In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, and especially
differential and
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, K-stability is an
algebro-geometric stability condition, for
complex manifold
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic.
The term complex manifold is variously used to mean a com ...
s and
complex algebraic varieties
In algebraic geometry, a complex algebraic variety is an algebraic variety (in the scheme sense or otherwise) over the field of complex numbers. Parshin, Alexei N., and Igor Rostislavovich Shafarevich, eds. ''Algebraic Geometry III: Complex Algeb ...
. The notion of K-stability was first introduced by
Gang Tian
Tian Gang (; born November 24, 1958) is a Chinese mathematician. He is a professor of mathematics at Peking University and Higgins Professor Emeritus at Princeton University. He is known for contributions to the mathematical fields of Kähler g ...
and reformulated more algebraically later by
Simon Donaldson.
The definition was inspired by a comparison to
geometric invariant theory (GIT) stability. In the
special case
In logic, especially as applied in mathematics, concept is a special case or specialization of concept precisely if every instance of is also an instance of but not vice versa, or equivalently, if is a generalization of . A limiting case is ...
of
Fano varieties
In algebraic geometry, a Fano variety, introduced by Gino Fano in , is a complete variety ''X'' whose anticanonical bundle ''K''X* is ample. In this definition, one could assume that ''X'' is smooth over a field, but the minimal model program has ...
, K-stability precisely characterises the existence of
Kähler–Einstein metrics. More generally, on any compact complex manifold, K-stability is
conjectured to be equivalent to the existence of
constant scalar curvature Kähler metric In differential geometry, a constant scalar curvature Kähler metric (cscK metric), is (as the name suggests) a Kähler metric on a complex manifold whose scalar curvature is constant. A special case is Kähler–Einstein metric, and a more general ...
s (cscK metrics).
History
In 1954,
Eugenio Calabi
Eugenio Calabi (born 11 May 1923) is an Italian-born American mathematician and the Thomas A. Scott Professor of Mathematics, Emeritus, at the University of Pennsylvania, specializing in differential geometry, partial differential equations and ...
formulated a conjecture about the existence of Kähler metrics on compact
Kähler manifold
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnold ...
s, now known as the
Calabi conjecture. One formulation of the conjecture is that a compact Kähler manifold
admits a unique
Kähler–Einstein metric in the class
. In the particular case where
, such a Kähler–Einstein metric would be
Ricci flat In the mathematics, mathematical field of differential geometry, Ricci-flatness is a condition on the curvature of a Riemannian manifold. Ricci-flat manifolds are a special kind of Einstein manifold. In theoretical physics, Ricci-flat Lorentzian man ...
, making the manifold a
Calabi–Yau manifold
In algebraic geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical physics. Particularly in superstring ...
. The Calabi conjecture was resolved in the case where
by
Thierry Aubin and
Shing-Tung Yau, and when
by Yau. In the case where
, that is when
is a
Fano manifold, a Kähler–Einstein metric does not always exist. Namely, it was known by work of
Yozo Matsushima
was a Japanese mathematician.
Early life
Matsushima was born on February 11, 1921, in Sakai City, Osaka Prefecture, Japan. He studied at Osaka Imperial University (later named Osaka University) and graduated with a Bachelor of Science degree in ...
and
André Lichnerowicz that a Kähler manifold with
can only admit a Kähler–Einstein metric if the
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
is
reductive. However, it can be easily shown that the
blow up of the
complex projective plane at one point,
is Fano, but does not have reductive Lie algebra. Thus not all Fano manifolds can admit Kähler–Einstein metrics.
After the resolution of the Calabi conjecture for
attention turned to the loosely related problem of finding canonical metrics on ''
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
s'' over complex manifolds. In 1983, Donaldson produced a new proof of the
Narasimhan–Seshadri theorem In mathematics, the Narasimhan–Seshadri theorem, proved by , says that a holomorphic vector bundle over a Riemann surface is stable if and only if it comes from an irreducible projective unitary representation of the fundamental group.
The main c ...
. As proved by Donaldson, the theorem states that a
holomorphic vector bundle over a compact
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
is
stable
A stable is a building in which livestock, especially horses, are kept. It most commonly means a building that is divided into separate stalls for individual animals and livestock. There are many different types of stables in use today; the ...
if and only if it corresponds to an irreducible unitary
Yang–Mills connection. That is, a unitary connection which is a
critical point of the Yang–Mills functional
:
On a Riemann surface such a connection is projectively flat, and its
holonomy gives rise to a projective unitary representation of the
fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
of the Riemann surface, thus recovering the original statement of the theorem by
M. S. Narasimhan
Mudumbai Seshachalu Narasimhan (7 June 1932 – 15 May 2021) was an Indian mathematician. His focus areas included number theory, algebraic geometry, representation theory, and partial differential equations. He was a pioneer in the study of m ...
and
C. S. Seshadri. During the 1980s this theorem was generalised through the work of Donaldson,
Karen Uhlenbeck and Yau, and
Jun Li and Yau to the
Kobayashi–Hitchin correspondence In differential geometry, algebraic geometry, and gauge theory, the Kobayashi–Hitchin correspondence (or Donaldson–Uhlenbeck–Yau theorem) relates stable vector bundles over a complex manifold to Einstein–Hermitian vector bundles. The corres ...
, which relates stable holomorphic vector bundles to
Hermitian–Einstein connections over arbitrary compact complex manifolds. A key observation in the setting of holomorphic vector bundles is that once a holomorphic structure is fixed, any choice of Hermitian metric gives rise to a unitary connection, the
Chern connection In mathematics, a Hermitian connection \nabla is a connection on a Hermitian vector bundle E over a smooth manifold M which is compatible with the Hermitian metric
\langle \cdot, \cdot \rangle on E, meaning that
: v \langle s,t\rangle = \langle \na ...
. Thus one can either search for a Hermitian–Einstein connection, or its corresponding Hermitian–Einstein metric.
Inspired by the resolution of the existence problem for canonical metrics on vector bundles, in 1993 Yau was motivated to conjecture the existence of a Kähler–Einstein metric on a Fano manifold should be equivalent to some form of algebro-geometric stability condition on the variety itself, just as the existence of a Hermitian–Einstein metric on a holomorphic vector bundle is equivalent to its stability. Yau suggested this stability condition should be an analogue of
slope stability
Slope stability analysis is a static or dynamic, analytical or empirical method to evaluate the stability of earth and rock-fill dams, embankments, excavated slopes, and natural slopes in soil and rock. Slope stability refers to the condition of i ...
of vector bundles.
In 1997, Tian suggested such a stability condition, which he called ''K-stability'' after the
K-energy functional In mathematics, and especially complex geometry, the Mabuchi functional or K-energy functional is a functional on the space of Kähler potentials of a compact Kähler manifold whose critical points are constant scalar curvature Kähler metrics. The ...
introduced by
Toshiki Mabuchi. The ''K'' originally stood for ''kinetic'' due to the similarity of the K-energy functional with the kinetic energy, and for the
German
German(s) may refer to:
* Germany (of or related to)
**Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ger ...
''
kanonisch'' for the
canonical bundle. Tian's definition was analytic in nature, and specific to the case of Fano manifolds. Several years later Donaldson introduced an algebraic condition described in this article called K-stability, which makes sense on any polarised variety, and is equivalent to Tian's analytic definition in the case of the polarised variety
where
is Fano.
Definition
In this section we work over the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s
, but the essential points of the definition apply over any field. A polarised variety is a pair
where
is a complex
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
and
is an
ample line bundle on
. Such a polarised variety comes equipped with an embedding into projective space using the
Proj construction,
:
where
is any positive integer large enough that
is
very ample In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ...
, and so every polarised variety is
projective. Changing the choice of ample line bundle
on
results in a new embedding of
into a possibly different projective space. Therefore a polarised variety can be thought of as a projective variety together with a fixed embedding into some projective space
.
Hilbert–Mumford criterion
K-stability is defined by analogy with the
Hilbert–Mumford criterion from finite-dimensional
geometric invariant theory. This theory describes the stability of ''points'' on polarised varieties, whereas K-stability concerns the stability of the polarised variety itself.
The Hilbert–Mumford criterion shows that to test the stability of a point
in a projective algebraic variety
under the action of a
reductive algebraic group , it is enough to consider the one parameter subgroups (1-PS) of
. To proceed, one takes a 1-PS of
, say
, and looks at the limiting point
:
This is a fixed point of the action of the 1-PS
, and so the line over
in the
affine space is preserved by the action of
. An action of the multiplicative group
on a one dimensional vector space comes with a
weight
In science and engineering, the weight of an object is the force acting on the object due to gravity.
Some standard textbooks define weight as a Euclidean vector, vector quantity, the gravitational force acting on the object. Others define weigh ...
, an integer we label
, with the property that
:
for any
in the fibre over
. The Hilbert-Mumford criterion says:
* The point
is semistable if
for all 1-PS
.
* The point
is stable if
for all 1-PS
.
* The point
is unstable if
for any 1-PS
.
If one wishes to define a notion of stability for varieties, the Hilbert-Mumford criterion therefore suggests it is enough to consider one parameter deformations of the variety. This leads to the notion of a test configuration.
Test Configurations
![Test Configuration](https://upload.wikimedia.org/wikipedia/commons/4/44/Test_Configuration.png)
A test configuration for a polarised variety
is a pair
where
is a
scheme A scheme is a systematic plan for the implementation of a certain idea.
Scheme or schemer may refer to:
Arts and entertainment
* ''The Scheme'' (TV series), a BBC Scotland documentary series
* The Scheme (band), an English pop band
* ''The Schem ...
with a
flat morphism
In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism ''f'' from a scheme ''X'' to a scheme ''Y'' is a morphism such that the induced map on every stalk is a flat map of rings, i.e.,
:f_P\colon \mathcal_ \t ...
and
is a relatively ample line bundle for the morphism
, such that:
#For every
, the
Hilbert polynomial
In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a field are three strongly related notions which measure the growth of the dimension of the homoge ...
of the fibre
is equal to the Hilbert polynomial
of
. This is a consequence of the flatness of
.
#There is an action of
on the family
covering the standard action of
on
.
#For any (and hence every)
,
as polarised varieties. In particular away from
, the family is trivial:
where
is projection onto the first factor.
We say that a test configuration
is a product configuration if
, and a trivial configuration if the
action on
is trivial on the first factor.
Donaldson–Futaki Invariant
To define a notion of stability analogous to the Hilbert–Mumford criterion, one needs a concept of weight
on the fibre over
of a test configuration
for a polarised variety
. By definition this family comes equipped with an action of
covering the action on the base, and so the fibre of the test configuration over
is fixed. That is, we have an action of
on the central fibre
. In general this central fibre is not smooth, or even a variety. There are several ways to define the weight on the central fiber. The first definition was given by using Ding-Tian's version of generalized Futaki invariant. This definition is differential geometric and is directly related to the existence problems in Kähler geometry. Algebraic definitions were given by using Donaldson-Futaki invariants and CM-weights defined by intersection formula.
By definition an action of
on a polarised scheme comes with an action of
on the ample line bundle
, and therefore induces an action on the vector spaces
for all integers
. An action of
on a complex vector space
induces a direct sum decomposition
into ''weight spaces'', where each
is a one dimensional subspace of
, and the action of
when restricted to
has a weight
. Define the total weight of the action to be the integer
. This is the same as the weight of the induced action of
on the one dimensional vector space
where
.
Define the weight function of the test configuration
to be the function
where
is the total weight of the
action on the vector space
for each non-negative integer
. Whilst the function
is not a polynomial in general, it becomes a polynomial of degree
for all
for some fixed integer
, where
. This can be seen using an equivariant Riemann-Roch theorem. Recall that the Hilbert polynomial
satisfies the equality
for all
for some fixed integer
, and is a polynomial of degree
. For such
, let us write
:
The Donaldson-Futaki invariant of the test configuration
is the rational number
:
In particular
where
is the first order term in the expansion
:
The Donaldson-Futaki invariant does not change if
is replaced by a positive power
, and so in the literature K-stability is often discussed using
-line bundles.
It is possible to describe the Donaldson-Futaki invariant in terms of
intersection theory, and this was the approach taken by Tian in defining the CM-weight. Any test configuration
admits a natural compactification
over
(e.g.,see ), then the CM-weight is defined by
:
where
. This definition by intersection formula is now often used in algebraic geometry.
It is known that
coincides with
, so we can take the weight
to be either
or
.
The weight
can be also expressed in terms of the Chow form and hyperdiscriminant.
In the case of Fano manifolds, there is an interpretation of the weight in terms of new
-invariant on valuations found by Chi Li and Kento Fujita.
K-stability
In order to define K-stability, we need to first exclude certain test configurations. Initially it was presumed one should just ignore trivial test configurations as defined above, whose Donaldson-Futaki invariant always vanishes, but it was observed by Li and Xu that more care is needed in the definition. One elegant way of defining K-stability is given by
Székelyhidi using the norm of a test configuration, which we first describe.
[G. Székelyhidi. An introduction to extremal Kähler metrics, volume 152 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2014.]
For a test configuration
, define the norm as follows. Let
be the infinitesimal generator of the
action on the vector space
. Then
. Similarly to the polynomials
and
, the function
is a polynomial for large enough integers
, in this case of degree
. Let us write its expansion as
:
The norm of a test configuration is defined by the expression
:
According to the analogy with the Hilbert-Mumford criterion, once one has a notion of deformation (test configuration) and weight on the central fibre (Donaldson-Futaki invariant), one can define a stability condition, called K-stability.
Let
be a polarised algebraic variety. We say that
is:
*K-semistable if
for all test configurations
for
.
*K-stable if
for all test configurations
for
, and additionally
whenever
.
*K-polystable if
is K-semistable, and additionally whenever
, the test configuration
is a product configuration.
*K-unstable if it is not K-semistable.
Yau–Tian–Donaldson Conjecture
K-stability was originally introduced as an algebro-geometric condition which should characterise the existence of a Kähler–Einstein metric on a Fano manifold. This came to be known as the Yau–Tian–Donaldson conjecture (for Fano manifolds). The conjecture was resolved in the 2010s in works of
Xiuxiong Chen
Xiuxiong Chen () is a Chinese-American mathematician whose research concerns differential geometry and differential equations. A professor at Stony Brook University since 2010, he was elected a Fellow of the American Mathematical Society in ...
,
Simon Donaldson, and
Song Sun
Song Sun (, born in 1987) is a Chinese mathematician whose research concerns geometry and topology. A Sloan Research Fellow, he is a professor at the Department of Mathematics of the University of California, Berkeley, where he has been since 2018 ...
, and of
Gang Tian
Tian Gang (; born November 24, 1958) is a Chinese mathematician. He is a professor of mathematics at Peking University and Higgins Professor Emeritus at Princeton University. He is known for contributions to the mathematical fields of Kähler g ...
. The strategy is based on a continuity method with respect to the cone angle of a Kähler–Einstein metric with cone singularities along a fixed anticanonical divisor, as well as an in-depth use of the Cheeger–Colding–Tian theory of Gromov–Hausdorff limits of Kähler manifolds with Ricci bounds.
Theorem (Yau–Tian–Donaldson conjecture for Kähler–Einstein metrics): A Fano Manifold admits a Kähler–Einstein metric in the class of if and only if the pair is K-polystable.
Chen, Donaldson, and Sun have alleged that Tian's claim to equal priority for the proof is incorrect, and they have accused him of academic misconduct. Tian has disputed their claims. Chen, Donaldson, and Sun were recognized by the
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
's prestigious 2019
Veblen Prize as having had resolved the conjecture.
The
Breakthrough Prize
The Breakthrough Prizes are a set of international awards bestowed in three categories by the Breakthrough Prize Board in recognition of scientific advances. The awards are part of several "Breakthrough" initiatives founded and funded by Yuri Mi ...
has recognized Donaldson with the
Breakthrough Prize in Mathematics
The Breakthrough Prize in Mathematics is an annual award of the Breakthrough Prize series announced in 2013.
It is funded by Yuri Milner and Mark Zuckerberg and others. The annual award comes with a cash gift of $3 million. The Breakthrough Prize ...
and Sun with the
New Horizons Breakthrough Prize, in part based upon their work with Chen on the conjecture.
More recently, a proof based on the "classical" continuity method was provided by Ved Datar and Gabor Székelyhidi, followed by a proof by Chen, Sun, and Bing Wang using the Kähler–Ricci flow. Robert Berman, Sébastien Boucksom, and Mattias Jonsson also provided a proof from the variational approach.
Extension to constant scalar curvature Kähler metrics
It is expected that the Yau–Tian–Donaldson conjecture should apply more generally to cscK metrics over arbitrary smooth polarised varieties. In fact, the Yau–Tian–Donaldson conjecture refers to this more general setting, with the case of Fano manifolds being a special case, which was conjectured earlier by Yau and Tian. Donaldson built on the conjecture of Yau and Tian from the Fano case after his definition of K-stability for arbitrary polarised varieties was introduced.
Yau–Tian–Donaldson conjecture for constant scalar curvature metrics: A smooth polarised variety admits a constant scalar curvature Kähler metric in the class of if and only if the pair is K-polystable.
As discussed, the Yau–Tian–Donaldson conjecture has been resolved in the Fano setting. It was proven by Donaldson in 2009 that the Yau–Tian–Donaldson conjecture holds for
toric varieties In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be no ...
of complex dimension 2. For arbitrary polarised varieties it was proven by Stoppa, also using work of Arezzo and Pacard, that the existence of a cscK metric implies K-polystability. This is in some sense the easy direction of the conjecture, as it assumes the existence of a solution to a difficult partial differential equation, and arrives at the comparatively easy algebraic result. The significant challenge is to prove the reverse direction, that a purely algebraic condition implies the existence of a solution to a PDE.
Examples
Smooth Curves
It has been known since the original work of
Pierre Deligne and
David Mumford that smooth
algebraic curves are asymptotically stable in the sense of geometric invariant theory, and in particular that they are K-stable. In this setting, the Yau–Tian–Donaldson conjecture is equivalent to the
uniformization theorem. Namely, every smooth curve admits a Kähler–Einstein metric of constant scalar curvature either
in the case of the
projective line ,
in the case of
elliptic curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
s, or
in the case of compact Riemann surfaces of genus
.
Fano varieties
The setting where
is ample so that
is a
Fano manifold is of particular importance, and in that setting many tools are known to verify the K-stability of Fano varieties. For example using purely algebraic techniques it can be proven that all Fermat hypersurfaces
are K-stable Fano varieties for
.
[Tian, G., 1987. On Kähler-Einstein metrics on certain Kähler manifolds withc 1 (M)> 0. ''Inventiones mathematicae'', ''89''(2), pp.225-246.]
Toric Varieties
K-stability was originally introduced by Donaldson in the context of
toric varieties In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be no ...
.
In the toric setting many of the complicated definitions of K-stability simplify to be given by data on the moment polytope
of the polarised toric variety
. First it is known that to test K-stability, it is enough to consider ''toric test configurations'', where the total space of the test configuration is also a toric variety. Any such toric test configuration can be elegantly described by a convex function on the moment polytope, and Donaldson originally defined K-stability for such convex functions. If a toric test configuration
for
is given by a convex function
on
, then the Donaldson-Futaki invariant can be written as
:
where
is the
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
on
,
is the canonical measure on the boundary of
arising from its description as a moment polytope (if an edge of
is given by a linear inequality
for some affine linear functional h on
with integer coefficients, then
), and
. Additionally the norm of the test configuration can be given by
:
where
is the average of
on
with respect to
.
It was shown by Donaldson that for toric surfaces, it suffices to test convex functions of a particularly simple form. We say a convex function on
is piecewise-linear if it can be written as a maximum
for some affine linear functionals
. Notice that by the definition of the constant
, the Donaldson-Futaki invariant
is invariant under the addition of an affine linear functional, so we may always take one of the
to be the constant function
. We say a convex function is simple piecewise-linear if it is a maximum of two functions, and so is given by
for some affine linear function
, and simple rational piecewise-linear if
has rational cofficients. Donaldson showed that for toric surfaces it is enough to test K-stability only on simple rational piecewise-linear functions. Such a result is powerful in so far as it is possible to readily compute the Donaldson-Futaki invariants of such simple test configurations, and therefore computationally determine when a given toric surface is K-stable.
An example of a K-unstable manifold is given by the toric surface
, the first
Hirzebruch surface
In mathematics, a Hirzebruch surface is a ruled surface over the projective line. They were studied by .
Definition
The Hirzebruch surface \Sigma_n is the \mathbb^1-bundle, called a Projective bundle, over \mathbb^1 associated to the sheaf\mathca ...
, which is the
blow up of the
complex projective plane at a point, with respect to the polarisation given by
, where
is the blow up and
the exceptional divisor.
![Moment polytope of first Hirzebruch surface](https://upload.wikimedia.org/wikipedia/commons/c/ca/Moment_polytope_of_first_Hirzebruch_surface.png)
The measure
on the horizontal and vertical boundary faces of the polytope are just
and
. On the diagonal face
the measure is given by
. Consider the convex function
on this polytope. Then
:
and
:
Thus
:
and so the first Hirzebruch surface
is K-unstable.
Alternative Notions
Hilbert and Chow Stability
K-stability arises from an analogy with the Hilbert-Mumford criterion for finite-dimensional geometric invariant theory. It is possible to use geometric invariant theory directly to obtain other notions of stability for varieties that are closely related to K-stability.
Take a polarised variety
with Hilbert polynomial
, and fix an
such that
is very ample with vanishing higher cohomology. The pair
can then be identified with a point in the
Hilbert scheme
In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space (or a more general projective scheme), refining the Chow variety. The Hilbert scheme is a d ...
of subschemes of
with Hilbert polynomial
.
This Hilbert scheme can be embedded into projective space as a subscheme of a Grassmannian (which is projective via the
Plücker embedding In mathematics, the Plücker map embeds the Grassmannian \mathbf(k,V), whose elements are ''k''-dimensional subspaces of an ''n''-dimensional vector space ''V'', in a projective space, thereby realizing it as an algebraic variety.
More precisely ...
). The general linear group
acts on this Hilbert scheme, and two points in the Hilbert scheme are equivalent if and only if the corresponding polarised varieties are isomorphic. Thus one can use geometric invariant theory for this group action to give a notion of stability. This construction depends on a choice of
, so one says a polarised variety is asymptotically Hilbert stable if it is stable with respect to this embedding for all
sufficiently large, for some fixed
.
There is another projective embedding of the Hilbert scheme called the Chow embedding, which provides a different linearisation of the Hilbert scheme and therefore a different stability condition. One can similarly therefore define asymptotic Chow stability. Explicitly the Chow weight for a fixed
can be computed as
:
for
sufficiently large. Unlike the Donaldson-Futaki invariant, the Chow weight changes if the line bundle
is replaced by some power
. However, from the expression
:
one observes that
:
and so K-stability is in some sense the limit of Chow stability as the dimension of the projective space
is embedded in approaches infinity.
One may similarly define asymptotic Chow semistability and asymptotic Hilbert semistability, and the various notions of stability are related as follows:
Asymptotically Chow stable
Asymptotically Hilbert stable
Asymptotically Hilbert semistable
Asymptotically Chow semistable
K-semistable
It is however not know whether K-stability implies asymptotic Chow stability.
[J. Ross and R. Thomas. A study of the Hilbert-Mumford criterion for
the stability of projective varieties. J. Algebraic Geom., 16(2):201–255,
2007.]
Slope K-Stability
It was originally predicted by Yau that the correct notion of stability for varieties should be analogous to slope stability for vector bundles. Julius Ross and
Richard Thomas developed a theory of slope stability for varieties, known as slope K-stability. It was shown by Ross and Thomas that any test configuration is essentially obtained by blowing up the variety
along a sequence of
invariant ideals, supported on the central fibre.
This result is essentially due to David Mumford. Explicitly, every test configuration is dominated by a blow up of
along an ideal of the form
:
where
is the coordinate on
. By taking the support of the ideals this corresponds to blowing up along a
flag
A flag is a piece of fabric (most often rectangular or quadrilateral) with a distinctive design and colours. It is used as a symbol, a signalling device, or for decoration. The term ''flag'' is also used to refer to the graphic design empl ...
of subschemes
:
inside the copy
of
. One obtains this decomposition essentially by taking the weight space decomposition of the invariant ideal
under the
action.
In the special case where this flag of subschemes is of length one, the Donaldson-Futaki invariant can be easily computed and one arrives at slope K-stability. Given a subscheme
defined by an
ideal sheaf In algebraic geometry and other areas of mathematics, an ideal sheaf (or sheaf of ideals) is the global analogue of an ideal in a ring. The ideal sheaves on a geometric object are closely connected to its subspaces.
Definition
Let ''X'' be a t ...
, the test configuration is given by
:
which is the
deformation to the normal cone of the embedding
.
If the variety
has Hilbert polynomial
, define the slope of
to be
:
To define the slope of the subscheme
, consider the
Hilbert-Samuel polynomial of the subscheme
,
:
for
and
a rational number such that
. The coefficients
are polynomials in
of degree
, and the K-slope of
with respect to
is defined by
:
This definition makes sense for any choice of real number
where
is the
Seshadri constant In algebraic geometry, a Seshadri constant is an invariant of an ample line bundle ''L'' at a point ''P'' on an algebraic variety. It was introduced by Jean-Pierre Demailly, Demailly to measure a certain ''rate of growth'', of the tensor powers of ' ...
of
. Notice that taking
we recover the slope of
. The pair
is slope K-semistable if for all proper subschemes
,
for all
(one can also define slope K-stability and slope K-polystability by requiring this inequality to be strict, with some extra technical conditions).
It was shown by Ross and Thomas that K-semistability implies slope K-semistability. However, unlike in the case of vector bundles, it is not the case that slope K-stability implies K-stability. In the case of vector bundles it is enough to consider only single subsheaves, but for varieties it is necessary to consider flags of length greater than one also. Despite this, slope K-stability can still be used to identify K-unstable varieties, and therefore by the results of Stoppa, give obstructions to the existence of cscK metrics. For example, Ross and Thomas use slope K-stability to show that the
Projective bundle#The projective bundle of a vector bundle, projectivisation of an unstable vector bundle over a K-stable base is K-unstable, and so does not admit a cscK metric. This is a converse to results of Hong, which show that the projectivisation of a stable bundle over a base admitting a cscK metric, also admits a cscK metric, and is therefore K-stable.
Filtration K-Stability
Work of Apostolov–Calderbank–Gauduchon–Tønnesen-Friedman shows the existence of a manifold which does not admit any extremal metric, but does not appear to be destabilised by any test configuration. This suggests that the definition of K-stability as given here may not be precise enough to imply the Yau–Tian–Donaldson conjecture in general. However, this example ''is'' destabilised by a limit of test configurations. This was made precise by
Székelyhidi, who introduced filtration K-stability.
A filtration here is a filtration of the coordinate ring
:
of the polarised variety
. The filtrations considered must be compatible with the grading on the coordinate ring in the following sense: A filtation
of
is a chain of finite-dimensional subspaces
:
such that the following conditions hold:
#The filtration is ''multiplicative''. That is,
for all
.
#The filtration is compatible with the grading on
coming from the graded pieces
. That is, if
, then each homogenous piece of
is in
.
#The filtration exhausts
. That is, we have
.
Given a filtration
, its
Rees algebra In commutative algebra, the Rees algebra of an ideal ''I'' in a commutative ring ''R'' is defined to be R t\bigoplus_^ I^n t^n\subseteq R The extended Rees algebra of ''I'' (which some authors refer to as the Rees algebra of ''I'') is defined asR t, ...
is defined by
:
We say that a filtration is finitely generated if its Rees algebra is finitely generated. It was proven by David Witt Nyström that a filtration is finitely generated if and only if it arises from a test configuration, and by Székelyhidi that any filtration is a limit of finitely generated filtrations.
[D. Witt Nyström. Test configurations and Okounkov bodies. Compos. Math., 148(6):1736–1756, 2012.] Combining these results Székelyhidi observed that the example of Apostolov-Calderbank-Gauduchon-Tønnesen-Friedman would not violate the Yau–Tian–Donaldson conjecture if K-stability was replaced by filtration K-stability. This suggests that the definition of K-stability may need to be edited to account for these limiting examples.
See also
*
Kähler manifold
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnold ...
*
Kähler–Einstein metric
*
K-stability of Fano varieties
*
Geometric invariant theory
*
Calabi conjecture
*
Kobayashi–Hitchin correspondence In differential geometry, algebraic geometry, and gauge theory, the Kobayashi–Hitchin correspondence (or Donaldson–Uhlenbeck–Yau theorem) relates stable vector bundles over a complex manifold to Einstein–Hermitian vector bundles. The corres ...
*
Stable curve
In algebraic geometry, a stable curve is an algebraic curve that is asymptotically stable in the sense of geometric invariant theory.
This is equivalent to the condition that it is a complete connected curve whose only singularities are ordinary ...
References
Notes
{{DEFAULTSORT:K-Stability
Differential geometry
Algebraic geometry