Periodic Reporting for period 1 - HPDR (Holomorphic Partial Differential Relations)
Reporting period: 2023-01-01 to 2025-06-30
A complex manifold is said to be an Oka manifold if the homotopy principle holds for holomorphic maps to it from Stein manifolds - every continuous map is homotopic to a holomorphic map, with approximation on suitable compact subsets and extension of maps from closed complex subvarieties. Similar properties must hold for continuous families of maps. Hence, complex analytic problems which can be expressed in terms of maps from Stein manifolds to Oka manifolds have only topological obstructions.
The classical part of the theory is the Oka-Grauert principle. Modern Oka theory was initiated by Mikhail Gromov who introduced more dominating holomorphic and algebraic sprays into the picture and showed that their existence imply Oka properties. The PI characterised Oka manifolds by a much simpler convex approximation property and showed that many natural Oka properties are equivalent. This unified the theory and expanded the class of Oka manifolds. Recently, Kusakabe showed that the Oka property is Zariski local in the holomorphic sense, and he gave many new examples.
The algebraic counterpart of the Oka principle holds only rarely. The closest analogue of algebraic Oka manifolds are the algebraically elliptic manifolds, which satisfy a relative Oka principle for maps from affine manifolds. In particular, every holomorphic map from an affine manifold which is homotopic to an algebraic map is a limit of algebraic maps.
For complete hermitian or Kähler manifolds, there seems to be a relationship between the Oka property and metric positivity, mirroring the connection between hyperbolicity and metric negativity. Mok's solution of the generalized Frankel conjecture implies that every compact Kähler manifold with nonnegative holomorphic bisectional curvature is Oka. It is an open problem whether the positivity of holomorphic sectional curvature is also related to Oka properties.
Another major open problem concerns the relationship between Oka properties of domains in complex manifolds and pseudoconcavity.
Oka theory has major applications in the theory of minimal and superminimal surfaces and in complex contact geometry. The main point concerning minimal surfaces in Euclidean spaces is that their derivatives have range in the null-quadric, which is an algebraically elliptic manifold. The algebraic aspect of Oka theory found applications to the construction of complete minimal surfaces of finite total curvature, which are of major interest in the field.
The classical part of the theory is the Oka-Grauert principle. Modern Oka theory was initiated by Mikhail Gromov who introduced more dominating holomorphic and algebraic sprays into the picture and showed that their existence imply Oka properties. The PI characterised Oka manifolds by a much simpler convex approximation property and showed that many natural Oka properties are equivalent. This unified the theory and expanded the class of Oka manifolds. Recently, Kusakabe showed that the Oka property is Zariski local in the holomorphic sense, and he gave many new examples.
The algebraic counterpart of the Oka principle holds only rarely. The closest analogue of algebraic Oka manifolds are the algebraically elliptic manifolds, which satisfy a relative Oka principle for maps from affine manifolds. In particular, every holomorphic map from an affine manifold which is homotopic to an algebraic map is a limit of algebraic maps.
For complete hermitian or Kähler manifolds, there seems to be a relationship between the Oka property and metric positivity, mirroring the connection between hyperbolicity and metric negativity. Mok's solution of the generalized Frankel conjecture implies that every compact Kähler manifold with nonnegative holomorphic bisectional curvature is Oka. It is an open problem whether the positivity of holomorphic sectional curvature is also related to Oka properties.
Another major open problem concerns the relationship between Oka properties of domains in complex manifolds and pseudoconcavity.
Oka theory has major applications in the theory of minimal and superminimal surfaces and in complex contact geometry. The main point concerning minimal surfaces in Euclidean spaces is that their derivatives have range in the null-quadric, which is an algebraically elliptic manifold. The algebraic aspect of Oka theory found applications to the construction of complete minimal surfaces of finite total curvature, which are of major interest in the field.
We found several new Oka manifolds. Among the new examples are Markov cubic surfaces in C3. Their algebraic automorphism groups are known to be discrete and act transitively on the set of points with integer components. By methods of Oka theory we showed that the tangent bundles of these surfaces are generated by algebraic vector fields, so their holomorphic automorphism groups are non-discrete and act transitively.
We found surprisingly small pseudoconcave Oka domains in Euclidean spaces Cn, n > 1. Indeed, most concave domains in Cn which are only slightly bigger than a halfspace are Oka. If E is a closed set in Cn whose projective closure avoids a hyperplane L and is polynomially convex in the complement of L, then the complement of E in Cn is an Oka domain. If E is a closed convex set in Cn contained in a closed halfspace H such that E ∩ bH is nonempty and bounded then Cn \E contains images of proper holomorphic maps X → Cn from any Stein manifold X of dimension < n. We constructed proper holomorphic embeddings of any Stein manifold of dimension n ≥ 1 in a complex Euclidean space of dimension 2n+1 with surprisingly small limit sets.
We studied Oka properties of disc tubes in semipositive hermitian line bundles (E,h) on a compact complex manifold X of dimension >1. Assuming that each point x ∈ X admits a divisor D whose complement X \ D is a Stein neighbourhood of x with the density property, we proved that the disc bundle {e ∈ E : |e| < 1} is Oka. This holds for any ample line bundle on a rational homogeneous manifold of dimension > 1, a class of manifolds including all projective spaces, Grassmannians, and flag manifolds. This contributes to the heuristic principles that the Oka property is related to metric positivity and that Oka domains are pseudoconcave.
Complex curves are of special importance in complex geometry. We introduced a new class of Oka-1 manifolds satisfying the Oka properties for maps from all open Riemann surfaces. We showed that a complex manifold densely dominable by a Euclidean space is an Oka-1 manifold. In particular, all Kummer surfaces and all elliptic K3 surfaces are Oka-1 manifolds. The class of Oka-1 manifolds is invariant under Oka maps inducing a surjective homomorphism of fundamental groups. We studied the algebraic version of the Oka-1 condition, showing that it is a birational invariant for compact algebraic manifolds and holds for all rational manifolds.
We developed the Oka theory for maps from families of complex structures on smooth open surfaces to any Oka manifold. We expect that similar results can be obtained for suitably tame families of Stein manifolds of higher dimension. These new techniques connect Oka theory to Teichmüller theory. They enable the construction of continuous or smooth families of directed holomorphic immersions, minimal surfaces, holomorphic Legendrian curves, etc.
We obtained the homotopy principle for group equivariant minimal surfaces in Euclidean spaces Rn for discrete groups acting on an open Riemann surface by holomorphic automorphisms, and acting on Rn by rigid maps. This seems to be the first results of its kind for surfaces of higher genus. The main problem was to control periods of their derivative maps in connection with the group equivariance property.
A new Kobayashi-type intrinsic metric on domains in Euclidean spaces was introduced by using conformal minimal discs. It describes the fastest rate of growth of minimal surfaces in the domain. We obtained new results on Gromov hyperbolicity for the Kobayashi and minimal metrics. It was shown that every bounded strongly minimally convex domain in Rn is Gromov hyperbolic.
We found surprisingly small pseudoconcave Oka domains in Euclidean spaces Cn, n > 1. Indeed, most concave domains in Cn which are only slightly bigger than a halfspace are Oka. If E is a closed set in Cn whose projective closure avoids a hyperplane L and is polynomially convex in the complement of L, then the complement of E in Cn is an Oka domain. If E is a closed convex set in Cn contained in a closed halfspace H such that E ∩ bH is nonempty and bounded then Cn \E contains images of proper holomorphic maps X → Cn from any Stein manifold X of dimension < n. We constructed proper holomorphic embeddings of any Stein manifold of dimension n ≥ 1 in a complex Euclidean space of dimension 2n+1 with surprisingly small limit sets.
We studied Oka properties of disc tubes in semipositive hermitian line bundles (E,h) on a compact complex manifold X of dimension >1. Assuming that each point x ∈ X admits a divisor D whose complement X \ D is a Stein neighbourhood of x with the density property, we proved that the disc bundle {e ∈ E : |e| < 1} is Oka. This holds for any ample line bundle on a rational homogeneous manifold of dimension > 1, a class of manifolds including all projective spaces, Grassmannians, and flag manifolds. This contributes to the heuristic principles that the Oka property is related to metric positivity and that Oka domains are pseudoconcave.
Complex curves are of special importance in complex geometry. We introduced a new class of Oka-1 manifolds satisfying the Oka properties for maps from all open Riemann surfaces. We showed that a complex manifold densely dominable by a Euclidean space is an Oka-1 manifold. In particular, all Kummer surfaces and all elliptic K3 surfaces are Oka-1 manifolds. The class of Oka-1 manifolds is invariant under Oka maps inducing a surjective homomorphism of fundamental groups. We studied the algebraic version of the Oka-1 condition, showing that it is a birational invariant for compact algebraic manifolds and holds for all rational manifolds.
We developed the Oka theory for maps from families of complex structures on smooth open surfaces to any Oka manifold. We expect that similar results can be obtained for suitably tame families of Stein manifolds of higher dimension. These new techniques connect Oka theory to Teichmüller theory. They enable the construction of continuous or smooth families of directed holomorphic immersions, minimal surfaces, holomorphic Legendrian curves, etc.
We obtained the homotopy principle for group equivariant minimal surfaces in Euclidean spaces Rn for discrete groups acting on an open Riemann surface by holomorphic automorphisms, and acting on Rn by rigid maps. This seems to be the first results of its kind for surfaces of higher genus. The main problem was to control periods of their derivative maps in connection with the group equivariance property.
A new Kobayashi-type intrinsic metric on domains in Euclidean spaces was introduced by using conformal minimal discs. It describes the fastest rate of growth of minimal surfaces in the domain. We obtained new results on Gromov hyperbolicity for the Kobayashi and minimal metrics. It was shown that every bounded strongly minimally convex domain in Rn is Gromov hyperbolic.
Our achievements include a wealth of new methodologies and interdisciplinary developments. One is a new method for constructing group equivariant minimal surfaces in Euclidean spaces by using a combination of nontrivial methods from Oka theory, group theory, and convex integration theory. Another one is the introduction and study of the new class of Oka-1 manifolds which admit plenty of noncompact complex curves. A newly observed phenomenon is that the construction of holomorphic maps from open Riemann surfaces can be localized. This observation, and the gluing methods from Oka theory, were used to show that every densely dominable complex manifold is Oka-1. This yields several interesting classes of Oka-1 surfaces which are not known to be Oka. The third one is the development of Oka theory for families of holomorphic maps from families of open Riemann surfaces to any Oka manifold. The results promise numerous applications to the construction of families of holomorphic curves and minimal surfaces with prescribed conformal types, satisfying various additional properties. This also establishes a connection between Oka theory and Teichmüller theory.lishes the first known connection between Oka theory and the Teichmüller theory, which will be explored.