# Topics in Differential geometry $\cap$ Algebraic geometry

I find (both: differential and algebraic) geometry fascinating. I'm just beginning my graduate studies, but I'd like to know some topics/theorems in the intersection of these two (since I don't really know where to go).

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Love the title! –  Gamma Function Feb 28 '14 at 5:25
I am the one who asked this question. I don't know how it happened, but I lost my account. I would like to thank the people that took some time to answer, I really appreciate it. (Please move this to a comment.) –  user132208 Feb 28 '14 at 22:17

1) One very nice topic is Hodge theory.
It allows you to decompose the De Rham cohomology vector spaces of a smooth projective complex variety into a direct sum of vector spaces having an algebro-geometric interpratation: $$H^r(X,\mathbb C)=\oplus_{p+q=r} H^q(X,\Omega^p)$$ where $\Omega^p$ is the bundle of algebraic differential $p$-forms.

2) A more restricted topic is Kodaira's embedding theorem which tells you when a compact complex manifold $X$ is projective algebraic in terms of a differential 2-form associated to a so-called Kähler metric on $X$.

3) Finally (for this post!) there are powerful "vanishing theorems" which tell you that certain line bundles on projective smooth complex varieties have zero positive dimensional cohomology as soon as you can put a metric on these bundles with certain "positivity" properties.
The best known of these results is the Kodaira-Nakano theorem: for a positive line bundle $L$ on the smooth complex projective $n$-dimensional variety $X$ we have $$H^i(X,\Omega^j\otimes L)=0 \;\text{for} \;i+j\gt n$$

Griffiths-Harris's book is a very rich reference for all of these topics and many more which are at the intersection you require.

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One topic that lies right at the intersection of these two fields is the study of the Gauss map of subvarieties in projective space.

An $k$-dimensional manifold embedded in Euclidean space $\mathbf R^n$ has a Gauss map to the Grassmannian $Gr(k,n)$, taking $p$ to the tangent space at $p$. Gauss studied this map for surfaces in $\mathbf R^3$, and this is one of the basic objects of interest in the differential geometry of surfaces.

To see examples of how this can be applied to algebraic geometry, you could look at the long paper of Griffiths–Harris studying the Gauss map of smooth subvarieties of projective space. I don't know the paper well enough to say much about it, but in summary they use this differential-geometric approach to obtain a number of classification results about subvarieties in "special position" in projective space. For example, they prove that a nondegenerate smooth surface in $\mathbf P^5$ with degenerate secant variety but nondegenerate tangent variety must be the Veronese surface.

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I would suggest that you have a look at Serre's paper GAGA. In this paper, he constructs an analytification functor from the category of complex algebraic varieties to the category of complex analytic spaces. It preserves the underlying point-set of a variety, but changes the topology from the Zariski-topology to a classical analytic one. If the variety happens to be smooth, its analytification is a complex manifold.

This functor is implicitly used when differential geometers talk about smooth varieties. They really mean to refer to the associated complex manifolds. It is often fruitful to study manifolds coming from varieties in this way, as they tend to be a bit more tame than arbitrary manifolds.

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In a completely different direction, you might wanna look at Arakelov Geometry. Arakelov geometry is an area of geometry that is motly interrested in Diophantine problems. The idea behind it is that in order to grasp arithmetic properties, it won't be sufficient to only study the algebraic geometry of the objects over $\text{Spec } \mathbb Z$, because $\text{Spec } \mathbb Z$ is not proper (compact if you prefer), so we have to compactify the situation, and to do so, we need to add an "analytic fiber" (which is a smooth complex manifold), and endow it with hermitian data.

We then have to find analogues of statements and theorems in classical algebraic geometry in this new algebro-hermitian setting, which involves of course algebraic geometry on the variety itself, and complex differential geometry on the "fiber at infinity", and to treat them in an unified setting.

This relates to deep differential concepts and question, such as index theory for instance, or questions related to the curvature on specific bundles (e.g the quillen curvature on the determinant of cohomology) and of course analytic torsion.

Maybe to give you a taste of the mixing of the two styles of geometry, let me state an example. On a riemann surface, classical algebraic geometry tells you that is a good idea to study cycles over it. In the arakelov context you study so called "arithmetic cycles", which consists of pairs $(Z,g)$ where Z is a cycle on the surface (the geometric part), and g is a so called green function for $Z$, which you might think of as a potential for the electric field generated by the uniformly charged distribution of charges supported by $Z$ (you put charges at every point of $Z$ and you $g$ is then a potential for the electric field generated).

You can take a look at these notes, or those ones, by Soulé, but they're in French

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Oh, and a nice way to see Arakelov theory is as a generalisation of intersection theory, so it includes three areas of your title, algebraic geometry, differential geometry and $\cap$. =) –  A.Rod Mar 1 '14 at 11:14
Could you explain why $\text{Spec } \mathbb Z$ is not compact? By the way, proper is a relative concept and it doesn't make sense to say that $\text{Spec } \mathbb Z$ is not proper. –  Georges Elencwajg Mar 2 '14 at 18:38
It's a manner of speaking. The long-discovered analogy is that Spec Z should be the analog of a proper curve over a field k. In the context of arakelov geometry, Spec Z is the analog of an affine open subset in a "proper" object. A basic observation why it should be so is the product formula which is considered the analog of the fact that the degree of a principal divisor is 0 on a proper variety. The arithmetic degree of a principal divisor is 0 if you add up the "infinite fiber". This is an indication that this should be considered as the compact object, and Spec Z is an open subset if it. –  A.Rod Mar 4 '14 at 19:05
More generally, if you want reasonnable analogs of classical resulats in projective geometry to work (Riemann-Roch, degree formula etc...) you need to add up that infinite fiber that compactify $\text{Spec }\mathbb{Z}$. It's a purely "interpretative" statement. To my knowledge, nobody has ever created a theory where one actually endows $X\coprod X(\mathbb C)$ with a topology, and develop a theory on this space. –  A.Rod Mar 4 '14 at 19:06
Thanks for your comments, @A.Rodriguez. –  Georges Elencwajg Mar 4 '14 at 20:52