Recent developments in toric geometry.

*(English)*Zbl 0899.14025
Kollár, János (ed.) et al., Algebraic geometry. Proceedings of the Summer Research Institute, Santa Cruz, CA, USA, July 9–29, 1995. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 62(pt.2), 389-436 (1997).

The title of the paper says everything about its contents. While every section from §2–§10 is devoted to a special subject in toric geometry, §1 contains a short summary of the basic notions, and §11 collects short remarks concerning recent developments not covered by the remaining text. The whole paper gives a wonderful survey which will be useful for everybody interested in toric varieties. The contents of the sections is the following:

§2 presents the construction of toric varieties as a quotient of the affine space minus an exceptional set by a torus \(G\). The advantage of this construction is that the group and its action does not depend on the fan, but only on its one-dimensional cones. Moreover, one has a homogeneous coordinate ring allowing to define toric subvarieties by global equations.

Let \(\Sigma\) be a simplicial, projective fan. In §3, the author describes the Kähler cone contained in \(H^2(X,\mathbb{R})\). Identifying the latter with the space of functions \(\Sigma^{(1)}\to\mathbb{R}\) divided out by the linear ones, the Kähler cone consists of those functions inducing, via linear continuation on the cones, a strong convex map \(|\Sigma|\to\mathbb{R}\).

In §4 toric varieties \(X\) are constructed via the moment map of a Hamiltonian action of the maximal compact subgroup of \(G\) on a symplectic manifold (“symplectic reduction”). In particular, on \(X\) there is still a Hamiltonian action of the real torus left. Its moment map provides the defining polytope as its image.

There is a rather straight way of defining projective toric varieties directly from the polytope: Assign to each lattice point a homogeneous variable and translate the affine relations between the lattice points into binomial equations in these variables. This approach, being introduced in §5, is very convenient for studying hypersurfaces in tori. Their natural compactifications are contained in the toric varieties assigned to their Newton polyhedra. §6 deals with the mixed Hodge structure of those hypersurfaces. §7 is devoted to a combinatorial object, the secondary fan. It collects all the Kähler cones of toric varieties \(X_\Sigma\) built from different fans \(\Sigma\), but with fixed set \(\Sigma^{(1)}\) of one-dimensional cones. Moreover, the paper describes the relations of the secondary fan to the notion of the secondary polytope of a given polytope.

In §8, the reader finds a summary of Batyrev’s famous work on reflexive polytopes and mirror symmetry [cf. V. V. Batyrev, J. Algebra Geom. 3, No. 3, 493-535 (1994; Zbl 0829.14023)]. Reflexive polytopes are defined as those polytopes \(P\) such that both \(P\) and its polar \(P^\vee\) have only lattice vertices. These polytopes give rise to Fano varieties, and the corresponding hypersurfaces are Calabi-Yau. Now, the polarity between the polytopes turn into mirror symmetry between these Calabi-Yau varieties, at least in the sense of the symmetry of Hodge numbers. This subject will be continued in §10 with a more general discussion of the ideas and conjectures concerning mirror symmetry. The subject of §9 are discriminants and resultants assigned to a fixed set \(\mathcal A\) of monomials. While the \(\mathcal A\)-discriminants are closely related to the generalized \(\mathcal A\)-hypergeometric functions of Gelfand-Kapranov-Zelevinsky, the combinatorial description of the monomials occurring in the \(\mathcal A\)-resultant leads to the concept of secondary polytopes again.

Finally, here are some of the topics discussed briefly in §11: Embeddings into toric varieties, intersection theory on toric varieties, counting lattice points, rational points on toric varieties defined over number fields, residues on toric varieties, deformations and resolutions of toric singularities.

For the entire collection see [Zbl 0882.00033].

§2 presents the construction of toric varieties as a quotient of the affine space minus an exceptional set by a torus \(G\). The advantage of this construction is that the group and its action does not depend on the fan, but only on its one-dimensional cones. Moreover, one has a homogeneous coordinate ring allowing to define toric subvarieties by global equations.

Let \(\Sigma\) be a simplicial, projective fan. In §3, the author describes the Kähler cone contained in \(H^2(X,\mathbb{R})\). Identifying the latter with the space of functions \(\Sigma^{(1)}\to\mathbb{R}\) divided out by the linear ones, the Kähler cone consists of those functions inducing, via linear continuation on the cones, a strong convex map \(|\Sigma|\to\mathbb{R}\).

In §4 toric varieties \(X\) are constructed via the moment map of a Hamiltonian action of the maximal compact subgroup of \(G\) on a symplectic manifold (“symplectic reduction”). In particular, on \(X\) there is still a Hamiltonian action of the real torus left. Its moment map provides the defining polytope as its image.

There is a rather straight way of defining projective toric varieties directly from the polytope: Assign to each lattice point a homogeneous variable and translate the affine relations between the lattice points into binomial equations in these variables. This approach, being introduced in §5, is very convenient for studying hypersurfaces in tori. Their natural compactifications are contained in the toric varieties assigned to their Newton polyhedra. §6 deals with the mixed Hodge structure of those hypersurfaces. §7 is devoted to a combinatorial object, the secondary fan. It collects all the Kähler cones of toric varieties \(X_\Sigma\) built from different fans \(\Sigma\), but with fixed set \(\Sigma^{(1)}\) of one-dimensional cones. Moreover, the paper describes the relations of the secondary fan to the notion of the secondary polytope of a given polytope.

In §8, the reader finds a summary of Batyrev’s famous work on reflexive polytopes and mirror symmetry [cf. V. V. Batyrev, J. Algebra Geom. 3, No. 3, 493-535 (1994; Zbl 0829.14023)]. Reflexive polytopes are defined as those polytopes \(P\) such that both \(P\) and its polar \(P^\vee\) have only lattice vertices. These polytopes give rise to Fano varieties, and the corresponding hypersurfaces are Calabi-Yau. Now, the polarity between the polytopes turn into mirror symmetry between these Calabi-Yau varieties, at least in the sense of the symmetry of Hodge numbers. This subject will be continued in §10 with a more general discussion of the ideas and conjectures concerning mirror symmetry. The subject of §9 are discriminants and resultants assigned to a fixed set \(\mathcal A\) of monomials. While the \(\mathcal A\)-discriminants are closely related to the generalized \(\mathcal A\)-hypergeometric functions of Gelfand-Kapranov-Zelevinsky, the combinatorial description of the monomials occurring in the \(\mathcal A\)-resultant leads to the concept of secondary polytopes again.

Finally, here are some of the topics discussed briefly in §11: Embeddings into toric varieties, intersection theory on toric varieties, counting lattice points, rational points on toric varieties defined over number fields, residues on toric varieties, deformations and resolutions of toric singularities.

For the entire collection see [Zbl 0882.00033].

Reviewer: Klaus Altmann (Berlin)

##### MSC:

14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |