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I am not an expert, but there seems to be an enormous technical difference between algebraic geometry and differential/metric geometry stemming from the fact that there is apparently no such thing as curvature in the former context while curvature is everywhere in the latter (indeed, it is hard to produce nontrivial results in Riemannian geometry that DON'T involve curvature). Of course it seems unreasonable to just port definitions of curvature into an algebraic context, but maybe there are constructions that play the same role in algebraic geometry that curvature does in other kinds of geometry. Here are two specific ways the notion of curvature shows up which I can imagine making sense in more general contexts.

Algebraic Chern - Weil Theory? In differential geometry one uses the curvature of a connection on a vector bundle to produce explicit cohomology classes. Does this have an algebraic analogue?

Algebraic Curvature Bounds? One supremely important theme in modern geometry involves proving theorems that depend only on the large scale geometry of a space, and the main strategy is to compare the space to a simpler one with the same large scale properties. This reminds me a little bit of tropical geometry wherein one replaces an algebraic variety with a simple combinatorial proxy, but from what little I know the analogy seems to stop there.

Any thoughts? I hope this question is not too vague, but it seems worthwhile and part of the problem is that I can't formulate a precise question along these lines. Thanks in advance!

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Personally, I find the question about Algebraic Chern - Weil Theory very interesting. I've wondered for a while now whether any such theory is possible, and I've been unable to find any literature on the subject. –  Dan Ramras Mar 25 '10 at 15:02
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The case of Riemann surfaces is instructive. The category of algebraic curves is equivalent to the category of conformal surfaces. However conformal surfaces do not have a unique Riemannian metric structure, so this means that if we want a notion of curvature for algebraic curves, this has to be above and beyond the algebro-geometric structure of a variety. –  Colin Tan Mar 25 '10 at 15:37
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I think this is one of the main questions of Algebraic Geometry. Connections are a lot of times algebraic, and then, as it happens in Hodge Theory, one can sometimes circumvent metrics and curvatre by some smart spectral sequences. However, in my opinion this is a very accidental, case by case method. –  Zsolt Patakfalvi Mar 25 '10 at 21:16
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6 Answers 6

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An algebraic analog of Chern-Weil theory (explicitly taking symmetric polynomials of curvature) is given by the Atiyah class. Given a vector bundle $E$ on a smooth variety we can consider the short exact sequence $$ 0\to End(E) \to A(E) \to T_X\to 0$$ where $T_X$ is the tangent sheaf and $A(E)$ is the "Atiyah algebroid" --- differential operators of order at most one acting on sections of $E$, whose symbol is a scalar first order diffop (hence the map to the tangent sheaf). A (holomorphic or algebraic) connection is precisely a splitting of this sequence, and a flat connection is a Lie algebra splitting. Now algebraically such splittings will often not exist (having a holomorphic connection forces your characteristic classes to have type $(p,0)$ rather than the $(p,p)$ you want..) but nonetheless we can define the extension class, which is the Atiyah class $$a_E\in H^1(X, End(E)\otimes \Omega^1_X).$$ This is the analog of the curvature form in the Riemannian world -- we now can take symmetric polynomials in the $End(E)$ factor to get the characteristic classes of $E$ in $H^p(X,\Omega_X^p)$ as desired.

This answer and Mariano's agree of course in the sense that Atiyah classes can be interpreted via Hochschild and cyclic (co)homology and generalized to arbitrary coherent sheaves (or complexes) on varieties (or stacks) (let me stick to characteristic zero to be safe). Namely the Atiyah class of the tangent sheaf can be used to define a Lie algebra structure (or more precisely $L_\infty$) on the shifted tangent sheaf $T_X[-1]$, and Hochschild cohomology is its enveloping algebra. This Lie algebra acts as endomorphisms of any coherent sheaf (which is another way to say Hochschild cohomology is endomorphisms of the identity functor on the derived category), and one can take characters for these modules, recovering the characteristic classes defined concretely above.

(In fact the notion of characters is insanely general... for example an object of any category - with reasonable finiteness - defines a class (or "Chern character") in the Hochschild homology of that category, which is cyclic and so descends to cyclic homology. An example of this is the category of representations of a finite group, whose HH is class functions, recovering usual characters, or coherent sheaves on a variety, recovering usual Chern character. or one can go more general.)

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"reasonable finiteness" = dualizable? –  Kevin H. Lin Jan 7 '11 at 11:28
    
Yes, at least in the formulation I know (which is in the revised version of my Character Theory paper with Nadler, out in a week or two) - a dualizable module over an associative algebra object in a symmetric monoidal ∞-category has a Chern character in cyclic homology. –  David Ben-Zvi Jan 7 '11 at 15:26
    
That's a really nice answer! –  David Speyer Jan 7 '11 at 16:23
    
I think this answer (and Mariano's) settle my question about Chern-Weil theory in the most satisfying possible way. And from what I have learned since I asked this question long ago, it seems that Joel Fine's answer gives about as much as there is to say about "algebraic curvature bounds". I've learned a lot from these answers - thanks everyone! –  Paul Siegel Mar 27 '11 at 17:56
    
Here you wrote that $HH^*$ is $End(Id)$ in the derived category: do you have a reference for this fact? –  Mariano Suárez-Alvarez Apr 28 '13 at 8:17
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One well-studied example is ampleness of an invertible sheaf and positive curvature for a holomorphic line bundle. A theorem of Kodaira says that a holomorphic line bundle over a complex manifold is ample if and only if it admits a metric of positive curvature.

Related to this, in the special case when the line bundle is the canonical bundle and also invoking Yau's proof of the Calabi conjecture, one has the following equivalences:

  • A compact complex manifold admits a Kähler metric of positive Ricci curvature only if it is Fano i.e, has ample anticanonical bundle.

  • A compact complex manifold admits a Kähler metric of negative Ricci curvature if and only if it has ample canonical bundle.

  • A compact complex manfiold admits a Ricci flat Kähler metic if and only if it has torsion first Chern class.

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I thought we only had "only if" for the case of positive Ricci curvature? I think there's a conjecture of Yau saying that a manifold admits a Ricci-positive Kahler-Einstein metric iff its tangent bundle is stable. –  Gunnar Magnusson Jan 3 '11 at 2:16
    
Oops, for someone who earns his living in this area that's a massive mistake to make! Of course you're right - there are certainly Fanos which do not have Kahler-Einstein metrics. As for the conjecture of Yau, I believe that the current state of affairs is that one believes that a Fano admits a Kahler-Einstein metric iff it is "K-stable". This is an algebro-geometric stability condition, related to the Hilber-Mumford criterion for a certain infinite dimensional GIT quotient. You can find details in an article of Donaldson called "Scalar curvature and stability of toric varieties", in JDG. –  Joel Fine Jan 7 '11 at 0:56
    
Just to add, I think the conjecture concerning K-stability should be attributed to Donaldson-Tian-Yau in reverse chronological order, but in the correct order of precision. Also, I've edited my answer to correct the mistake you've pointed out. –  Joel Fine Jan 7 '11 at 0:58
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Here is a short overview by Brian Osserman on curvature in algebraic gemetry, incl. the very interesting "p-curvature" which had been used much by N.Katz and Ogus.

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But p-curvature are usually used in the context where one has a flat vector bundle, which means the curvature vanish and the de Rham "thing" really gives a de Rham "complex". –  temp Jun 5 '12 at 16:46
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The algebraic Chern-Weyl theory exists. You can find expositions of it in, say, Jean-Louis Loday's Cyclic Homology or in Max Karoubi's Homologie Cyclique Astérisque. It constructs characteristic classes for finitely generated projective modules, and more generally on elements of $K_0$, with values in the cyclic homology of the coordinate algebra, and/or the usual variations of this homology theory. This precisely generalizes the usual case, in view of the Swan-Serre theorem that states that finitely generated projective modules over $C^\infty(M)$ are exactly the same thing as vector bundles over a manifold $M$, and the computation of the (periodic) cyclic homology of $C^\infty(M)$, which turns out to be de Rham cohomology of $M$.

One can also get characteristic cyclic classes on the higher algebraic $K$-groups, and this works more or less automatically not only for commutative algebras but for algebras in general.

If you want non-affine schemes, you can generalize everything with sufficient care to that context.

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Do the characteristic classes constructed in this way have some relationship with etale chern classes? –  Dan Ramras Mar 25 '10 at 16:54
    
The Chern-Weyl theory gives you explicit forms, not just cohomology classes. Hence there is a lot more information in it. Do you get anything like that with cyclic cohomology? –  Zsolt Patakfalvi Mar 25 '10 at 21:21
    
The computation of the classes follows exactly the same pattern as in the classical case, exhibiting specific cycles which represent them. –  Mariano Suárez-Alvarez Mar 25 '10 at 21:35
    
Interesting... what you describe sounds virtually identical to Connes' construction of the Chern character in noncommutative geometry. I'm a little surprised that everything works in the algebraic context given the amount of analysis involved in Connes' case, but I guess the analysis must secretly not be all that important. I wonder which idea inspired the other. –  Paul Siegel Mar 27 '10 at 3:22
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I believe it's Chern-Weil, not Chern-Weyl. –  José Figueroa-O'Farrill Apr 16 '10 at 9:32
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Ivey and Landsberg's book "Cartan for Beginners" covers material which you may find interesting. In particular, chapter 3 presents ideas on the geometry of projective varieties from the point of view of moving frames.

The rough idea (which I may be misstating, as this is stuff I haven't absorbed yet) is that differential invariants (various forms of curvature) of maps into homogeneous spaces (e.g. submanifolds of E^3 or projective varieties) come from pulling back the Maurer-Cartan form from the Lie group of transformations under which two maps would be considered equivalent. Thus I think these ideas are more about "extrinsic" differential geometry, but it still may be useful.

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For varities containing sufficiently many rational curves, the collection of rational curves of minimal degree determine the global geometry to some extent. This collection is known as the variety of minimal rational tangents, or VMRT. The rational curves passing through a general point serve as an alternative to a tangent space.

If the variety has symmetry, for instance a Hermitian symmetric space, then the tangent curves passing through two different points can be compared. This will give the notion of curvature if you ask for. I am aware that Hwang and Mok has done work in this area.

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