Hodge-theory: Abel to Deligne - IAS Video Lectures

77 Pages·2013·0.7 MB·English

Hodge-theory:
Abel to Deligne
Phillip Griffiths
Talk given at the IAS, October 14, 2013.
(cid:73) this will be mostly an historical talk — it is not intended
as an overview nor a current state of affairs of Hodge
theory — it will also be informal — intent is to give a
sense of things and how they fit together, not to present
precise statements
(cid:73) the central point of the talk will be the analogy
(cid:73) Abel took a subject (elliptic integrals) in which there
were very interesting special cases, and set it in a new
and general context that opened up the subject
(cid:73) similarly, Deligne’s theory of mixed Hodge structures
took the subject of classical Hodge theory for smooth,
projective varieties and recast it for arbitrary complex
varieties and maps between them, thus similarly opening
up the subject of Hodge theory
(cid:73) will begin by recounting Abel’s theorem in its original
form and then trace some history, emphasizing especially
the work of Picard which is perhaps less familiar, up
through Lefschetz, Hodge and Deligne’s theory,
concluding with some comments on the recent work of
M. Saito who took mixed Hodge theory and variations of
Hodge structure and fused them into the currently
penultimate state of the subject
(cid:73) my presentation of the historical development is
subjective and personal and does not reflect a researched
account of the development of the subject — I will also
draw on some recollections of what was told to me as a
student by Lefschetz, Kodaira and Spencer, and later by
Hodge when I visited him in Cambridge (UK)
(cid:73) finally, as we all know there are issues about what was by
current standards actually proved by the classical
mathematicians — here I will draw three levels of
distinction
(cid:73) an argument that can readily be made into a modern
proof (e.g., Abel’s theorem or Picard’s version of the
algebraic de Rham theorem)
(cid:73) an argument that is at least to me convincing but I have
not written out all the details (the existence of the
Picard variety of ∞q rationally inequivalent curves on an
algebraic surface S, where q = h0(Ω1))
S
(cid:73) an argument that is incomplete but which was later done
(e.g., Lefschetz’s for the hard Lefschetz theorem and
Hodge’s for the Hodge theorem)
(cid:104)◦(cid:105)
Abel
(cid:73) 18th century mathematicians were interested in integrals
of algebraic functions
ˆ
r(x,y(x))dx, f(x,y(x)) = 0;
they arise in geometry
ˆ ˆ
(cid:112) dx
dx2 +dy2 = √ for x2 +y2 = 1
1−x2
and in mechanics
ˆ
(cid:112)
p(x)dx for y˙(x)2 = p(x)
(cid:73) of particular interest were elliptic integrals
x2/a2 +y2/b2 = 1, a > b
ˆ ˆ
(cid:112) (1−k2sin2θ)dθ
a2cos2θ+b2sin2θdθ = a
(cid:112)
1−k2sin2θ
ˆ
(1−k2x2)dx
= a
(cid:112)
(1−x2)(1−k2x2)
where k2 = (a2 −b2)/a2 and x = sinθ; by analogy with
the formulas for sin2θ,cos2θ for doubling the arc length
of a circle, the mathematicians of that period (Count
Fagnano, Euler, Lagrange, Gauss, ...) found formulas for
doubling the arc length of the ellipse;
(cid:73) also much studied was the arclength of lemniscate =
locus of a point p = (x,y) such that product of distances
from two fixed points p ,p is constant
1 2
((cid:114)x,y)
(cid:114)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:8)(cid:114)(cid:72)(cid:72)(cid:72)(cid:72)(cid:72)(cid:72)(cid:72)(cid:72)(cid:114)
p2=(−a,0) (0,0) p1=(a,0)
(cid:114) (cid:114)
p2 p1
ˆ ˆ
(cid:112) dx
dx2 +dy2 = √
1−x4
Here we have taken 2a2 = 1
(cid:73) The integral above was understood as the “function” of
the upper limit ξ of integration and by choosing a path γ
of integration from x to ξ together with a branch of the
0
multi-valued function y(x) along γ
ˆ
ξ
F(ξ) = r(x,y(x))dx
x0
(cid:115)
ξ
∗
∗
∗ ∗ = branch points
γ
(cid:115)
x
0
A different choice of path changes F(ξ) by a period
(which is where we now think of as Hodge theory started).
When f(x,y) ∈ Q[x,y] and ξ ∈ Q the transcendence of
F(ξ), as well as of the periods, is a much studied and
very beautiful subject (Siegel, Bombieri, etc.)
(cid:73) when degf = 1 such integrals of algebraic functions may
be expressed in terms of elementary functions (partial
fractions), and when degf = 2 they may also be explicitly
evaluated, essentially because conics are rational curves
(x(t),y(t))
t
but when degf (cid:61) 3, e.g., f(x,y) = y2 −p(x) where
degp(x) (cid:61) 3, these integrals were of a different character;