced the concepts of Rie- face that is homotopic to the complex projective mann surfaces and Riemannian metrics. His map- 2 2 ping theorem motivated people to endow each Rie- plane CP is biholomorphic to CP . mann surface with a canonical Riemannian metric, 2. Bogomolov-Miyaoka-Yau Chern number inequal- and from there one of the greatest theorems in the ities and their associated uniformization theo- history of mathematics, the uniformization theorem rems in higher dimensions. n for Riemann surfaces and surfaces with Riemannian 3. Uniqueness of Kähler complex structure on CP . metrics, took shape. 4. Torelli theorem for K3 surfaces. What happened to higher dimensional Rieman- 5. Every K3 surface is Kähler. nian manifolds? What kind of geometric structures 6. The existence of Hermitian-Yang-Mills connec- tions in stable vector bundles (Donaldson- * Mathematical Sciences Center, Tsinghua University, Beijing, Uhlenbeck-Yau theorem). China 7. The existence of complete Kähler-Einstein met- Email: [email protected] † Department of Mathematics, University of Michigan, Ann rics on pseudoconvex domains. Arbor, MI, U.S.A. 8. Calabi-Yau manifolds are used in the context of Email: [email protected] string theory to make models of particle physics ‡ Higher Education Press, Beijing, China unified with quantum gravity. Email: [email protected] 9. Construction of non-flat compact simply- § Department of Mathematics, University of Pittsburg, PA, U.S.A. connected Kähler manifolds whose Ricci curva- Email: [email protected] ture is identically zero. 90 NOTICES OF THE ICCM VOLUME 2,NUMBER 2 10. Any compact Kähler manifold with non-negative SP(n) by reducing the construction of them to alge- first Chern class is covered holomorphically by braic conditions. the product of Cn and a compact simply con- 4. Algebraic geometry: nected Kähler manifold. (a) The second Chern class of such manifolds is nu- 11. The vanishing of c and c [ [ ]n−2 for a Kähler 1 2 w merically positive in the sense that the cup prod- manifold implies that the manifold admits a met- uct of it with the top number of products of any ric with zero curvature and is covered by the flat positive (1,1) class is positive unless it is covered torus. by the torus. 12. As conjectured by Yau, the existence of Kähler- (b) Calabi-Yau manifolds are stable and their tangent Einstein metrics on Fano manifolds has intimate bundles are also stable with respect to any polar- connections to the stability of manifolds. Impor- ization. tant progress has been made recently. (c) Counting the number of algebraic curves in Such beautiful results are appreciated not only by Calabi-Yau manifolds has become quite success- mathematicians but by physicists as well. For, accord- ful since the introduction of the concept of mir- ing to string theory, Calabi-Yau manifolds provide the ror symmetry, which has opened up many new hidden inner spaces of our universe. directions in mathematics. It is true that 60 is a magic number, but 65 is also (d) It gives a structure theorem for Compact Kähler a milestone in one’s life. Here is a poem by Profes- manifolds with non-negative first Chern class. sor Yau reflecting his experience with geometry and 5. Nonlinear partial differential equations: The physics. proof of Calabi conjecture gave the first general method to solve the nonlinear Complex Ampere equa- Forty years of geometry, tion on a complex manifold. The equation solves a equation creates structure, very important inverse problem: given any volume el- analysis blends perfection and beauty. ement on any compact Kähler manifold, any Kähler metric can be deformed to a new Kähler metric whose Thirty years of physics, volume form is the given one up to a constant. The strings and branes drifting through methods of estimates have been fundamental to all hyperspace, constructions of Kähler-Einstein metrics, including breaking supersymmetry as they go. Kähler metrics with prescribed singularities. The sin- Quantum connects to gravity, gular metrics were used by Donaldson et al. to solve and nature is whole again. a famous conjecture of Yau on existence of Kähler- S.-T. Yau Einstein metric on Stable Fano manifolds. The Power of Calabi-Yau Manifolds Perspectives of Experts When Yau turned 65 this year, we asked experts From the Uniformization (Klein, Poincare, Koebe, all over the world, in both math and physics, to com- Brouwer) to Riemman-Hilbert Correspondence, ment on the applications and impacts of Calabi-Yau Picard-Fuchs Equations, and Kähler-Einstein manifolds and on the work of Yau. What followed is Metrics (Calabi-Yau) the mathematical garden of Yau and Calabi-Yau man- 1. Number theory: Calabi-Yau manifolds are nat- ifolds as seen through their eyes. We hope you will ural, higher-dimensional generalizations of elliptic enjoy a walk through this joyful, colorful, and mind- curves—the theory of which dominated number the- expanding garden. ory in the 20th century. Many believe that Calabi-Yau manifolds will be the main tool in number theory in In Mathematics… the 21st century. The related questions of modular forms and quasi-modular forms are now being devel- Calabi-Yau manifold is the treasure of human oped. civilization, although many of its mysteries are yet 2. Algebra: There is a natural category, which ap- to be explored. In view of its surrounding theo- peared in algebra, where the Calabi-Yau condition rems, tools, questions and conjectures, it has been plays an important role. It is used in classifying topo- an extremely active and important research field. logical field theories. Lo Yang 3. Differential geometry: The construction of the Academy of Mathematics and System Sciences, Calabi-Yau metric solves the old problem of classifi- Chinese Academy of Sciences cation of manifolds with holonomy group SU(n) and (Hua Loo-Keng Mathematics Prize 1997) DECEMBER 2014 NOTICES OF THE ICCM 91 There is no doubt that Calabi-Yau manifolds shown to be of great importance in physics, and have become justly famous. The mere fact that one can speculate that their arithmetic mysteries, they are now abbreviated to CY manifolds testifies still largely untouched by number-theorists, could to their importance. some day throw light on even some of the most In both geometry and physics the Kummer sur- intractable ancient problems of the subject. face and its relatives have been of central interest In addition to his own mathematical research, since the 19th century. They are the natural suc- Yau has played an immense role since the middle cessors to elliptic curves, and look how long these 1990’s in the rebuilding of mathematics in China. curves have fascinated mathematicians. Yau has His creation and organization of the ICCM’s has pioneered many important ideas in global geomet- brought together mathematicians of Chinese ori- ric analysis and has influenced much of the math- gin from all over the world, and built up a new es- ematics and physics of our time. prit de corps amongst them. He has known how Michael Atiyah to persuade the highest political voices in China University of Edinburgh that high-level mathematical research is of great (Fields Medal 1966; Abel Prize 2004; importance for the future of China, and deserves King Faisal International Prize 1987) good financial support. Above all, Yau has striven always to foster the very finest mathematical re- search, and to give gifted young Chinese mathe- Calabi-Yau manifolds form a very specific class maticians every chance to develop their talents. of mathematical objects, which play an increas- ingly important role in various branches of math Just one of many examples of Yau’s legacy in this and physics. They are fundamental in algebraic ge- direction, which I enormously appreciate, has been ometry, where they stand somehow in the middle his creation of an outstanding group of number- of the complexity hierarchy of algebraic varieties: theorists in the Chinese Academy of Sciences in not too simple, but still accessible. In differential Beijing. It seems no exaggeration to say that Yau’s geometry they provide the simplest examples of myriad contributions to all aspects of the mathe- the so-called Kähler-Einstein manifolds, a very par- matical world must place him amongst the greatest ticular and very rich structure.
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