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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeFeb 18th 2015
• (edited Feb 18th 2015)

Am working on the entry higher Cartan geometry. Started writing a Motivation section.

This is just the first go, need to quit now, will polish tomorrow.

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeFeb 19th 2015

had a second go through the Motivation, now the flow of its contents should be stable, but I suppose I need to streamline a bit more…

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeFeb 20th 2015

I further edited the Motivation, now trying to improve the exposition by producing plenty of standout formulas and standout boxes, for guiding the eye and catch-wording the discussion.

There is still clearly much further room for further polishing, but for the moment I will have to leave it as is.

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeMar 28th 2015
• (edited Mar 30th 2015)

I see that people are finding a use for Cartan geometry within algebraic geometry, Mori geometry meets Cartan geometry, Jun-Muk Hwang’s ICM2014 talk.

I suppose there might be a higher version :)

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJun 28th 2017

The thesis by Felix Wellen on Formalizing higher Cartan geometry in modal homotopy type theory is available now. I have added a pointer to the entry.

• CommentRowNumber6.
• CommentAuthorDavid_Corfield
• CommentTimeJun 28th 2017

Has he been examined yet?

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeJun 28th 2017

The defense is in two weeks. But he already got a position: with Steve Awodey at CMU!

• CommentRowNumber8.
• CommentAuthorDavid_Corfield
• CommentTimeJun 29th 2017
• (edited Jun 29th 2017)

Three typos:

goups; can not (one word); thereoms

I can’t say I’m a fan of the no space/no indentation approach to paragraphs. If the last line of a paragraph happens to reach the right side of the page, there’s no way of knowing it has ended.

One more

depedent

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeJun 29th 2017

Does this refer to Felix’s thesis? I don’t think he is reading here.

• CommentRowNumber10.
• CommentAuthorDavid_Corfield
• CommentTimeJun 29th 2017

Yes, it does.

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeJun 29th 2017

Okay, I have emailed him.

• CommentRowNumber12.
• CommentAuthortrent
• CommentTimeJun 30th 2017

has anyone investigated higher “mori-cartan” geometry since David’s 2015 comment? (very much looking forward to reading the wellen thesis btw).

• CommentRowNumber13.
• CommentAuthorDavid_Corfield
• CommentTimeJul 1st 2017

Due to reactivation of this thread I had a look again at that paper, Mori geometry meets Cartan geometry. I’m always a sucker for that kind of sweeping narrative opening:

Lines have been champion figures in classical geometry. Together with circles, they dominate the entire geometric contents of Euclid. Their dominance is no less strong in projective geometry. Classical projective geometry is full of fascinating results about intricate combinations of lines. As geometry entered the modern era, lines evolved into objects of greater flexibility and generality while retaining all the beauty and brilliance of classical lines. As Euclidean geometry developed into Riemannian geometry, for example, lines were replaced by geodesics which then inherited all the glory of Euclidean lines.

In the transition from classical projective geometry to complex projective geometry, real lines have been replaced by complex lines. Lines over complex numbers have all the power of lines in classical projective geometry and even more: results of greater elegance and harmony are obtained over complex numbers. A large number of results on lines and their interactions with other varieties have been obtained in complex projective geometry,their dazzling beauty no less impressive than that of classical geometry. But as complex projective geometry develops further into complex geometry and abstract algebraic geometry, which emphasize intrinsic properties of complex manifolds and abstract varieties, the notion of lines in projective space seems to be too limited for it to keep its leading role.

Firstly, to be useful in intrinsic geometry of projective varieties in projective space, lines should lie on the projective varieties. But most projective varieties do not contain lines. Even when a projective manifold contains lines, the locus of lines is, often, small and then such a locus is usually regarded as an exceptional part. Of course, there are many important varieties that are covered by lines, but they belong to a limited class from the general perspective of classification theory of varieties. In short,

($\ast$) the class of projective manifolds covered by lines seems to be too special from the perspective of the general theory of complex manifolds or algebraic varieties.

Secondly, many of the methods employed to use lines on varieties in projective space depend on the extrinsic geometry of ambient projective space. They do not truly belong to intrinsic geometry of the varieties. Such geometric arguments are undoubtedly useful in fathoming deeper geometric properties of varieties which are described explicitly, at least to some extent. But can such methods yield results on a priori unknown varieties, defined abstractly by intrinsic conditions? In short,

($\ast$$\ast$) tools employed in line geometry are not intrinsic enough to handle intrinsic problems on abstractly described varieties.

These concerns show that lines in projective space have a rather limited role in the modern development of complex algebraic geometry. Is there a more general and more powerful notion in complex algebraic geometry that can replace the role of lines, as geodesics do in Riemannian geometry? No serious candidate had emerged until Mori’s groundbreaking work [37].

In the celebrated paper [37], Mori shows that a large class of projective manifolds, including all Fano manifolds, are covered by certain intrinsically defined rational curves that behave like lines in many respects. Let us call these rational curves ‘minimal rational curves’. If a projective manifold embedded in projective space is covered by lines, these lines are minimal rational curves of the projective manifold, so the notion of minimal rational curves can be viewed as an intrinsic generalization of lines.

The main tool here is the deformation theory of curves, a machinery of modern complex algebraic geometry-somewhat reminiscent of the use of variational calculus in the local study of geodesics in Riemannian geometry. An example is the property that a minimal rational curve cannot be deformed when two distinct points on the curve are fixed. This result generalizes the fundamental postulate of classical geometry that “two points determine one line”. The important point is that such a classical property of lines can be recovered by modern deformation theory in an abstract setting.

Hwang goes on to explain how he came upon Cartan’s $G$-structures on p.5.

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeJul 1st 2017
• (edited Jul 1st 2017)

has anyone investigated higher “mori-cartan” geometry since David’s 2015 comment?

I haven’t looked into it, and I don’t hear people around me speak about it, so I don’t know. But of course the application of the concept of Cartan geometries and G-structures in areas other than the ordinary differential geometry that they were originally conceived in, say in higher and in algebraic geometry, this is precisely what a synthetic formulation as in Felix Wellen’s thesis is supposed to be good for.