The Construction Problem in Kähler Geometry

Abstract

One of the most surprising things in algebraic geometry is the fact that algebraic varieties over the complex numbers benefit from a collection of metric properties which strongly influence their topological and geometric shapes. The existence of a Kähler metric leads to all sorts of Hodge theoretical restrictions on the homotopy types of algebraic varieties. On the other hand, a sparse collection of examples shows that the remaining liberty is nontrivially large. Paradoxically, with all of this information, the research field remains as wide open as it was many decades ago, because the gap between the known restrictions, and the known examples of what can occur, only seems to grow wider and wider the more closely we look at it.In spite of the differential-geometric nature of the questions and methods, the origins of the situation are very algebraic. We look at subvarieties of projective space over the complex numbers. The main over-arching problem in algebraic geometry is to understand the classification of algebro-geometric objects. The topology of the usual complex-valued points of a variety plays an important role, because the topological type is a locally constant function on any classifying space. Thus the partitioning of the classification problem according to topological type provides a coarse, and more calculable, alternative to the partition by connected components. Furthermore, the topology of a variety strongly influences its geometric properties. A sociological observation is that the quest for understanding the topology of algebraic varieties has led to a rich set of techniques which found applications elsewhere, even in physics.Perhaps a word about the choice of ground field is appropriate. One could also look at the shapes of the real points of real algebraic varieties. However, by Weierstrass approximation, pretty much anything can arise if you let the degree get big enough. Thus the classical question in this case is "which shapes can occur for a given degree?" This is so much more difficult 1

On the construction problem for Hodge numbers

“…In [20] it is speculated that the middle Hodge numbers should probably dominate the outer ones. In our third main theorem of this paper, we classify all nontrivial dominations among Hodge numbers in any given dimension.…”

“…It is a very difficult and wide open problem to determine all universal inequalities among Hodge numbers in a fixed dimension, see [20]. In Theorem 5 we basically solved this problem for inequalities of the form (1.5).…”

The construction problem for Hodge numbers modulo an integer

“…(2) While Hodge theory places severe restrictions on the geometry and topology of Kähler manifolds, Simpson points out in [Sim04] that very little is known to which extent the theoretically possible phenomena actually occur. This leads to the following construction problem for Hodge numbers: Question 1.…”

Cohomology algebras in symplectic, Kähler and algebraic geometry

“…Let us consider a K3 surface S. It is a compact Kähler surface with b 1 = 0, b 2 = 22 and the signature of the intersection on H 2 (S, R) is (3,19).…”