Complex and Differential Geometry

In the last years there have been several new constructions of surfaces of general type with p _g = 0, and important progress on their classification. The present paper presents the status of the art on surfaces of general type with p _g = 0, and gives an updated list of the existing surfaces, in the case where K ² = 1,...,7. It also focuses on certain important aspects of this classification.

The usual structures of symplectic geometry (symplectic, contact, Poisson) make sense for complex manifolds; they turn out to be quite interesting on projective, or compact Kähler, manifolds. In these notes we review some of the recent results on the subject, with emphasis on the open problems and conjectures.

We introduce the notion of Kähler manifolds that are almost Einstein and we define a generalized mean curvature vector field along submanifolds in them. We prove that Lagrangian submanifolds remain Lagrangian, when deformed in direction of the generalized mean curvature vector field. For a Kähler manifold that is almost Einstein, and which in addition has a trivial canonical bundle, we show that the generalized mean curvature vector field of a Lagrangian submanifold is the dual vector field associated to the Lagrangian angle.

Let C be a cone in the space of algebraic curvature tensors. Moreover, let (M,g) be a compact Einstein manifold with the property that the curvature tensor of (M,g) lies in the interior of the cone C at each point on M. We show that (M,g) has constant sectional curvature if the cone C satisfies certain structure conditions.

Let (M,g(t)) be a solution to the Ricci flow on a closed Riemannian manifold. In this paper, we prove differential Harnack inequalities for positive solutions of nonlinear parabolic equations of the type We also comment on an earlier result of the first author on positive solutions of the conjugate heat equation under the Ricci flow.

In this paper we study the behaviour of the degree of the Fulton–Johnson class of a complete intersection under a blow–up with a smooth center under the assumption that the strict transform is again a complete intersection. Our formula is a generalization of the genus formula for singular curves in smooth surfaces.

A set of points in the projective plane is said to be Cremona special if its orbit with respect to the Cremona group of birational transformations consists of finitely many orbits of the projective group. This notion was extended by A. Coble to sets of points in higher–dimensional projective spaces and by S. Mukai to sets of points in the product of projective spaces. No classification of such sets is known in these cases. In the present article we survey Coble’s examples of Cremona special points in projective spaces and initiate a search for new examples in the case of products of projective spaces. We also extend to the new setting the classical notion of associated points sets.

We introduce an algebra of Schouten–commuting holomorphic polyvector fields on the moduli space of stable G-bundles over a curve by using invariant forms on the Lie algebra. The generators begin in degree three – we prove a vanishing theorem for degree two in the case of G = GL(n).

We show that a general n-dimensional polarized abelian variety (A,L) of a given polarization type and satisfying \(h^0(A,L)\geq \frac{8^n}{2}. \frac{n^n}{n !}\) is projectively normal. In the process, we also obtain a sharp lower bound for the volume of a purely onedimensional complex analytic subvariety in a geodesic tubular neighborhood of a subtorus of a compact complex torus.

Classifying Kähler-Einstein manifolds has progressed very far for compact manifolds. In the non-compact setting, a lot of encouraging results have been obtained, with the greatest gap of knowledge for the Ricci-flat case. This article wants to present the state of the art of classification and explain current problems and questions with respect to existence and uniqueness of complete Ricci-flat Kähler metrics.

The paper is a survey of some results about Weil algebras applicable in differential geometry, especially in some classification questions on bundles of generalized velocities and contact elements. Mainly, a number of claims concerning the form of subalgebras of fixed points of various Weil algebras are demonstrated.

In this note, I provide some detailed computation of constructing translating solutions from self-similar solutions for Lagrangian mean curvature flow discussed in [6] and explore the related geometric meanings. This method works for all mean curvature flows and has great potential to find other new translating solutions.

This is an expository article, which gives an overview about aspects of the theory of conformal holonomy. In particular, we announce a complete geometric description of compact Riemannian conformal manifolds with decomposable conformal holonomy representation. Furthermore, we discuss the relation to almost Einstein structures and generalised Fefferman constructions. Generically, the latter conformal geometries have irreducible conformal holonomy. Reduced conformal holonomy is related to the existence of solutions of certain overdetermined conformally covariant PDE systems. We explain this relation in a unified approach using BGG-sequences.

We consider spaces of plane curves in the setting of algebraic geometry and of singularity theory. On one hand there are the complete linear systems, on the other we consider unfolding spaces of bivariate polynomials of Brieskorn-Pham type. For suitable open subspaces we can define the bifurcation braid monodromy taking values in the Zariski resp. Artin braid group. In both cases we give the generators of the image. These results are compared with the corresponding geometric monodromy. It takes values in the mapping class group of braided surfaces. Our final result gives a precise statement about the interdependence of the two monodromy maps. Our study concludes with some implication with regard to the unfaithfulness of the geometric monodromy ([W]) and the – yet unexploited – knotted geometric monodromy, which takes the ambient space into account.

We survey recent results about the Torelli question for holomorphicsymplectic varieties. Following are the main topics. A Hodge theoretic Torelli theorem. A study of the subgroup WExc, of the isometry group of the weight 2 Hodge structure, generated by reflection with respect to exceptional divisors. A description of the birational Kähler cone as a fundamental domain for the WExc action on the positive cone. A proof of a weak version of Morrison’s movable cone conjecture. A description of the moduli spaces of polarized holomorphic symplectic varieties as monodromy quotients of period domains of type IV.

In 1965, Feder proved using a cohomological identity that any holomorphic immersion \(\tau : {\mathbb{P}^n} \rightarrow {\mathbb{P}^m}\) between complex projective spaces is necessarily a linear embedding whenever m < 2n. In 1991, Cao-Mok adapted Feder’s identity to study the dual situation of holomorphic immersions between compact complex hyperbolic space forms, proving that any holomorphic immersion \( f : X \rightarrow Y \) from an n-dimensional compact complex hyperbolic space form X into any m-dimensional complex hyperbolic space form Y must necessarily be totally geodesic provided that m < 2n. We study in this article singularity loci of generically injective holomorphic immersions between complex hyperbolic space forms. Under dimension restrictions, we show that the open subset U over which the map is a holomorphic immersion cannot possibly contain compact complex-analytic subvarieties of large dimensions which are in some sense sufficiently deformable. While in the finitevolume case it is enough to apply the arguments of Cao-Mok, the main input of the current article is to introduce a geometric argument that is completely local. Such a method applies to \( f : X \rightarrow Y \) in which the complex hyperbolic space form X is possibly of infinite volume. To start with we make use of the Ahlfors-Schwarz Lemma, as motivated by recent work of Koziarz-Mok, and reduce the problem to the local study of contracting leafwise holomorphic maps between open subsets of complex unit balls. Rigidity results are then derived from a commutation formula on the complex Hessian of the holomorphic map.

The cotangent bundle of a non-uniruled projective manifold is generically nef, due to a theorem of Miyaoka. We show that the cotangent bundle is actually generically ample, if the manifold is of general type and study in detail the case of intermediate Kodaira dimension. Moreover, manifolds with generically nef and ample tangent bundles are investigated as well as connections to classical theorems on vector fields on projective manifolds.

We discuss the known evidence for the conjecture that the Dolbeault cohomology of nilmanifolds with left-invariant complex structure can be computed as Lie-algebra cohomology and also mention some applications.

We show that if X is a smooth rationally connected threefold and C is a smooth projective curve then C can be embedded in X. Furthermore, a version of this property characterises rationally connected varieties of dimension at least 3. We give some details about the toric case.

We describe various classes of submanifolds of a Poisson manifold M, both in terms of tensors on M and of constraints: coisotropic submanifolds, Poisson- Dirac submanifolds (which inherit a Poisson structure), and the very general class of pre-Poisson submanifolds. We discuss embedding results for these classes of submanifolds, quotient Poisson algebras associated to them, and their relationship to subgroupoids of the symplectic groupoid of M.