Pluripotential-theoretic methods in K-stability and the space of Kähler metrics
Doktorsavhandling, 2022

It is a natural problem, dating back to Calabi, to find canonical metrics on complex manifolds. In the case of polarized compact Kähler manifolds, a natural candidate is a metric with constant scalar curvature (cscK metric).
Since the 80s, Yau, Tian, Donaldson among others proposed that the existence of these special metrics are equivalent to an algebrico-geometric notion of K-stability.

There are several known approaches to the study of K-stability and canonical metrics, using various tools from the theory of PDEs, algebraic geometry and non-Archimedean geometry for example. In this thesis, we study a different approach, based on pluripotential theory.
In geometric terms, pluripotential theory is the study of positively curved metrics on vector bundles. For the purpose of K-stability, we only need pluripotential theory on an ample line bundle. In this case, pluripotential theory can be identified with the study of quasi-plurisubharmonic functions on the manifold.

The application of pluripotential theory in K-stability is not completely new, but previously, people are principally interested in the regular (or mildly singular) quasi-plurisubharmonic functions. In this thesis, we put more emphasis on the role of singular quasi-plurisubharmonic functions and their singularities.

In Paper 1 and Paper 2, we prove a criterion for the existence of canonical metrics on Fano manifolds in terms of quasi-plurisubharmonic functions. In Paper 3, we are concerned with the case when there are no canonical metrics, we prove that there is always an optimal destabilizer to the K-stability.

quasi-plurisubharmonic functions

K-stability

cscK metrics

Room Pascal, Department of mathematics
Opponent: Ruadhaí Dervan, University of Cambridge, UK

Författare

Mingchen Xia

Chalmers, Matematiska vetenskaper, Algebra och geometri

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In 1687, Issac Newton published the celebrated book Philosophiae Naturalis Principia Mathematica. In Principia, Newton developed a theory of mechanics, a theory later known as Newton mechanics. According to Newton, particles propagate in the 3-dimensional space $\mathbb{R}^3$. In the absence of external forces, particles move in straight lines. In general, the acceleration of a particle is proportional to the force.
In addition to the three spacial axes, there is a fourth axis: time. Time and space are irrelevant to each other.

According to Newton's theory, the speed of light depends on the observer. If one observer moves in the direction of the propagation of light, he will observe a lower speed of light than one observer who move in the opposite direction.

In the 19th century, with the development of electromagnetism, or more precisely electricity and magnetism according to the pre-Maxwell terminology, people begin to understand that light is nothing but the electromagnetic wave.
In 1873, James Clerk Maxwell published A Treatise on Electricity and Magnetism, proposing the laws governing electromagnetism. These laws are written in a number of equations, called Maxwell equations. A surprising consequence of Maxwell equations is that the speed of light is constant, independent of the choice of the observer! Hence time and space must be dependent.
This observation is in stark contradiction with Newton mechanics. It suggests the need of a new theory of mechanics.

The solution was proposed in the beginning of 20th century by Minkowski, Poincaré and Einstein among others, leading to the theory of special relativity. In special relativity, space and time are no longer independent axes. Two observers moving at different speeds feel different time and space simultaneously. The inseparable object consisting of space and time is known as the spacetime. In the mathematical terminology, the spacetime in special relativity is a flat 4-dimensional space with a Lorentz metric. Particles propagate along geodesics, the shortest path in spacetime.
A prominent feature of relativity is that the speed of objects are bounded from above: nothing can move faster than light.

It turns out that the gravity does not fit into the theory of relativity, as the propogators of the gravity force would have to move infinitely fast. The disastrous situation was solved by Albert Einstein in 1915, leading to a vast generalization of the theory of relativity, known as general relativity. Roughly speaking, in general relativity, the spacetime is no longer flat, it is curved instead. Matters in the spacetime make the spacetime curved.
Gravity is no longer a force, instead it is the curvature of the spacetime. In the curved spacetime, particles still tend to move along the shortest paths, namely the geodesics, which are no longer straight lines. This movement is effectively the gravity.

When no matter is presented, the spacetime is a vacuum. The equation describing the vacuum is the celebrated vacuum Einstein's field equation. In mathematical terms, this means the Ricci curvature of the spacetime vanishes. We will explain the notion of Ricci curvature in detail in later chapters. Here we stress that a space with vanishing Ricci curvature is not necessarily flat. In fact, there is an abundance of such spaces.
More generally, in the presence of a cosomological constant, the vacuum is described by the fact that the Ricci curvature is constant.

The spacetimes in general relativity are 4-dimensional with the Lorentzian signature. However, the quest for quantum gravity suggests that higher dimensional manifolds (spacetimes) with the Riemannian signature are also important. Moreover, in the presence of supersymmetry, one needs to put an extra structure on the manifold, namely, the Kähler structure.

In short, this thesis concerns the existence of vacua of the Riemannian signature in all even dimensions when an extra Kähler structure is presented.
In mathematical terms, we are studying the Kähler--Einstein metrics.

Ämneskategorier

Matematik

Fundament

Grundläggande vetenskaper

ISBN

978-91-7905-712-1

Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 5178

Utgivare

Chalmers

Room Pascal, Department of mathematics

Online

Opponent: Ruadhaí Dervan, University of Cambridge, UK

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Senast uppdaterat

2023-03-30