Stability and its Ramifications

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Stability and its Ramifications

• M.S. Narasimhan

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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
• 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 Dirichlets
  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.Andre Weil, emphasised  and popularised
  this analogy,was fond of the Rosetta stone
  analogy

Dedekind
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
DEEP RESULTS IN ALGEBRAIC GEOMETRY
NUMBER THEORY
ALGEBRAIC GEOMETRY
PHYSICS
ALGEBRAIC GEOMETRY
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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
• 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.

Hilbert
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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-semi-stable 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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Mumfords 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 lt 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.