# Meromorphic extension of local defining equations of a complex submanifold

let $M$ be a smooth compact complex manifold of dimension $m$ and $N\subset M$ a smooth complex submanifold of dimension $1\leq n \leq m-2$. Covering $N$ with well chosen open sets of $M$ we can always find $m-n$ holomorphic functions that locally cut $N$ in $M$.

My question(s) is (are) the following: can we extend these holomorphic functions defined on open sets of $M$ to global meromorphic functions on $M$? If we can't find global meromorphic functions can we extend these holomorphic functions defined on open sets of $M$ to meromorphic functions on an open neighborhood in $M$ of $N$?

Does the answer change if the manifold is Kahler?

Thank you in advance.

## 1 Answer

There are manifolds without non-constant global meromorphic functions, such as generic K3 or a torus. Among these K3 surfaces, there are ones with (-2)-curves, which give a counterexample to the question. It is not hard to see that a K3 admits a -2-curve if and only if it has a vector with square -2 in its Picard group, and no non-constant meromorphic functions if its Picard lattice is negative definite. Global Torelli theorem implies that any sublattice of $H_2^3+(-E_8)^2$ is realised as a Picard lattice, allowing one to find such an example.

Edited the answer. If you require your subvariety to be not a divisor, there are still many examples of such, say, a product of two tori or to K3 K3 with no subvarieties and no global meromorphic functions. For a more interesting example, you may take a Lagrangian projective subspace in a hyperkahler manifold and deform the complex structures in such a way that this subvariety remains complex analytic, and no global meromorphic functions remain. This is explained in this paper: http://arxiv.org/abs/1401.0479

Edited again. There are subvarieties in complex manifolds admitting no global holomorphic functions in any neighbourhood, though I don't know any examples for Kahler situation. Examples of such subvarieties (in fact, rational curves) for twistor spaces are given in this paper: http://arxiv.org/abs/1211.5765

• Thanks for the answer, but in this case the submanifold is a divisor $n=m-1$, are there counterexamples anyway? In the case of K3 surface with (-2)-curve can we find meromorphic extensions on a "small" neighborhood of the (-2)-curve? – Italo May 20 '14 at 10:36
• edited the answer. there are still many examples. – Misha Verbitsky May 20 '14 at 10:59