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### Stability and its Ramifications

M.S. Narasimhan

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…

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

withf (x,y) = 0,

where fis 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

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

Proofs (contd.)

- Later Hilbert proved the Dirichlet principle.
- Direct methods of calculation of variations.
- Initiated Hilbert space methods in PDE.

Hilbert

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.
- Fromthis 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

The Rosetta Stone Analogy, & the Role of Analogies

hieroglyphs

Number theory

demotic

function fields

Greek

Riemann

surfaces

Andre Weil

The Rosetta Stone

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.

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 geometrictechniques to prove deep "vanishing theorems" in algebraic geometry, which play a key role.
- Work of Kodairaand Spencer.
- Riemann-Roch theorem (in algebraic geometry) and Atiyah-Singer theorem onindex of linear elliptic operators (theorem on PDE).

Picard Poincare Hodge Kodaira Spencer Atiyah Singer

ALGEBRAIC GEOMETRY

COMPLEX MANIFOLDS

DIFFERENTIAL ANALYSIS ON MANIFOLDS

PDE &

DIFFERENTIAL GEOMETRY

DEEP RESULTS IN

ALGEBRAIC GEOMETRY

ALGEBRAIC GEOMETRY

NUMBER THEORY

ALGEBRAIC GEOMETRY

PHYSICS

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

InvariantTheory

- 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.

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

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

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 -abeliangeneralisation 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

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) holomorphicsubbundle is < 0 (resp. ≤ 0 )(MUMFORD)

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 Hitchinand Kobayashi.
- Hermitian -Einstein metrics and Stability.
- Proved by Donaldson, Uhlenbeck-Yau.
- Solve a non-linear PDE.

Seshadri Hitchin Kobayashi Donaldson

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

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.

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.
- Hitchinhamiltonian on the moduli space of Hitchin-(Higgs) bundles and "Fundamental Lemma“.

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