Vojta: Big semistable vector bundles

Bigness

Throughout this talk, is a field of characteristic zero, algebraically closed unless otherwise specified.

A variety is an integral scheme, separated and of finite type over a field.

Throughout this talk, is a complete variety over .

Definition: Let be a line sheaf on . We say that is big if there is a constant such that for all sufficiently large and divisible .

Lemma. (Kodaira) Let be a line sheaf and an ample line sheaf on . Then is big iff has a (nonzero) global section for some .

Proof:  '' is obvious.

'': Write with a reduced effective very ample divisor. It will suffice to show that has a global section for some . Consider the exact sequence

The middle term has rank , but the rightmost term has rank , for divisible.

Definition: A vector sheaf of rank on is big if there is a such that

for all divisible.

Equivalently, is big iff on is big.

Essential base locus

Definition: Assume that is projective, and let be a (big) line sheaf on . The essential base locus of is the subset

for any ample divisor on (it is independent of ). The essential base locus of a vector sheaf on is the set , where is the essential base locus of on and is the canonical morphism.

Question: If is a big vector sheaf, is its essential base locus properly contained in ?

Answer: No. Example: Unstable over curves.

Question: What if is big and semistable?

Curves

Throughout this section, is a (projective) curve.

Definition: (Mumford) A vector sheaf on is semistable if, for all short exact sequences

of nontrivial vector sheaves on ,

or (equivalently)

Theorem. Let be a big semistable vector sheaf on . Then is ample (i.e., is ample on ). In particular, the essential base locus of is empty.

Proof: By Kleiman's criterion for ampleness, the sum of an ample and a nef divisor is again ample, so by Kodaira's lemma it suffices to show that if is a semistable vector sheaf on , then all effective divisors on are nef.

So, let be an effective divisor and a curve on . We want to show:

Since is semistable, so is (proof later).

Therefore we may assume that is a section of , and that is a prime divisor.

Since is a section, it corresponds to a surjection . Moreover, . By semistability, therefore,

Now consider . Let be the degree of on fibers of ; . Then for some . Thus corresponds to a section of , hence we have an injection

with locally free quotient.

Since is semistable, so is (proof later); hence

Let ; then has rank . The diagram

commutes for all diagonal matrices, hence for all diagonalizable matrices, hence for all matrices. Thus

and therefore by (**),

Thus by (*),

Higher Dimensional Varieties

Let again be a complete variety of arbitrary dimension.

Construction: Given a vector sheaf on of rank and a representation

we can construct a vector sheaf on of rank by applying to the transition matrices of . Equivalently, if corresponds to , then corresponds to .

Examples of this include , , and .

Definition: (Bogomolov) A vector sheaf of rank on is unstable if there exists a representation of determinant 1 (i.e., factoring through ) such that has a nonzero section that vanishes at at least one point. It is semistable if it is not unstable.

Theorem. (Bogomolov) If is a curve, then Bogomolov's definition of semistability agrees with Mumford's.

Remark: If has determinant then , but not conversely.

Indeed, the representation , , has image contained in but its does not factor through .

To see that the (true) converse holds, first show that the vanishing of the determinant defines an irreducible subset of ; this is left as an exercise for the reader. Now suppose that is a representation that factors through , and suppose also that its image is not contained in . Then is a nonconstant regular function , hence it determines a nonconstant rational function on with zeros and poles contained in . But the latter is irreducible, so it can't have both zeroes and poles there, contradiction.

So now we can pose:

Question: If is a projective variety and is a big, semistable vector sheaf on , then is the essential base locus of a proper subset of ?

Remark: We can't conclude that is ample in the above, as the following example illustrates. Let be a projective variety of dimension , let be a big semistable vector sheaf on of rank , let be the blowing-up of at a closed point, and let be the exceptional divisor. Then the essential base locus of must contain .

My Mitteljahrentraum

The question of an essential base locus being a proper subset comes up in Nevanlinna theory, and I hope to be able to use it in number theory, as well. Here's how.

Bogomolov has shown that is semistable for a smooth surface . One would hope to generalize this, to for a normal crossings divisor on , and also to higher dimensions. Then it would suffice to prove that one of these bundles is big to get arithmetical consequences.

Moreover, Bogomolov's definition of semistability can be generalized to defining semistability of higher jet bundles. These are not vector bundles, because they correspond to elements of for a group other than . But, one can make the same definition, using those representations of having the appropriate kernel: again (Green-Griffiths), or a certain bigger group (Semple-Demailly). Probably the latter.

Bigness is easy to define in this context, and then one hopefully can use the two properties to talk about the exceptional base locus. Already the proof of Bloch's theorem in Nevanlinna theory can probably be recast in this mold.

Is Semistability Really Necessary?

The proof of the main theorem of this talk didn't really need the full definition of semistability; it only used the condition on the degrees of subbundles for subbundles of rank 1 and corank 1. Would the following definition make sense, and would it be preserved under pull-back and symmetric power?

Definition: Let be a projective curve and let be a vector sheaf of rank on . Then is -semistable if the condition on degrees and ranks of subbundles holds for all full subbundles of rank and corank .

Again, what would be a reasonable representation-theoretic formulation of this definition?

Loose Ends

In the proof of the main theorem it remains to show that semistability is preserved under pull-back and under taking .

To show the first assertion, let be generically finite, and let be a semistable vector sheaf on . Suppose that is unstable. Let be a representation such that has a nonzero global section that vanishes somewhere. Let . Then taking norms gives a global section of