The aim of this question is to understand the interaction between projective algebraic varieties (over the complex or real numbers), Riemannian manifolds, and projective space, through shared concepts – that is, to understand which concepts can be defined intrinsically in more than one of these structures, and whether these concepts agree when a space admits more than one of these structures.
I'll start with motivation: In the usual Euclidean geometry of the plane, one has the concept of straight lines. It is well known that these arise from the Riemannian structure of the Euclidean plane as geodesics – that is, if one looks at the Euclidean plane only as a Riemannian manifold (without the vector space structure, etc.) one can still recover the straight lines as locally distance-minimizing curves. Interestingly, however, one can also forget the metrical\Riemannian structure and look at the projective plane, and somehow still talk about straight lines – straight lines still make sense in projective geometry, even though there is no metric structure and thus no geodesics.
Projective varieties are in some sense a generalization of projective geometry to spaces that are not necessarily 'linear'. In projective varieties, is there a concept generalizing straight lines or geodesics? Another generalization of projective geometry to curved spaces is manifolds with projective connections; in these spaces, there is indeed an appropriate notion of geodesics. So, do smooth projective varieties (at least over the real numbers) have a natural projective connection that somehow agrees with the variety structure? If there was such a natural connection, this would provide a natural extensions of the concept of straight lines to projective varieties.
The two questions that I'd like to ask are:
- Which concepts from projective geometry and from Riemannian geometry have natural analogs in the intrinsic geometry of projective varieties?
- Is there a relationship between the structure of a (smooth) projective variety and the structure of a manifold with a projective connection? Are there spaces which have both of these structures and they somehow agree with each other?
Some concepts I already know that generalize from projective geometry to general algebraic varieties are dimension, the automorphism group of the variety (which for projective space is the projective general linear group), subvarieties, polynomial functions, and for one-dimensional varieties, the cross-ratio (this has a generalization to any Riemann surface). As raised above, what about straight lines? Do these have a generalization to projective surfaces/varieties? What about the degree of an embedded subvariety (for projective space this is defined using straight lines/hypersurfaces)? And are there Riemannian concepts which have natural analogs on algebraic varieties, such as curvature? connection? geodesics? holonomy?
This question is rather broad, but I'll appreciate any partial answers, perspectives on this, or even references or names that can point me to relevant material.
Best Answer
Let me try to answer your question to some extent; given the vague nature of the question, there will be no canonical answer.
You should think in terms of three different areas of mathematics:
Classical projective geometry (PG)
Riemannian geometry (RG)
Algebraic geometry (AG)
These areas have different agendas, different tools, different "favorite toys." These areas, however, share some common toys, such as projective spaces of various dimensions and their projective subspaces.
For instance, the real-projective space ${\mathbb R}P^n$ (and its complex analogue ${\mathbb C}P^n$) appears as one of these common toys. In order to become one, some further choices have to be made in RG and AG. From RG viewpoint, one needs to choose a Riemannian metric, the most important one is the Fubini-Study metric. From the AG viewpoint, one needs to make the projective space $P$ into an algebraic variety. The standard choice is to identify $P$ with ${\mathbb R}P^n$ (resp. ${\mathbb C}P^n$). But other important choices are Veronese embeddings. They are all isomorphic to the "standard" projective space but the choice of an embedding is important.
Moreover, from the AG viewpoint, it is also important to work with other fields and rings, e.g. say with the projective spaces over $p$-adic numbers, over finite fields, over rings of polynomials, etc. These will have increasingly less and less in common with PG and RG.
As a projective geometer, you can think of all projective lines in the projective plane (or space) as congruent, but images of the Veronese embeddings of the projective line in the projective planes are genuinely different ones. From the AG viewpoint, projective lines in the projective plane are the degree 1 rational curves. Veronese curves/surfaces have higher degree.
From PG viewpoint, "everything takes place inside of a fixed projective space." To some extent, this is also true from the AG viewpoint, except you may have to work with different projective embeddings and some (or many, depending on what you do) of the objects of AG are not projective varieties and do not embed as subsets in a projective space. But from the RG viewpoint, one usually considers Riemannian manifolds abstractly and not as embedded isometrically in a projective space. (Caveat: There are many exceptions to this rule, for instance, much of the theory of minimal surfaces deals with surfaces in the Euclidean 3-space.)
I did not even get started on Kahler geometry that links (complex) AG and RG, but, again, AG and RG agendas are different here.
As for some of your more specific questions:
i. A question in a comment
the answer is that this is not quite true, as Tabes says. For one thing, you have to work "over the real numbers" (i.e. with real projective spaces). Secondly, you have to remove a point from the projective line to make it an affine line. Or, if you like, you remove the line at infinity from the projective plane, which will make it into an affine plane and then affine lines become complete Euclidean geodesics (after you choose a metric!). With these caveats, the fact that you have mention is true but is an accident of geometry of a space of constant curvature. "Most" Riemannian manifolds do not have constant sectional curvature and do not admit even a local a "model" where their geodesics are (pieces of) Euclidean straight lines. (OK, I am lying a bit here, this is not a complete accident and spherical, Euclidean and hyperbolic geometries were and continue to be a source of inspiration of RG.)
ii.
No, in general they do not. Once you fix an embedding in a projective space then, yes, you can induce the ambient connection from the FS metric. But there are many different projective embeddings of the same variety, so you do not have a canonical choice. As for "somehow agrees with the variety structure," I do not even know how to interpret this. Connections, curvature, Chern classes, etc, play important role in AG, but one should definitely not limit oneself to the tangent bundle here. Frequently, other bundles and their connections are more informative. For instance, sometimes you work with the canonical or anticanonical line bundle, sometimes you work with all line bundles simultaneously, or work with all stable vector bundles, etc.
iii.
I would say that the curvature sign (more precisely, the sign of one of your favorite curvatures or even positivity of the curvature operator) and the notion of "positivity" in algebraic geometry which manifests itself in different ways. Assuming that a particular curvature is positive/negative, semipositive, etc., usually makes it possible to prove interesting theorems in RG. Ditto in AG:
Lazarsfeld, Robert, Positivity in algebraic geometry. I. Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 48. Berlin: Springer (ISBN 3-540-22533-1/hbk). xviii, 387 p. (2004). ZBL1093.14501.
Lazarsfeld, Robert, Positivity in algebraic geometry. II. Positivity for vector bundles, and multiplier ideals., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 49. Berlin: Springer (ISBN 3-540-22534-X/hbk). xviii, 385 p. (2004). ZBL1093.14500.
Also the key dichotomy: Positive curvature-negative curvature (RG), rational variety-general type variety (AG).
iv.
As far as I am concerned, this is too vague to be answerable. One can say that these are rational curves. (Which play an important role in huge chunk of AG.) But in many cases one works with varieties that simply do not contain rational curves. Then you consider curves of higher genus: Thinking of all these curves at once is frequently quite useful. Or, you become a complex-analyst and work with the Kobayashi metric; the Kobayashi-extremal disks can be regarded as analogues of straight lines.
I declare myself done here....
The bottom line is: Treat the three geometries as separate areas of math. Occasionally, they meet in one place and share their toys, tools and theorems.