Kähler Geometry of Toric Varieties and Extremal Metrics

Abstract

1991 Mathematics Subject Classification: Primary 53C55, Secondary 14M25 53C25 58F05 A (symplectic) toric variety X, of real dimension In, is completely determined by its moment polytope A C K n . Recently Guillemin gave an explicit combinatorial way of constructing "toric" Kahler metrics on X, using only data on A. In this paper, differential geometric properties of these metrics are investigated using Guillemin's construction. In particular, a nice combinatorial formula for the scalar curvature R is given, and the Euler-Lagrange condition for such "toric" metrics being extremal (in the sense of Calabi) is proven to be R being an affine function on A C 1". A construction, due to Calabi, of a 1-parameter family of extremal Kahler metrics of non-constant scalar curvature on CP 2 jjCP 2 is recast very simply and explicitly using Guillemin's approach. Finally, we present a curious combinatorial identity for convex polytopes A C R n that follows from the wellknown relation between the total integral of the scalar curvature of a Kahler metric and the wedge product of the first Chern class of the underlying complex manifold with a suitable power of the Kahler class.

Convergence of Kähler to real polarizations on flag manifolds via toric degenerations

“…Starting from a Delzant polytope, we construct a symplectic toric manifold in Subsection 5.1 and a complex toric manifold in Subsection 5.2. We identify them according to a choice of symplectic potentials due to [Ab1,Ab2,Gu1,Gu2] in Subsection 5.3. We also review certain deformation of toric Kähler structures by changing symplectic potentials, which was introduced in [BFMN].…”

“…Then the principle can be understood naturally if there is a family {P Js } s∈[0,∞) of Kähler polarizations on M with P J 0 = P J which converges to P µ in the sense that there exists a basis {σ m s } m∈BS(Pµ) of H(P Js ) for each s ∈ [0, ∞) such that, for each m ∈ BS(P µ ), σ m s converges to a delta-function section supported on the Bohr-Sommerfeld fiber µ −1 (m) as s goes to ∞. In [BFMN], the authors carried out such a construction in the case of a non-singular projective toric variety by changing symplectic potentials, an important notion in the deformation theory of toric Kähler structures due to Guillemin [Gu1,Gu2] and Abreu [Ab1,Ab2].…”

Melting of the Euclidean Metric to Negative Scalar Curvature

“…This is similar to the solution of the equation (2). Again, one can properly choosec small and α i j so that U ij become smooth functions with compact support in µ −1 (S c ) and thatḡ t , t > 0, is an almost Kähler metric which is non-Kähler, i.e., 1 2 |DJ| 2 = s * − s = 0 somewhere. Indeed, either by direct computation on a component of DJ or by an argument using [5, Section 4], one can find {U ij } so that near some chosen pointḡ t is non-Kähler for any small t.…”

“…In this geometry, for the canonical hermitian connection ∇ determined by J we have the corresponding hermitian scalar curvature s ∇ . It proves to be equal to 1 2 (s * + s), where s * is the starscalar curvature. It is known that s * − s = 1 2 |DJ| 2 , where D is the Levi-Civita connection.…”

“…Example ( [1]). Consider the complex projective space CP n with the FubiniStudy metric g F S in homogeneous coordinates [z 0 , z 1 , .…”