What are some open problems in algebraic geometry?

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

The Jacobian conjecture

Answer 4

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.

Answer 5

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

Answer 6

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)

Answer 7

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?)

Answer 8

The Maximal Rank Conjecture is a major outstanding problem in Brill-Noether theory, although recent advances in tropical techniques might point the way to a solution; see https://arxiv.org/abs/1505.05460.

EDIT: The Maximal Rank Conjecture was proved by Eric Larson in his PhD thesis; see: https://arxiv.org/abs/1711.04906

Answer 9

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.

Answer 10

Motives and Algebraic cycles: A selection of conjectures and open questions, Joseph Ayoub.

Open problems in Algebraic Geometry, S.J. Edixhoven (editor), B.J.J. Moonen (editor), F. Oort (editor).

Answer 11

The Tate conjecture: Let $k$ be a finitely generated field, $X/k$ a smooth projective geometrically integral variety and $\ell$ invertible in $k$. Then the cycle class map $$\mathrm{CH}^r(X) \otimes_\mathbf{Z} \mathbf{Q}_\ell \to \mathrm{H}^{2r}(\bar{X},\mathbf{Q}_\ell(r))^{G_k}$$ is surjective.

It is e.g. proved for $r=1$ and Abelian varieties, a deep theorem. See http://www.math.harvard.edu/~chaoli/doc/TateConjecture.html.

This is analogous to the Hodge conjecture for complex varieties.