Moving frames and exterior differential systems Euclidean geometry and Riemannian geometry Projective geometry Cartan-Kahler I: Linear algebra and constant-coefficient homogeneous systems Cartan-Kahler II: The Cartan algorithm for linear Pfaffian systems Applications to PDE Cartan-Kahler III: The general case Geometric structures and connections Linear algebra and representation theory Differential forms Complex structures and complex manifolds Initial value problems Hints and answers to selected exercises Bibliography Index

Cartan for beginners

We characterize complete intersections in terms of local differential geometry. Let X ⊂ CPn+a be a variety. We first localize the problem; we give a criterion for X to be a complete intersection that is testable at any smooth point of X. We rephrase the criterion in the language of projective differential geometry and derive a sufficient condition for X to be a complete intersection that is computable at a general point x ∈ X. The sufficient condition has a geometric interpretation in terms of restrictions on the spaces of osculating hypersurfaces at x. When this sufficient condition holds, we are able to define systems of partial differential equations that generalize the classical Monge equation that characterizes conic curves in CP2. Using our sufficent condition, we show that if the ideal of X is generated by quadrics and a < n−(b+1)+3 3 , where b =dimXsing, then X is a complete intersection. §0. Introduction Local and global geometry Projective differential geometry has been used to study the local geometry of subvarieties of projective space by various authors (e.g. [C], [F], [GH], [JM], [T]). However, there are few examples where global conclusions are drawn from the local picture. One (global) fact about projective varieties that has been encoded into the infinitesimal geometry is the following: if there is a line on a variety X ⊂ CP along which the embedded tangent space is constant, then X must be singular. Griffiths and Harris realized this fact had implications for the projective second fundamental form (that its singular locus must be empty at general points) which 1991 Mathematics Subject Classification. primary 14e335, secondary 533a20.

Differential-geometric characterizations of complete intersections

This is an expanded and updated version of a lecture series I gave at Seoul National University in September 1997. It is in some sense an update of the 1979 Griffiths and Harris paper with a similar title. I discuss: 
Homogeneous varieties, Topology and consequences Projective differential invariants, Varieties with degenerate Gauss images, When can a uniruled variety be smooth?, Dual varieties, Linear systems of bounded and constant rank, Secant and tangential varieties, Systems of quadrics with tangential defects, Recognizing uniruled varieties, Recognizing intersections of quadrics, Recognizing homogeneous spaces, Complete intersections. 
This is a preliminary version, so please send me comments, corrections and questions.

Algebraic geometry and projective differential geometry, Seoul National University concentrated lecture series, 1997

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On the local differential geometry of complete intersections

p. 17, Def. 1.6.4, line 2 change ω|e to ωe p. 17, line -1 change Ada−1 to Ad(a −1) p. 18, Def. 1.6.9 right-hand side of displayed equation should read [ω(X), θ(Y )] − [ω(Y ), θ(X)] (not +) p. 19, Proof of 1.6.10, last line the “uniqueness” referred to is in the Frobenius Theorem p. 20, Exercise 1.6.14.3 change “complete the proof of” to “prove” p. 20, Exercise 1.6.14.6 should begin “On a connected Lie group” and displayed equation should read dωe(X,Y ) = −[X,Y ] (not +)

Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems, Second Edition

In his paper [1] J.M. Landsberg proves, among others, an elementary characterization of complete intersections [1, Proposition 1.2; see also Proposition 1.3 below]. The proof of this proposition uses method of moving frames. The aim of this note is to give an elementary proof of Landsberg's criterion that is valid over any ground field. In fact, we prove a little more, since we do not assume the variety in question to be irreducible. A preliminary version of this note was posted as an eprint at publicat ions.math.duke.edu/alg-geom/9408006. The author is grateful to J.M. Landsberg for his concern and useful correspondence and to the referee for useful suggestions.

On Landsberg's criterion for complete intersections

In his preprint ``Differential-Geometric Characterizations of Complete Intersections'' (alg-geom/9407002), J.M.Landsberg introduces an elementary characterization of complete intersections. The proof of this criterion uses the method of moving frames. The aim of this note is to present an elementary proof of Landsberg's criterion that is valid over any ground field.

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On Landsberg's criterion for complete intersections

On Landsberg's criterion for complete intersections