Stability and its Ramifications
M.S. Narasimhan
 R( x, y)dx
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Interaction between Algebraic Geometry and
Other Major Fields of Mathematics & Physics
• Main theme:
notion of “stability”, which arose in moduli
problems in algebraic geometry (classification
of geometric objects), and its relationship with
topics in partial differential equations,
differential geometry, number theory and
physics…
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• These relationships were already present in
the work of Riemann on abelian integrals,
which started a new era in modern
algebraic geometry:
• A problem in integral calculus: study of
abelian integrals
 R( x, y)dx with f (x,y) = 0,
where f is a polynomial in two variables,
and R a rational function of x and y.
• Riemann studied the problem of existence
of abelian integrals (differentials) with given
singularities and periods on the Riemann
surface associated with the algebraic curve
f(x,y)=0. (Period is the integral of the
differential on a loop on the surface).
Riemann
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• For the proof, Riemann used Dirichlet’s principle.
• Construction of a harmonic function on a domain
with given boundary values.
• The harmonic function is obtained as the function
which minimises the Dirichlet integral of functions
with given boundary values.
• The existence of such a minimising function is not
clear.
• Proofs of the existence theorem were given by
Schwarz and Carl Neumann by other methods.
• The methods invented by them to solve the
relevant differential equations, e.g., the use of
potential theory, were to play a role in the theory
of elliptic partial differential equations.
Dirichlet
Schwarz
Carl Neumann
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Proofs (contd.)
• Later Hilbert proved the Dirichlet principle.
• Direct methods of calculation of variations.
• Initiated Hilbert space methods in PDE.
Hilbert
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The Algebraic Study of Function Fields
• Dedekind and Weber:
purely algebraic treatment of the work of
Riemann
(avoiding analysis)
• The algebraic study of function fields.
• From this point of view, the profound
analogies between algebraic geometry and
algebraic number theory.
Dedekind
Andre Weil, emphasised and popularised
this analogy,
was fond of the “Rosetta stone” analogy…
Weber
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The Rosetta Stone Analogy,
& the Role of Analogies
hieroglyphs
Number theory
demotic
function fields
Greek
Riemann
surfaces
Andre Weil
The Rosetta Stone
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Algebraic Geometry & Number Theory
• Problems in number theory have given
rise to development of techniques and
theories in algebraic geometry.
• These provided in turn tools to solve
problems in number theory.
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Transcendental Methods in Higher Dimensions
• Work of Picard and Poincare in algebraic geometry,
largely part of complex analysis; partly a motivation for
Poincare for developing topology ("Analysis situs").
• Work of Hodge on harmonic forms and the application
to the study of the topology of algebraic varieties.
• Work of Kodaira using harmonic forms and differential
geometric techniques to prove deep "vanishing
theorems" in algebraic geometry, which play a key
role.
• Work of Kodaira and Spencer.
• Riemann-Roch theorem (in algebraic geometry) and
Atiyah-Singer theorem on index of linear elliptic
operators (theorem on PDE).
Picard
Poincare
Hodge
Kodaira
Spencer
Atiyah
Singer
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ALGEBRAIC GEOMETRY
COMPLEX MANIFOLDS
DIFFERENTIAL ANALYSIS ON MANIFOLDS
PDE &
DIFFERENTIAL GEOMETRY
ALGEBRAIC GEOMETRY
ALGEBRAIC GEOMETRY
DEEP RESULTS IN
ALGEBRAIC GEOMETRY
NUMBER THEORY
PHYSICS
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Now restrict to:
Particular area of
ALGEBRAIC GEOMETRY:
“STABILITY”
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• Notion of semi-stability occurs in the celebrated work of
Hilbert on invariant theory.
• Proved basic theorems in commutative algebra:
– HILBERT BASIS THEOREM
– HILBERT NULLSTELLEN SATZ
– SYZYGIES
Hilbert
• INVARIANT THEORY
Suppose the (full or special) linear group) G acts linearly
on a vector space V and S(V) the algebra of polynomial
functions on V .
HILBERT: The ring of G-invariants in S(V) is finitely
generated.
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Invariant Theory
• Criticism: no explicit generators.
“It is not mathematics; it is theology.”
• Partly to counter this, non-semi-stable points
were introduced by him. He called them “Null
forms”.
– Null form or NON-SEMI-STABLE point: a
(non-zero) point in V is said to be non-semistable if all ( non-constant , homogeneous)
invariants vanish at this point.
– SEMI-STABLE := not a null form.
– STABLE: an additional condition.
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Hilbert-Mumford
Numerical Criterion for semi stability
• NS := set of non-semi stable points and [NS} the
corresponding set in the projective space (P(V)
associated to V.
• Knowledge of the variety [NS] gives information
about the generators of the ring of invariants.
Mumford, 1975
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Mumford’s Geometric Invariant Theory
• CONSTRUCTION OF QUOTIENT SPACES IN
ALGEBRAIC GEOMETRY; A SUBTLE PROBLEM.
• A topological quotient may exist , but quotient as an
algebraic variety may not.
• MUMFORD:
P(ss) the set of semi-stable points in P(V).
Then a "good " quotient of P(ss) by the group exists (and
is a projective variety, compact, in particular)
GIT quotient
Mumford
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Moduli
MODULI Problems
-- classification problem in algebraic geometry .
• Compact Riemann surfaces/curves of a given genus.
• Ruled surfaces .
• Holomorphic vector bundles on a compact Riemann
surfaces .
• (Non -abelian generalisation of Riemann's theory)
• Subvarieties of a projective (up to projective
equivalence).
In order to get moduli spaces one has to restrict to the
class of good objects
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Moduli and GIT
• CONSTRUCTION OF MODULI SPACES REDUCED
TO CONSTRUCTION OF QUOTIENTS .
• GIVES A WAY OF IDENTIFYING "GO0D OBJECTS“.
• THESE ARE OBJECTS CORRESPONDING TO STABLE
POINTS.
• CALCULATION OF STABLE POINTS IS NOT EASY.
• A holomorphic vector bundle of degree zero on a
a Riemann surface is stable (resp. semi stable) if
the degree of all (proper) holomorphic subbundle
is < 0 (resp. ≤ 0 )(MUMFORD)
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Stability, Differential Geometry & PDEs
• THEOREM: A vector bundle of degree o on a compact
Riemann surface arises from an irreducible unitary
representation of the fundamental group of the surface
if and only if it is stable. (M.S.N & Seshadri)
• Formulation in terms of flat unitary bundles.
• A generalisation for bundles on higher dimensional
manifolds was conjectured by Hitchin and Kobayashi.
• Hermitian -Einstein metrics and Stability.
• Proved by Donaldson, Uhlenbeck-Yau.
• Solve a non-linear PDE.
Seshadri
Hitchin
Kobayashi
Donaldson
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• The problem of existence of a Kahler- Einstein
metric on a Fano manifold (anti -canonical bundle
ample) is related to a suitable notion of stability.
• The problem of the existence of a "good metric"
on a projective variety is also tied to a notion of
stability.
• Kahler metric with constant scalar curvature in a
Kahler class.
• Active research.
• Speculation: PDE and stability
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Physics
• Yang-Mills on Riemann surfaces and stable
bundles.
• STABLE BUNDLES ON ALGEBRAIC SURFACES
AND (anti-)SELF DUAL CONNECTIONS.
• Moduli spaces of stable bundles and
conformal field theory.
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Number Theory
• ROSETTA STONE ANALOGY
• Usual Integers (more generally integers in a number
field) augmented by valuations of the field - analogue
of a compact Riemann surface.
• Can study analogues of stable bundles-”arithmetic
bundles“.
• Many interesting questions.
• Canonical filtrations on arithmetic bundles used to
study the space of all bundles (not necessarily semi stable ones) by partitioning the space by degree of
instability.
• Hitchin hamiltonian on the moduli space of Hitchin(Higgs) bundles and "Fundamental Lemma“.
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