Yau’s Conjecture with positive first Chern class was solved by the joint effort from Professor CHEN Xiuxiong, a Thousand Talents in the School of Mathematics Sciences of USTC, Fields Medal Laureate Sir Donaldson from Britain, together with CHEN’s ex-student Dr. SUN Song. Their three series papers were published in a top international mathematics journal Journal of the American Mathematical Society.
Einstein established General Theory of Relativity in 1916 in order to explain the nature of Gravity. He tried to measure Gravity with a set of two-order nonlinear differential equations. The famous Kähler-Einstein metrics is the solution of the equation. Latter physicists go further to string theory--in which our universe is a ten-dimension space-time: a normal four-dimension space time plus a tiny six-dimension space-time. These complex high-dimensional space-time must be Kähler-Einstein metrics, which only exists in the calculation of theoretical physicists and mathematicians.
Calabi, a famous Italian geometrician proposed a great conjecture in 1954 for the exploration of high-dimensional space-time, that complex high-dimensional space is composed of several simple hyperspaces. Considered that simple hyperspaces could be analyzed by existing mathematical tools, high-dimensional space would be constructed by the assembling of multiple simple geometric models. This is the Calabi Conjecture-- a conjecture about the uniformization of high-dimensional space in complex geometry, and also a conjecture concerning the verification of existence of Kähler-Einstein metrics on high-dimensional space.
Calabi conjecture can be divided into three cases, namely with the first Chern class being negative, vanishing and positive (A great mathematician Prof. Shing Shen Chern discovered in 1945 that there is an invariant on complex manifolds reflecting the global nature of manifolds. It was named Chern characteristic class later on, referred as Chern class in short, and exerted profound and extensive impact on the development of mathematics and theoretical physics). The case of vanishing and negative Chern classes was settled by Shing-Tung Yau until twenty years later (the case of negative Chern classes was proved by Yau and French mathematician Aubin separately), and Yau won the Fields Medal-the Nobel Prize for mathematics in 1982 for his contribution. Vanishing first Chern class is called Calabi-Yau manifold now, in which the three-dimensional complex Calabi-Yau manifold is the six-dimensional space that string theorists are looking for.
Much efforts by mathematicians reveal that Kahler-Einstein metrics exists only when the positive Chern class in high-dimensional space satisfies certain conditions. The problem is thus difficult and it has puzzled the academics for decades. Yau proposed a conjecture that the existence of Kähler-Einstein metrics in high-dimensional space with positive first Chern class could be transformed to the problem of stability in algebraic geometry. And this is considered to be the “the most important issue following the solution of Calabi conjecture in complex geometry”.
In a series of papers of Chen-Donaldson-Sun (CHEN Xiuxiong, Simon Donaldson and SUN Song), they gave a complete proof of Yau's conjecture on the existence of Kähler-Einstein metrics. Based on the research program launched by Donaldson in 2008, they finally shed light to solving Yau’s conjecture with positive first Chern class, with a variety of new innovative methods and techniques in several mathematical branches including differential geometry, algebraic geometry, several complex variables and metric geometry.
Chen-Donaldson-Sun’s proof proved to be ground-breaking, not only solving a fundamental problem, but also developing novel and powerful tools to reveal the profound relationship between Kähler geometry, algebraic geometry and partial differential equations-- evaluated by a reviewer of Journal of the American Mathematical Society. It is undeniable that this progress has aroused significant impact worldwide, said Demailly, an internationally re-known mathematician. The success of this breakthrough depends on many fundamental developments by mathematicians in various fields for the past twenty years, which also marks the research of Kähler geometry reaching a new level. This breakthrough is hopefully to be applied to algebraic geometry and string theory in theoretical physics as well.
It is interesting that CHEN is the last Ph.D. student of Prof. Calabi, and the co-writer SUN Song was the Ph.D. student of CHEN. The relay and cooperation of three generation finally solved the Conjecture.
Link of relevant paper:
http://www.ams.org/journals/jams/0000-000-00/
A brief introduction of Prof. CHEN Xiuxiong:
Professor CHEN Xiuxiong was born in Qingtian county, Zhejiang province. Graduating from USTC’s Department of Mathematics in 1987, CHEN continued his study with Prof. PENG Jiagui, and received his master’s degree from Graduate School of USTC. In 1989, CHEN went to University of Pennsylvania for doctoral study and graduated in 1994, being the last Ph.D. student of the famous geometer Prof. Calabi. CHEN was invited by Sir Donaldson to study the existence of Kähler -Einstein metrics together in the summer of 2008, and they have been studying this issue jointly until now.
CHEN was once invited to deliver a 45-minute talk at the 24th International Mathematician Congress. He was hired as “Chair Professors in Cheung Kong Scholars” in 2008, first “Master Professors (II)” of USTC in 2009, and was elected as the “Thousand Talents” of the second batch. He has been involved in fostering USTC young talents and international academic exchange for a long time. He has organized Geometry Summer School nine years in a row since 2004, and initiated Pacific Rim Complex Geometry International Conference in USTC in 2006, making extraordinary contribution to the talent training and academic interchange of mathematics. His student, SUN Song, WANG Bin .etc. have grown up as outstanding mathematicians in the international academic circle.
(CHE Yifeng, USTC News Center)
