Calabi-Yau Volumes and Reflexive Polytopes

We consider families ${\cal F}(\Delta)$ consisting of complex $(n-1)$-dimensional projective algebraic compactifications of $\Delta$-regular affine hypersurfaces $Z_f$ defined by Laurent polynomials $f$ with a fixed $n$-dimensional Newton polyhedron $\Delta$ in $n$-dimensional algebraic torus ${\bf T} =({\bf C}^*)^n$. If the family ${\cal F}(\Delta)$ defined by a Newton polyhedron $\Delta$ consists of $(n-1)$-dimensional Calabi-Yau varieties, then the dual, or polar, polyhedr...

We study a variational problem whose critical point determines the Reeb vector field for a Sasaki-Einstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the Einstein-Hilbert action, restricted to a space of Sasakian metrics on a link L in a Calabi-Yau cone X, is the volume functional, which in fact is a function on the space of Reeb vector fields. We relate this function both to the Duistermaa...

We show that the complex projective space has maximal degree (volume) among all n-dimensional Kahler-Einstein Fano manifolds admitting a holomorphic C^*-action with a finite number of fixed points. The toric version of this result, translated to the realm of convex geometry, thus confirms Ehrhart's volume conjecture for a large class of rational polytopes, including duals of lattice polytopes. The case of spherical varieties/multiplicity free symplectic manifolds is also disc...

The diffeomorphism class of simply-connected smooth Calabi-Yau threefolds with torsion-free cohomology is determined via certain basic topological invariants: the Hodge numbers, the triple intersection form, and the second Chern class. In the present paper, we shed some light on this classification by placing bounds on the number of diffeomorphism classes present in the set of smooth Calabi-Yau threefolds constructed from the Kreuzer-Skarke list of reflexive polytopes up to P...

We show that the dual of the Cayley cone, associated to a Minkowski sum decomposition of a reflexive polytope, contains a reflexive polytope admitting a nef-partition. This nef-partition corresponds to a Calabi-Yau complete intersection in a Gorenstein Fano toric variety degenerating to an ample Calabi-Yau hypersurface in another Fano toric variety. Using the Batyrev-Borisov mirror symmetry construction, we found the mirror contraction of a Calabi-Yau complete intersection to...

It is shown that any affine toric variety Y, which is Q-Gorenstein, admits a conical Ricci flat Kahler metric, which is smooth on the regular locus of Y. The corresponding Reeb vector is the unique minimizer of the volume functional on the Reeb cone of Y. The case when the vertex point of Y is an isolated singularity was previously shown by Futaki-Ono-Wang. The proof is based on an existence result for the inhomogeneous Monge-Ampere equation in real Euclidean space with expon...

We present the last missing details of our algorithm for the classification of reflexive polyhedra in arbitrary dimensions. We also present the results of an application of this algorithm to the case of three dimensional reflexive polyhedra. We get 4319 such polyhedra that give rise to K3 surfaces embedded in toric varieties. 16 of these contain all others as subpolyhedra. The 4319 polyhedra form a single connected web if we define two polyhedra to be connected if one of them...

After providing an explicit K-stability condition for a $\mathbb{Q}$-Gorenstein log spherical cone, we prove the existence and uniqueness of an equivariant K-stable degeneration of the cone, and deduce uniqueness of the asymptotic cone of a given complete $K$-invariant Calabi-Yau metric in the trivial class of an affine $G$-spherical manifold, $K$ being the maximal compact subgroup of $G$. Next, we prove that the valuation induced by $K$-invariant Calabi-Yau metrics on affi...

These lecture notes are an introduction to toric geometry. Particular focus is put on the description of toric local Calabi-Yau varieties, such as needed in applications to the AdS/CFT correspondence in string theory. The point of view taken in these lectures is mostly algebro-geometric but no prior knowledge of algebraic geometry is assumed. After introducing the necessary mathematical definitions, we discuss the construction of toric varieties as holomorphic quotients. We d...

We generalize the standard combinatorial techniques of toric geometry to the study of log Calabi-Yau surfaces. The character and cocharacter lattices are replaced by certain integral linear manifolds described by Gross, Hacking, and Keel, and monomials on toric varieties are replaced with the canonical theta functions which GHK defined using ideas from mirror symmetry. We describe the tropicalizations of theta functions and use them to generalize the dual pairing between the ...