Asymptotically holomorphic embeddings in projective spaces and applications.
by
Vicente Muñoz
Universidad Autónoma de Madrid, Spain
Coauthors: Fran Presas (Univ. Complutense Madrid), Ignacio Sols (Univ. Complutense de Madrid)

Oral Communication

Suppose that we have the following set-up. Let (M, \omega) be a symplectic manifold with [\omega]/2\pi an integer cohomology class. Then we consider an hermitian line bundle L --> M with c₁(L)=[\omega]/2\pi and a connection on M with curvature -i\omega.

Endow M with a compatible almost complex structure J.

In his outbreaking work [D1] Donaldson has proved the existence of a sequence of sections s_k of L^\otimesk, for k large enough, which is asymptotically holomorphic (i.e. |[`(\partial)] s_k|=O(k^-1/2)) and transverse to zero in a controlled sense (i.e. there is some \eta > 0 such that |\partials_k| > \eta at the points where |s_k| < \eta).

The sets of zeroes of s_k, W_k=Z(s_k), are then nearly complex, in the sense that their tangent hyperplanes are within an angle O(k^-1/2) of being complex. In particular this gives a construction of symplectic codimension 2 submanifolds.

Later on, Auroux [A1] refined this circle of ideas to extend the result by introducing an hermitian rank r bundle E --> M, constructing sections s_k of E\otimesL^\otimesk which are asymptotically holomorphic and \eta-transverse to zero (meaning in this case that \partials_k multiplies the norms of tangent vectors by an amount at least \eta, for all the vectors in a suitable suplementary to the kernel).

This provides us with W_k=Z(s_k) which are nearly complex, hence symplectic, codimension 2r submanifolds of M.

In further work, Donaldson [D2] constructs two sequences of sections (s_k⁰, s_k¹) of L^\otimesk giving a map M --> CP¹ which is a symplectic Lefschetz fibration, and Auroux [A2] does the analogue with three sequences of sections to get a ramified asymptotically holomorphic covering M --> CP².

We extend the technique to the somewhat simpler case of constructing 2n+2 sequences of sections s_k=(s_k⁰, ... , s_k²ⁿ⁺¹) which yield an asymptotically holomorphic sequence of embeddings \phi_k:M \hookrightarrowCP²ⁿ⁺¹. For this we need to obtain an asymptotically holomorphic sequence s_k satisfying two conditions:

1. \gamma-projectizability: |s_k| >= \gamma at all points and for all k. This is necessary to define maps \phi_k=P(s_k) from M to CP²ⁿ⁺¹ and to control the behaviour of such maps uniformly in k.

2. \gamma-genericity: |\Lambdaⁿ d \phi_k| >= \gamma at all points and for all k. With this the differential d\phi_k is injective and moreover it multiplies the length of tangent vectors by an amount bounded below uniformly in k.

The method for achieving these conditions is an extension of that of Donaldson-Auroux. One starts with any asymptotically holomorphic sequence and perturb it to satisfy the requirements. One needs to construct local perturbations to see that the result can be obtained in a neighbourhood of any point and then one has to globalize carefully the perturbations to get a global sequence of sections.

In the case where we have an hermitian rank r bundle E --> M, one can construct sequences of sections s_k=(s_k¹, ... , s_k^N) of the bundles E\otimesL^\otimesk which produce asymptotically holomorphic embeddings \phi_k=Gr(s_k):M \hookrightarrow Gr(r, N) in the grassmannians.

We propose two main applications of constructions of sympletic submanifolds of M which extend constructions in the Kähler category:

1. One may embed M into CP²ⁿ⁺¹ with the asymptotically holomorphic maps constructed above and then cut down the image with a holomorphic submanifold N subset CP²ⁿ⁺¹. We prove that for a fixed holomorphic submanifold N one can perturb the sequence of sections s_k so as to obtain asymptotically holomorphic embeddings \phi_k=P(s_k) of M which cut N transversally by an estimated amount of transversality. Hence N_k=\phi_k^-1(N) subset M are nearly holomorphic, in particular symplectic, submanifolds.

2. Determinantal subvarieties. Let E and F be two hermitian bundles over M of ranks s and r respectively. We construct bundle maps f_k: E --> F\otimesL^\otimes2k, for large enough k, such that the loci of degeneracy of f,

Wi (f) = {x in M ê ê rank fk(x) <= i},

are smooth and symplectic submanifolds off W_i-1(f). For this embed M diagonally into the product of grassmannians Gr(r, N)×Gr(s, N) by using asymptotically holomorphic sequences of sections of E^* \otimesL^\otimesk and F \otimesL^\otimesk. Then fix a generic map between the universal bundles of the grassmannians and consider the loci of degeneracy of it. The embeddings can be perturbed to cut every stratum transversely, being careful of what happens near the singular points at each stratum.

This has applications to the case E=C^s, to considering the dependence loci of s sections of a rank r bundle F.

References

[A1] D. Auroux. Asymptotically holomorphic families of symplectic submanifolds. Geom. Funct. Anal., 7, 971-995 (1997).

[A2] D. Auroux. Théorèmes de structure des variétés symplectiques compactes via des techniques presque complexes. Ph. D. Thesis. (1999).

[D1] S. K. Donaldson. Symplectic submanifolds and almost-complex geometry. J. Diff. Geom., 44, 666-705 (1996).

[D2] S. K. Donaldson. Lefschetz pencils in Symplectic Geometry. Preprint (1999).

Date received: December 22, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadq-04.