Showing papers on "Toric variety published in 1998"
TL;DR: In this article, the authors introduce a method of constructing the virtual cycle of any scheme associated with a tangent-obstruction complex and apply this method to construct the virtual moduli cycle of the moduli of stable maps from n-pointed genus g curves to any smooth projective variety.
Abstract: We introduce a method of constructing the virtual cycle of any scheme associated with a tangent-obstruction complex. We apply this method to constructing the virtual moduli cycle of the moduli of stable maps from n-pointed genus g curves to any smooth projective variety. As an application, we give an algebraic definition of GW-invariants for any smooth projective variety.
610 citations
TL;DR: In this article, it was shown that toric geometry can be used to translate a brane configuration to geometry and that the skeletons of toric space are identified with the brane configurations.
Abstract: We show that toric geometry can be used rather effectively to translate a brane configuration to geometry. Roughly speaking the skeletons of toric space are identified with the brane configurations. The cases where the local geometry involves hypersurfaces in toric varieties (such as P^2 blown up at more than 3 points) presents a challenge for the brane picture. We also find a simple physical explanation of Batyrev's construction of mirror pairs of Calabi-Yau manifolds using T-duality.
441 citations
01 Jan 1998
TL;DR: In this paper, a generalized mirror conjecture for non-negative complete intersections in symplectic toric manifolds has been proved, where the PDE system describing quantum cohomology of such a manifold is expressed in terms of suitable hypergeometric functions.
Abstract: We prove a generalized mirror conjecture for non-negative complete intersections in symplectic toric manifolds. Namely, we express solutions of the PDE system describing quantum cohomology of such a manifold in terms of suitable hypergeometric functions.
432 citations
TL;DR: In this article, a 1-parameter family of extremal Kahler metrics of non-constant scalar curvature on convex polytopes is recast using Guillemin's approach.
Abstract: A (symplectic) toric variety X, of real dimension 2n, is completely determined by its moment polytope Δ ⊂ ℝn Recently Guillemin gave an explicit combinatorial way of constructing "toric" Kahler metrics on X, using only data on Δ 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 Δ ⊂ ℝn A construction, due to Calabi, of a 1-parameter family of extremal Kahler metrics of non-constant scalar curvature on is recast very simply and explicitly using Guillemin's approach Finally, we present a curious combinatorial identity for convex polytopes Δ ⊂ ℝn that follows from the well-known 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
248 citations
TL;DR: In this paper, the authors use local mirror symmetry in type IIA string compactifications on Calabi-Yau n + 1 folds Xn+1 to construct vector bundles on (possibly singular) elliptically fibered Calabi Yau n-folds Zn.
Abstract: We use local mirror symmetry in type IIA string compactifications on Calabi–Yau n + 1 folds Xn+1 to construct vector bundles on (possibly singular) elliptically fibered Calabi–Yau n-folds Zn. The interpretation of these data as valid classical solutions of the heterotic string compactified on Zn proves Ftheory/heterotic duality at the classical level. Toric geometry is used to establish a systematic dictionary that assigns to each given toric n+1-fold Xn+1 a toric n fold Zn together with a specific family of sheafs on it. This allows for a systematic construction of phenomenologically interesting d = 4 N = 1 heterotic vacua, e.g. on deformations of the tangent bundle, with grand unified and SU(3)× SU(2) gauge groups. As another application we find nonperturbative gauge enhancements of the heterotic string on singular Calabi–Yau manifolds and new non-perturbative dualities relating heterotic compactifications on different manifolds. November 1998 1 berglund@itp.ucsb.edu 2 Peter.Mayr@cern.ch
114 citations
TL;DR: The notion of genericity for lattice ideals was introduced in this paper, which includes ideals defining toric varieties, where the generators are generic with respect to their exponents, not their coefficients.
Abstract: Let S = k[x1, . . . , xn] be a polynomial ring over a field k and I a homogeneous ideal in S. A basic problem in commutative algebra is to construct the minimal free resolution FI of S/I over S. The resolution is nicely structured and simple when I is a complete intersection: in this case FI is the Koszul complex. Complete intersections are ideals whose generators have sufficiently general coefficients, so they might be regarded as generic among all ideals. Yet there is another, entirely different, notion of genericity: ideals whose generators are generic with respect to their exponents – not their coefficients. This point of view was developed for monomial ideals in [BPS]. In the present work we introduce a notion of genericity for lattice ideals, which include ideals defining toric varieties. If L is any sublattice of Z, then its associated lattice ideal in S is IL := 〈xa − x : a,b ∈ N and a− b ∈ L 〉, where monomials are denoted x = x1 1 · · ·xan n for a = (a1, . . . , an). We call a lattice ideal IL generic if it is generated by binomials with full support, i.e.,
94 citations
TL;DR: In this article, a general scheme for identifying fibrations in the framework of toric geometry and a large list of weights for Calabi-Yau 4-folds were presented.
83 citations
Posted Content•
TL;DR: In this paper, the authors use local mirror symmetry in type IIA string compactifications on Calabi-Yau n+1 folds to construct vector bundles on (possibly singular) elliptically fibered Calabi Yau n-folds Z_n.
Abstract: We use local mirror symmetry in type IIA string compactifications on Calabi-Yau n+1 folds $X_{n+1}$ to construct vector bundles on (possibly singular) elliptically fibered Calabi-Yau n-folds Z_n. The interpretation of these data as valid classical solutions of the heterotic string compactified on Z_n proves F-theory/heterotic duality at the classical level. Toric geometry is used to establish a systematic dictionary that assigns to each given toric n+1-fold $X_{n+1}$ a toric n fold Z_n together with a specific family of sheafs on it. This allows for a systematic construction of phenomenologically interesting d=4 N=1 heterotic vacua, e.g. on deformations of the tangent bundle, with grand unified and SU(3)\times SU(2) gauge groups. As another application we find non-perturbative gauge enhancements of the heterotic string on singular Calabi-Yau manifolds and new non-perturbative dualities relating heterotic compactifications on different manifolds.
74 citations
01 Jan 1998
TL;DR: In this paper, Batyrev and Borisov showed that a certain relative cohomology module H(T rel Zs−1) is a GKZ hypergeometric D-module which over an appropriate domain is isomorphic to the trivial Dmodule RA,T ⊗ OT, where OT is the sheaf of holomorphic functions on this domain.
Abstract: In Part I the Γ-series of [11] are adapted so that they give solutions for certain resonant systems of Gel’fand-Kapranov-Zelevinsky hypergeometric differential equations. For this some complex parameters in the Γseries are replaced by nilpotent elements from a ring RA,T . The adapted Γ-series is a function ΨT,β with values in the finite dimensional vector space RA,T ⊗ZC . Part II describes applications of these results in the context of toric Mirror Symmetry. Building on Batyrev’s work [2] we show that a certain relative cohomology module H(T rel Zs−1) is a GKZ hypergeometric D-module which over an appropriate domain is isomorphic to the trivial D-module RA,T ⊗ OT, where OT is the sheaf of holomorphic functions on this domain. The isomorphism is explicitly given by adapted Γ-series. As a result one finds the periods of a holomorphic differential form of degree d on a d-dimensional Calabi-Yau manifold, which are needed for the B-model side input to Mirror Symmetry. Relating our work with that of Batyrev and Borisov [3] we interpret the ring RA,T as the cohomology ring of a toric variety and a certain principal ideal in it as a subring of the Chow ring of a Calabi-Yau complete intersection. This interpretation takes place on the A-model side of Mirror Symmetry.
61 citations
TL;DR: In this article, the authors studied elliptic Calabi-Yau fourfolds for F-theory compactifications dual to E8 × E8 heterotic strings compactified to four dimensions with some choice of vector bundle.
Abstract: We study, as hypersurfaces in toric varieties, elliptic Calabi-Yau fourfolds for F-theory compactifications dual to E8 × E8 heterotic strings compactified to four dimensions on elliptic Calabi-Yau threefolds with some choice of vector bundle. We describe how to read off the vector bundle data for the heterotic compactification from the toric data of the fourfold. This map allows us to construct, for example, Calabi-Yau fourfolds corresponding to three generation models with unbroken GUT groups. We also find that the geometry of the Calabi-Yau fourfold restricts the heterotic vector bundle data in a manner related to the stability of these bundles. Finally, we study Calabi-Yau fourfolds corresponding to heterotic models with fivebranes wrapping curves in the base of the Calabi-Yau fourfolds. We find evidence of a topology changing extremal transition on the fourfold side which corresponds, on the heterotic side, to fivebranes wrapping different curves in the same homology class in the base.
57 citations
TL;DR: In this article, the moduli kspace of the D-brane world-volume gauge theory was investigated by using toric geometry and gauged linear sigma models, and it was shown that there are five phases, which are topologically distinct and connected by flops to each other.
Posted Content•
TL;DR: In this paper, the Picard lattice of certain K3 surfaces is computed using toric geometry, lattice theory, and elliptic surface techniques, and the results appear in a multipage table near the end of the paper.
Abstract: Using toric geometry, lattice theory, and elliptic surface techniques, we compute the Picard Lattice of certain K3 surfaces. In particular, we examine the generic member of each of M. Reid's list of 95 families of Gorenstein K3 surfaces which occur as hypersurfaces in weighted projective 3-spaces. The results appear in a multipage table near the end of the paper. As an application, we are able to determine whether the mirror family (in the sense of mirror symmetry for K3 surfaces) for each one is also on Reid's list.
01 Jan 1998
TL;DR: In this paper, Lakshmibai, Musili and Seshadri initiated a program to construct a basis for the space H 0(G/B, \( \mathcal{L} \) λ) with some particularly nice geometric properties.
Abstract: The aim of this article is to give an introduction to the theory of path models of representations and their associated bases. The starting point for the theory was a series of articles in which Lakshmibai, Musili and Seshadri initiated a program to construct a basis for the space H 0(G/B, \( \mathcal{L} \) λ) with some particularly nice geometric properties. Here we suppose that G is a reductive algebraic group defined over an algebraically closed field k, B is a fixed Borel subgroup, and \( \mathcal{L} \) λ is the line bundle on the flag variety G/B associated to a dominant weight λ. The purpose of the program is to extend the Hodge-Young standard monomial theory for the group GL(n) to the case of any semisimple linear algebraic group and, more generally, to Kac-Moody algebras. Apart from the independent interest of such a construction, the results have important applications to the combinatorics of representations as well as to the geometry of Schubert varieties. For the geometric applications note that standard monomial theory provides proofs of the vanishing theorems for the higher cohomology of effective line bundles on Schubert varieties, explicit bases for the rings of invariants in classical invariant theory, a proof of Demazure’s conjecture, a proof of the normality of Schubert varieties, another proof of the good filtration property, a determination of the singular locus of Schubert varieties [9], a deformation of SL(n)/B into a toric variety [2], etc.
TL;DR: In this article, the authors studied elliptic Calabi-Yau fourfolds for F-theory compactifications dual to E8xE8 heterotic strings compactified to four dimensions with some choice of vector bundle.
Abstract: We study, as hypersurfaces in toric varieties, elliptic Calabi-Yau fourfolds for F-theory compactifications dual to E8xE8 heterotic strings compactified to four dimensions on elliptic Calabi-Yau threefolds with some choice of vector bundle. We describe how to read off the vector bundle data for the heterotic compactification from the toric data of the fourfold. This map allows us to construct, for example, Calabi-Yau fourfolds corresponding to three generation models with unbroken GUT groups. We also find that the geometry of the Calabi-Yau fourfold restricts the heterotic vector bundle data in a manner related to the stability of these bundles. Finally, we study Calabi-Yau fourfolds corresponding to heterotic models with fivebranes wrapping curves in the base of the Calabi-Yau threefolds. We find evidence of a topology changing extremal transition on the fourfold side which corresponds, on the heterotic side, to fivebranes wrapping different curves in the same homology class in the base.
TL;DR: In this article, it was shown that Delzant's result does not generalize to the non-abelian case, and that these actions fail a criterion which Tolman [9] shows is necessary for the existence of a compatible Kahler structure.
Abstract: Multiplicity-free actions are symplectic manifolds with a very high degree of symmetry. Delzant [2] showed that all compact multiplicity-free torus actions admit compatible Kahler structures, and are therefore toric varieties. In this note we show that Delzant's result does not generalize to the non-abelian case. Our examples are constructed by applying U(2)-equivariant symplectic surgery to the flag variety U(3)/T
3. We then show that these actions fail a criterion which Tolman [9] shows is necessary for the existence of a compatible Kahler structure.
Book•
01 Jan 1998
TL;DR: The theory of toric varieties plays an important role as a bridge between algebraic geometry, combinatorial convex geometry, and commutative algebra, and the interpretation and application of many notions and methods of these three theories leads to fruitful examples and results as mentioned in this paper.
Abstract: The theory of toric varieties plays an important role as a bridge between algebraic geometry, combinatorial convex geometry, and commutative algebra. The interpretation and application of many notions and methods of these three theories in the case of toric varieties leads to fruitful examples and results. The relations between various algebraic, topological, and geometric properties of toric varieties ll extensive dictionaries. From the point of view of combinatorial convexity, all the structure of a toric variety can be encoded in a fan, i.e., a nite collection of strictly convex cones spanned by nitely many vectors with integral coordinates and satisfying very natural incidence relations. Now the obvious question is how to extract algebro-geometric or topological invariants of the associated toric variety from these data. A most striking example is given by the famous Theorem of Jurkiewicz-Danilov Da, Theorem 10.8 and Remark 10.9]: the integral cohomology ring of a smooth compact toric variety can be explicitly computed in terms of the associated fan. Allowing mild singularities, the analogous result holds with rational coeecients. For toric varieties with arbitrary singularities, but still in the compact case, there is a spectral sequence relating data of the fan to integral cohomology that has been investigated by Stephan Fischli in his dissertation Fi]. It admits the explicit computation of some integral cohomology groups in low and in high degrees; in particular, it yields complete results up to dimension three. In the work presented here, we also use spectral sequences to determine (co-) homological data of a toric variety in terms of the associated fan, but in a much more general setting: we investigate the homology with closed supports and the cohomology with compact supports and with arbitrary (constant) coeecients for not necessarily compact toric varieties with arbitrary singularities. We do not consider toric varieties over arbitrary elds but restrict ourselves to the complex case: The toric variety X associated to a fan in a vector space R n consists of (C) n-orbits O corresponding to the cones 2 ; in particular, the full-vi Preface dimensional cones 2 n correspond to xed points x. It turns out that the natural ltration of the toric variety X induced by its orbit structure provides convergent (co-)homology spectral sequences. An explicit calculation of the associated E 2-or E 2-terms yields formull | in low and in high degrees`| for the homology groups H cld`(X ; G), the cohomology groups …
TL;DR: In this paper, the authors studied the data needed to specify a morphism from a scheme to a toric variety in terms of its homogeneous coordinate ring and showed that simplicial toric and quasi-projective toric varieties have enough invariant effective Cartier divisors.
Abstract: The homogeneous coordinate ring of a toric variety was first introduced by Cox. In this paper, we study that of a toric variety with enough invariant effective Cartier divisors in detail. Here a toric variety is said to have enough invariant effective Cartier divisors if, for each nonempty affine open subset stable under the action of the torus, there exists an effective Cartier divisor whose support equals its complement. Both quasi-projective toric varieties and simplicial toric varieties have enough invariant effective Cartier divisors. In terms of the homogeneous coordinate ring, we describe the data needed to specify a morphism from a scheme to such a toric variety. As a consequence, we generalize a result of Cox, one of Oda and Sankaran, and one of Guest concerning data on morphisms. Introduction. Let A: be a field, N a free Z-module of rank r, M the Z-module dual to N, T: = Gm®N the algebraic torus of dimension r corresponding to N9 and Δ a (finite) fan of NQ. Let XΔ be the toric variety associated to Δ, Dp the closure of the Γ-orbit corresponding to a one-dimensional cone peΔ, σ(l) the set of one-dimensional cones contained in a cone σeΔ, and Pic(zl)>0 the monoid of linear equivalence classes of invariant effective Cartier divisors. A toric variety XΔ is said to have enough invariant effective Cartier divisors if, for each cone σeΔ, there exists an effective Γ-invariant Cartier divisor D with SuppD=(Jp ί έ σ ( 1 )Z)p. Both quasi-projective toric varieties and simplicial toric varieties have enough invariant effective Cartier divisors (cf. Remark 1.6(3)). Cox [1] introduced two homogeneous coordinate rings of a toric variety XΔ\\ one is the monoid algebra S of the monoid of effective Γ-invariant Weil divisors with Chow-grading, while the other is the subring SΔ of S with Pic-grading (see [1, p. 19, p. 35]). He constructed in [1] the toric variety XΔ as the quotient of an open subscheme of Spec S, and described in [2, Theorem 1.1] the data needed to specify a map from a scheme to an arbitrary smooth toric variety in terms of its homogeneous coordinate ring. The purpose of this paper is to generalize Cox's description to one for an arbitrary toric variety with enough invariant effective Cartier divisors by studying the latter homogeneous coordinate ring in detail (cf. Theorem 3.4 and Theorem 4.3). The contents of this paper are as follows: 1991 Mathematics Subject Classification. Primary 14M25; Secondary 14E99. Partly supported by the Grants-in-Aid for Encouragement of Young Scientists, the Ministry of Education, Science, Sports and Culture, Japan.
TL;DR: In this paper, an algorithm for obtaining the matter content of effective six-dimensional theories resulting from compactification of F-theory on elliptic Calabi-Yau threefolds which are hypersurfaces in toric varieties is presented.
TL;DR: In this paper, the authors investigate geometrical interpretations of various structure maps associated with the Landweber-Novikov algebra S and its integral dual S. In particular, they study the coproduct and antipode in S, together with the left and right actions of S on S which underly the construction of the quantum doubleD(S).
Abstract: We investigate geometrical interpretations of various structure maps associated with the Landweber{Novikov algebra S and its integral dual S . In particular, we study the coproduct and antipode in S , together with the left and right actions of S on S which underly the construction of the quantum (or Drinfeld) doubleD(S). We set our realizations in the context of double complex cobordism, utilizing certain manifolds of bounded flags which generalize complex projective space and may be canonically expressed as toric varieties. We discuss their cell structure by analogy with the classical Schubert decomposition, and detail the implications for Poincar e duality with respect to double cobordism theory; these lead directly to our main results for the Landweber{ Novikov algebra.
01 Jan 1998
TL;DR: In this paper, a detailed analysis of the GKZ (Gel'fand, Kapranov and Zelevinski) hypergeometric systems in the context of mirror symmetry of Calabi-Yau hypersurfaces in toric varieties is presented.
Abstract: We present a detailed analysis of the GKZ (Gel’fand, Kapranov and Zelevinski) hypergeometric systems in the context of mirror symmetry of Calabi-Yau hypersurfaces in toric varieties. As an application, we will derive a concise formula for the prepotential about large complex structure limits.
01 Jan 1998
TL;DR: In this paper, the Mirror Symmetry Conjecture for Schoen's Calabi-Yau 3-fold has been verified up to a high order using the toric mirror construction.
Abstract: In this paper, we verify a part of the Mirror Symmetry Conjecture for Schoen’s Calabi-Yau 3-fold,
which is a special complete intersection in a toric variety. We calculate a part of the prepotential
of the A-model Yukawa couplings of the Calabi-Yau 3-fold directly by means of a theta function
and Dedekind’s eta function. This gives infinitely many Gromov-Witten invariants, and equivalently
infinitely many sets of rational curves in the Calabi-Yau 3-fold. Using the toric mirror construction
[Ba-Bo, HKTY, Sti], we also calculate the prepotential of the B-model Yukawa couplings of the mirror
partner. Comparing the expansion of the B-model prepotential with that of the A-model prepotential,
we check a part of the Mirror Symmetry Conjecture up to a high order.
http://www.arxiv.org/abs/alg-geom/9709027
TL;DR: In this article, the resolution of conifold singularity by D-branes was studied by considering compactification of Dbranes on C 3 /( Z 2 × Z 2 ).
TL;DR: In this paper, the authors show that divisors contributing to the superpotential are always "exceptional" for the Calabi-Yau 4-fold X. In fact, this is a consequence of the (log-minimal model algorithm in dimension 4.
TL;DR: In this paper, the authors introduce the concepts of toric geometry that are necessary to understand applications in the context of string and F-theory dualities, and give both intuitive pictures and precise rules that enable the reader to work with the concepts presented here.
Abstract: This note is supposed to be an introduction to those concepts of toric geometry that are necessary to understand applications in the context of string and F-theory dualities. The presentation is based on the definition of a toric variety in terms of homogeneous coordinates, stressing the analogy with weighted projective spaces. We try to give both intuitive pictures and precise rules that should enable the reader to work with the concepts presented here.
TL;DR: In this article, the break through in deformation theory of (two-dimensional) quotient singularities Y was Kollar/Shepherd-Barron's discovery of the one-to-one correspondence between so-called P-resolutions, on the one hand, and components of the versai base space.
Abstract: (1.1) The break through in deformation theory of (two-dimensional) quotient singularities Y was Kollar/Shepherd-Barron’s discovery of the one-to-one correspondence between so-called P-resolutions, on the one hand, and components of the versai base space, on the other (cf. [KS], Theorem (3.9)). It generalizes the fact that all deformations admitting a simultaneous (RDP-) resolution form one single component, the Artin component.
Posted Content•
TL;DR: In this paper, the authors consider regular Calabi-Yau hypersurfaces in smooth toric varieties and construct a fibration over a sphere whose generic fibers are tori, and show that the monodromy transformation is induced by a translation of the $T^{N-1}$ fibration by a section.
Abstract: We consider regular Calabi-Yau hypersurfaces in $N$-dimensional smooth toric varieties. On such a hypersurface in the neighborhood of the large complex structure limit point we construct a fibration over a sphere $S^{N-1}$ whose generic fibers are tori $T^{N-1}$. Also for certain one-parameter families of such hypersurfaces we show that the monodromy transformation is induced by a translation of the $T^{N-1}$ fibration by a section. Finally we construct a dual fibration and provide some evidence that it describes the mirror family.
TL;DR: In this article, the relation between these vertex algebras for mirror Calabi-Yau manifolds and complete intersections in toric varieties is established, which can be used to rewrite the whole story of toric mirror symmetry in the language of sheaves of vertex algesbras.
Abstract: Mirror Symmetry for Calabi-Yau hypersurfaces in toric varieties is by now well established. However, previous approaches to it did not uncover the underlying reason for mirror varieties to be mirror. We are able to calculate explicitly vertex algebras that correspond to holomorphic parts of A and B models of Calabi-Yau hypersurfaces and complete intersections in toric varieties. We establish the relation between these vertex algebras for mirror Calabi-Yau manifolds. This should eventually allow us to rewrite the whole story of toric Mirror Symmetry in the language of sheaves of vertex algebras. Our approach is purely algebraic and involves simple techniques from toric geometry and homological algebra, as well as some basic results of the theory of vertex algebras. Ideas of this paper may also be useful in other problems related to maps from curves to algebraic varieties. This paper could also be of interest to physicists, because it contains explicit descriptions of A and B models of Calabi-Yau hypersurfaces and complete intersection in terms of free bosons and fermions.
Posted Content•
TL;DR: In this paper, the underlying spaces of all abelian quotient singularities which are embeddable as complete intersections of hypersurfaces in an affine space can be overall resolved by means of projective torus-equivariant crepant birational morphisms in all dimensions.
Abstract: It is known that the underlying spaces of all abelian quotient singularities which are embeddable as complete intersections of hypersurfaces in an affine space can be overall resolved by means of projective torus-equivariant crepant birational morphisms in all dimensions. In the present paper we extend this result to the entire class of toric l.c.i.-singularities. Our proof makes use of Nakajima's classification theorem and of some special techniques from toric and discrete geometry.
01 Jan 1998
TL;DR: In this paper, a generalisation and a new proof are given of a recent result of J. F. Thomsen (1996), which says that for L a line bundle on a smooth toric variety X over a field of positive characteristic, the direct image F*L under the Frobenius morphism splits into a direct sum of line bundles.
Abstract: A generalisation and a new proof are given of a recent result of J. F. Thomsen (1996), which says that for L a line bundle on a smooth toric variety X over a field of positive characteristic, the direct image F*L under the Frobenius morphism splits into a direct sum of line bundles. (The special case of projective space is due to Hartshorne.) Our method is to interpret the result in terms of Grothendieck differential operators Diff(1) (L, L) Homo (1) (F.L, F.L), and T-linearized sheaves.
Posted Content•
TL;DR: The interpretation of the classical Futaki invariant of Fano toric manifolds is extended to the case of the generalized Fano invariant, introduced by W. Ding and G. Tian.
Abstract: The interpretation, due to T. Mabuchi, of the classical Futaki invariant of Fano toric manifolds is extended to the case of the Generalized Futaki invariant, introduced by W. Ding and G. Tian, of almost Fano toric varieties. As an application it is shown that the real part of the Generalized Futaki invariant is positive for all degenerations of the Fano manifold V_{38}, obtained by intersection of the Veronese embedding of ${\bf P}^3\times{\bf P}^2 \subset {\bf P}^{11}$ with codimension-two hyperplanes.