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What are the open big problems in algebraic geometry and vector bundles?

More specifically, I would like to know what are interesting problems related to moduli spaces of vector bundles over projective varieties/curves.

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Why don't you read some of the literature on these topics to find out? Usually recent ICM talks, survey articles in the bulletin, and recently published advanced textbooks are good places to start for this kind of thing. –  Emerton Aug 30 '10 at 15:31
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This seems a perfectly good question. I would be interested to see some of the answers. –  Richard Borcherds Aug 30 '10 at 15:52
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MO questions like the rest of us need luck. This question was lucky enough that Richard Borcherds offered a very nice answer and potentially there will be further answers that we can enjoy and ultimately this will be a useful source. Let's keep it open! –  Gil Kalai Aug 30 '10 at 16:59
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We've had many discussions over at meta about whether a sufficient condition to be a good question is that it generates good answers. The overall consensus (that's too strong a word ... plurality opinion?) seems to be "no". If "too broad/vague" were a criterion on the list of reasons to close, I would vote to close. As of my comment, this question currently has four votes to close as "off topic", but it's certainly not that, it's just too vague. I do think it should be improved, though, and I will go in to fix capitalization. –  Theo Johnson-Freyd Aug 30 '10 at 18:39
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Theo, this is not a correct characterization of the discussions on meta. This was an issue where there were different opinions. My opinion was that just like in "real world mathematics" (and science) attracting good answers is a merit of a question. The answers can give prople some clues for what to look for in the ICM talks and bulletin articles Mathew referred to. In fact, good answers can give useful links to specific such papers. In any case, I have voted to reopen. –  Gil Kalai Aug 30 '10 at 21:39

8 Answers 8

up vote 21 down vote accepted

A few of the more obvious ones:

* Resolution of singularities in characteristic p
*Hodge conjecture
* Standard conjectures on algebraic cycles (though these are not so urgent since Deligne proved the Weil conjectures).
*Proving finite generation of the canonical ring for general type used to be open though I think it was recently solved; I'm not sure about the details.

For vector bundles, a longstanding open problem is the classification of vector bundles over projective spaces.

(Added later) A very old major problem is that of finding which moduli spaces of curves are unirational. It is classical that the moduli space is unirational for genus at most 10, and I think this has more recently been pushed to genus about 13. Mumford and Harris showed that it is of general type for genus at least 24. As far as I know most of the remaining cases are still open.

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You remember correctly, here's the paper: arxiv.org/abs/math/0610203 –  Charles Siegel Aug 30 '10 at 15:55
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At the end of her talk at the Hyderabad Congress, Claire Voisin was asked by someone whether she believed in the Hodge conjecture. Her answer was equivocal, if memory serves me right. –  Chandan Singh Dalawat Aug 31 '10 at 3:13
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Farkas proved that $\overline{M}_g$ is of general type for $g = 22$. –  Moon Aug 31 '10 at 7:01

Let me mention a couple of problems related to vector bundles on projective spaces.

  1. The Hartshorne conjecture. In its weak form it says that any rank 2 vector bundle on $\mathbf{P}^n_{\mathbf{C}},n>6$ is a direct sum of line bundles, which implies that any codimension 2 smooth subvariety whose canonical class is a multiple of the hyperplane sectionis a complete intersection. In a stronger form Hartshorne's conjecture says that any codimension $>\frac{2}{3}n$ subvariety of $\mathbf{P}^n_{k},k$ an algebraically closed field is a complete intersection. See Hartshorne, Varieties of small codimension in a projective space, Bull AMS 80, 1974. The weak conjecture fails for $n=3$ and $4$ -- there are examples (due to Horrocks and Mumford) of non-split vector bundles of rank 2 on $\mathbf{P}^4_{\mathbf{C}}$, but so far as I know the question if any such examples exist for $n>4$ is open. See here Evidences on Hartshorne's conjecture? References? for a discussion including some references.

  2. The existence of non-algebraic topological vector bundles on $\mathbf{P}^n_{\mathbf{C}}$. It is a classical result that any topological complex vector bundle on $\mathbf{P}^n_{\mathbf{C}}, n\leq 3$ is algebraic, see e.g. Okonek, Schneider, Spindler, Vector bundles on complex projective spaces, chapter 1, \S 6. It is strongly suspected that for $n>3$ there are topological complex vector bundles that are not algebraic. Good candidates are nontrivial rank 2 vector bundles on $\mathbf{P}^n_{\mathbf{C}}, n\geq 5$ all of whose Chern classes vanish which were constructed by E. Rees, see MR0517518. It is claimed there that these bundles do not admit a holomorphic structure, but later a gap was found in the proof. See here Complex vector bundles that are not holomorphic for some more information.

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There are examples of indecomposable rank $2$ vector bundles on $\mathbb{P}^5$ in characteristic $2$ due to Tango and Kumar-Peterson-Rao (independently). –  Mahdi Majidi-Zolbanin May 18 '12 at 19:33
    
Thanks, Mahdi, that's interesting. Does this generalize to projective spaces of higher dimensions? –  algori Jun 1 '12 at 21:24
    
No. If it did, then this problem would no longer be an open problem. –  Mahdi Majidi-Zolbanin Aug 2 '13 at 1:00

The Jacobian conjecture

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This problem is (in)famous. I've lost track of the number of false claims regarding this on the arxiv and elsewhere. –  Donu Arapura Aug 31 '10 at 14:24
    
For a good introduction to the subject, allow me to recommend the book Polynomial automorphisms and the Jacobian conjecture, by Arnoldus Richardus and Petrus van den Essen. Given the simplistic statement, how little is truly understood of that problem is simply shocking, and the first pages of the book really helped me dispel many misconception. –  Thierry Zell Aug 31 '10 at 16:13

We can also mention two other major open problems :

  • The abundance conjecture, stating that if a $K_X+\Delta$ is klt and nef, then it is semi-ample (a multiple has no base-point)

  • The Griffith's conjecture : if $E$ is an ample vector bundle over a compact complex manifold, then it is Griffith-positive. (this is known for line bundles of course)

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Henri, can you add links? –  Gil Kalai Aug 30 '10 at 17:20
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Well, Y.T Siu has recently claimed he had proved the abundance conjecture (in a version stating that the Kodaira dimension equals the numerical Kodaira dimension); here's the paper : arxiv.org/abs/0912.0576 –  Henri Aug 30 '10 at 20:54
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For some additional discussion of Siu's work, see the recent question mathoverflow.net/questions/31605/… –  Karl Schwede Aug 31 '10 at 0:51
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Griffiths conjecture is also known to be true for general vector bundles on curves! –  diverietti Jan 11 '12 at 16:59

There's also Fujita's conjecture.

Conjecture: Suppose $X$ is a smooth projective dimensional complex algebraic variety with ample divisor $A$. Then

  1. $H^0(X, \mathcal{O}_X(K_X + mA))$ is generated by global section when $m > \dim X$.
  2. $K_X + mA$ is very ample for $m > \dim X + 1$

It's also often stated in the complex analytic world.

Also there are many refinements (and generalizations) of this conjecture. For example, the assumption that $X$ is smooth is probably more than you need (something close to rational singularities should be ok). It also might even be true in characteristic $p > 0$.

It's known in relatively low dimensions (up to 5 in case 1. I think?)

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Only part 1 is known in low dimension - part 2 is open even in dimension 3. –  Arend Bayer May 21 '12 at 7:51

There's also the big open question (I think it's still open) about whether rationally connected varieties are always unirational. I think people believe the answer is NO, but they don't know an example.

Joe Harris had some slides a few years ago with regards to this Seattle 2005

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Linearization Conjecture. Every algebraic action of $\mathbb{C}^*$ on $\mathbb{C}^n$ is linear in some coordinates of $\mathbb{C}^n$. Open for $n>3$.

Cancellation Conjecture. If $X\times \mathbb{C}\cong \mathbb{C}^{m+1}$ then $X\cong \mathbb{C}^m$. Open for $m>2$.

Coolidge-Nagata Conjecture. A rational cuspidal curve in $\mathbb{P}^2$ is rectifiable, i.e. there exists a birational automorphism of $\mathbb{P}^2$ which transforms the curve into a line.

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In connection to vector bundles over $\mathbb{P}^n$, Hartshorne's paper from 1979 provides a list of open problems. The paper is "Algebraic vector bundles on projective spaces: A problem list" Topology, 18:117–128, 1979.

I don't know which of those problems are still open, but I would be interested in knowing how much progress has been made on those problems, since 1979.

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