Space Symmetries Dictate the Geometry

19 Nov 2015 NUS mathematicians discover the relation between space symmetries and the underlying geometric shape of the space.

It is known that a complex torus (or doughnut shaped space) has many space symmetries. Is the converse true?

A team led by Prof De-Qi ZHANG from the Department of Mathematics in NUS has conjectured that a compact complex space of dimension n at least 3 could have a maximal number of dynamically interesting commutative symmetries (i.e., n-1 of them) only when the space is generically identical (i.e. birationally isomorphic) to a complex torus or its quotient space. In the two papers in the references below, his team shows that the above mentioned conjecture holds true unless the space is rationally connected, i.e., covered by 2-dimensional spheres.

This is a breakthrough in understanding the dynamically interesting symmetries of compact spaces. For instance, the above mentioned results of the papers convincingly explain the reason why interesting examples of symmetries discovered so far by many mathematicians in the world are mostly constructed from complex tori by surgeries of blowups or blowdowns or by taking their quotient spaces. It also guides people toward the right spaces (i.e., tori and rationally connected spaces) to find varieties with dynamically interesting symmetries.

Of course, one still has to understand the symmetries on rationally connected spaces, and to find the criteria or obstructions for these special spaces to be generically complex tori or their quotients. The same team is currently working on this.

Figure shows the torus has horizontal and also vertical rotations as symmetries. [Image credit: http://www.horntorus.com/illustration/standard_dynamic_horn-torus_still.html]

References

1. Zhang DQ. “n-dimensional projective varieties with the action of an abelian group of rank n-1.” Transactions of the American Mathematical Society, arXiv:1412.5779.

2. F Hu, TC Dinh, DQ Zhang. “Compact Kähler manifolds admitting large solvable groups of automorphisms.” Advances in Mathematics 281 (2015) 33.