Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties

Victor V. Batyrev

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Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties

Victor V. Batyrev

alg-geom · Submitted 5 Oct 1993 · 45 references

Pith's verdict: The founding paper of toric mirror symmetry: reflexive polytopes, polar duality, and the Hodge-number exchange, with only cosmetic typos in the §4.4 derivation.

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CitationVictor V. Batyrev. "Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties." arXiv:alg-geom/9310003 (1993). https://pith.science/paper/WMAF7NW4

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Editor's note

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This is the paper that made toric mirror symmetry a construction rather than a collection of examples. If you work on Calabi-Yau hypersurfaces in toric varieties, everything downstream — Kreuzer-Skarke enumeration, the later Batyrev-Borisov work, the whole industry of computing Hodge numbers from polytope data — starts here. Read it, and have your students read §4.1 and §4.5 first, because those are the parts that stayed load-bearing for thirty years. The abstract's claim [abstract:s2] that the duality reproduces the physicists' mirror symmetry is exactly what the theorems deliver, and the CFT isomorphism is correctly flagged as a conjecture rather than smuggled in as a result [abstract:s3]. That distinction is honest and it is the right one; nobody has proved the CFT statement in general even now. What is actually proved is clean. Theorem 4.1.6 (polar of a reflexive pair is reflexive) is a two-line argument that is completely convincing. Theorem 4.3.7 and Theorem 4.4.2 are symmetric under Δ ↔ Δ*, and Corollary 4.5.6 (opposite Euler characteristics for dual 4-polytopes) falls out immediately. The quintic check in §5.1 reproduces the Candelas et al. one-parameter family independently, which is a real external benchmark, not a fit. The Galois correspondence in §5.3 and the reflexive-simplex analysis in §5.4-5.5 generalize Greene-Plesser and Roan in a way that is genuinely useful. Soft spots, in proportion. The §4.4 derivation of h^{1,1} is terse, and the printed proof has index typos: it writes H^{2n-k}_c where H^{n-k}_c is meant, and Theorem 4.4.3 states h^{n-1,1} where h^{n-2,1} is intended. These are cosmetic; the final formula is the standard correct one and is manifestly symmetric with 4.3.7. The stratification bookkeeping in §4.3 that a reader might worry about does hold up — the torus factors force the surviving contributions to be exactly the two claimed types. The acknowledgements disclose that an earlier version's h^{1,1} formula was wrong and was caught by Morrison, Greene and Aspinwall. That is disclosed, not hidden, and the corrected formula is what became standard, but it is a fair signal that §4.4 is the section a referee should check line by line. Who should read it: anyone doing toric geometry, mirror symmetry, or computational Calabi-Yau. It deserves a serious referee, and I would expect acceptance with minor corrections to the indices.

Referee report

Significance

If correct, this is a structural result of considerable importance: it gives a uniform, algorithmically computable source of mirror candidates for Calabi-Yau hypersurfaces in arbitrary toric varieties, subsumes the weighted-projective constructions as the special case of reflexive simplices (§5.4-5.5), and reduces the Hodge-number mirror symmetry to a purely combinatorial involution on a finite set of polytopes (Theorem 4.1.12). The paper is careful about what is proved and what is conjectured: the Hodge-number and Euler-characteristic statements are theorems, while the conformal-field-theory statement is explicitly labelled a conjecture in the abstract. Two features deserve particular credit. First, the derivation is parameter-free: no coefficient, triangulation, or normalization is tuned to produce the mirror relation, and the formulas of 4.3.7 and 4.4.2 are manifestly exchanged by Δ ↔ Δ*. Second, §5.1 provides a genuine external benchmark rather than a fit: the general construction, applied to Δ_n, reproduces the one-parameter mirror family of the quintic obtained by Candelas, de la Ossa, Green and Parkes, including the correct orbifold group (Z/5Z)^3. The prediction e(Ẑ_f) = −e

Major objections (2)

[§4.4, proof of Theorem 4.4.2] The exact-sequence argument that produces formula (2) is the load-bearing step for Theorem 4.4.3, and as printed its cohomological degrees are internally inconsistent. The text asserts that 'H^{2n−1}_c(T) is an n-dimensional space having the Hodge type (n−1,n−1)', but T is n-dimensional, so H^{2n−1}_c(T) = 0; the n-dimensional group of type (n−1,n−1) is H^{n−1}_c(T). The same doubling appears throughout the displayed sequence (H^{2n−4}_c(Z_f), H^{2n−4}_c(Ẑ_f), H^{2n−4}_c(Y), H^{2n−3}_c(Z_f), H^{2n−3}_c(Ẑ_f)), and it is not consistent with the sentence two paragraphs earlier that h^{1,1} equals h^{n−2,n−2} of H^{n−2}_c(Ẑ_f). The final formula (2) is correct and is visibly symmetric with 4.3.7 under Δ ↔ Δ*, so the theorem itself is not in doubt; but the proof as printed cannot be checked line by line. Please rewrite the sequence with consistent indices and add one sentence justifying the vanishing of the (n−1,n−3) part of H^{2n−4}_c(Y): Y = Ẑ_f ∖ Z_f is (n−2)-dimensional, so its top compactly supported cohomology is pure of type (n−2,n−2), which is what the argument needs and which the present text asserts in a single clause.
[Abstract, sentence s3; §4.5-§5] The abstract claims that the method 'allows to construct many new examples of Calabi-Yau 3-folds and new candidates for their mirrors which were previously unknown for physicists', but the body never exhibits one. Section 4.5 gives the general formulas for (h^{1,1}, h^{2,1}) and e(Ẑ_f), and §5.1 works out the quintic, which is precisely the case already known from Candelas et al.; §5.4-5.5 treats reflexive simplices, which reduce to weighted projective hypersurfaces and hence to the Greene-Plesser class. Since the novelty claim is one of the two contributions advertised in the abstract, it should be substantiated by at least one explicit worked example: a 4-dimensional reflexive polytope Δ whose associated families are not weighted-projective hypersurface quotients, together with the resulting (h^{1,1}, h^{2,1}) pair for Ẑ_f and Ẑ_g and the two Euler characteristics. A short table of several such polytopes would settle the point and would also let a reader check the arithmetic of 4.5.1 against 4.4.2.

Minor objections (10)

[Theorem 4.4.3] The statement reads 'the Hodge number h^{n−1,1}(Ẑ_f)', but the quantity computed in 4.3.7 and used in Corollary 4.5.1 is h^{n−2,1}; the index should be corrected in the theorem statement itself, since this is the paper's central formula.
[§4.2, opening paragraph, and Definition 4.2.4] The text says 'Take a maximal projective triangulation T of Δ', whereas Theorem 2.2.24 and Definition 4.2.1 both triangulate the dual polytope Δ*; similarly Definition 4.2.4 calls Θ* 'the face of the polyhedron Δ' where Δ* is meant. Both should be corrected, since the direction of the duality is exactly what a reader is tracking here.
[Proof of Theorem 4.2.5] The fibre isomorphism is stated for 'any closed point p ∈ T_{Θ*}', but the stratum in question is Z_{f,Θ} ⊂ T_Θ; and the vertical maps in the displayed diagram are described as 'divisions by the action of the torus T_Θ', which should read 'quotients by the action'.
[Corollaries 4.5.1 and 4.5.2] Both displays carry a stray '+' at the start of a continuation line (before 'h^{2,1}(Ẑ_f)' and before the second parenthesis respectively), which reads as a sign error in formulas whose signs matter.
[§5.1, Eq. (3) and the following line] The sum in Eq. (3) is written with upper limit 'd', but d is not defined at that point and the intended range is i = 1,...,n+1; the sentence after the equation also prints 'an+1' where 'a^{n+1}' is meant. Both affect the reading of the one-parameter family that is the paper's benchmark against Candelas et al.
[Theorem 5.4.3(iii) and Eq. (6) in §5.5] The formula for the weights is printed as 'w_i = l.c.m.(b_ii+1)b_ii+1' and, in Eq. (6), as 'w_i = l.c.m.(d_i)d_i'; in both places a division bar is missing and the intended expressions are l.c.m.(b_ii+1)/(b_ii+1) and l.c.m.(d_i)/d_i. These are the formulas a reader will use to compute examples, so they should be unambiguous.
[Definition 5.3.5] The definition of π_1(Δ) and π_1(Δ*) depends on the distinction between the maximal lattice N^{Δ*} and the minimal lattice N_{Δ*}, but the printed text renders both as 'N_{Δ*}', making the quotient trivial as written. Please make the superscript/subscript convention explicit here and in Proposition 5.2.8.
[§4.4, first paragraph of the proof of 4.4.2] The phrase 'Ẑ_f has only terminal Q-factorial singularities, i.e., Ẑ_f is quasi-smooth' uses 'quasi-smooth' in a sense stronger than what terminal Q-factorial singularities give in general. What the argument needs is that Ẑ_f is a rational homology manifold, which follows from 4.2.2 (singularities in codimension at least 4); a rephrasing would avoid an unnecessary terminological objection.
[References] Several entries need correction: [24] should read Hartshorne, not Hartshorn; [20] is conventionally cited as Gelfand-Kapranov-Zelevinsky, not Gelfand-Zelevinsky-Kapranov; [1] has a stray 'and' in the author list; [34] prints 'Nucl. Hys. B'; and [19] and [31] are both given as 'Nucl. Phys. B 389 (1993) 153', so at least one page or volume number is wrong.
[Abstract and §1] The abstract's conjecture on the isomorphism of the two conformal field theories is the paper's most consequential claim and deserves a numbered statement in the introduction, with a sentence on what evidence beyond the Hodge-number and Euler-characteristic agreement would test it. The authors are to be credited for labelling it a conjecture rather than a theorem.

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The paper

The paper claims

Batyrev proposes that polar duality of lattice polytopes is the combinatorial engine behind mirror symmetry: a reflexive Newton polytope and its dual pick out two families of Calabi-Yau hypersurfaces in toric varieties whose Hodge numbers $

The reading

The paper asks whether a purely combinatorial operation on lattice polytopes can reproduce mirror symmetry, the physicists' pairing of Calabi-Yau threefolds whose Hodge numbers $h^{1,1}$ and $h^{2,1}$ are exchanged. Batyrev's answer is yes: if an integral polytope $Delta$ is reflexive, meaning the origin lies inside and every facet hyperplane sits exactly one integral unit away from it, then the polar dual $Delta^*$ is reflexive too, and the two families of Calabi-Yau hypersurfaces cut out in the toric varieties $P_Delta$ and $P_{Delta^*}$ are mirror candidates. The evidence is a pair of dimension counts: for $n ge 4$ the Hodge number $h^{n-2,1}$ of a crepant partial resolution of a member of $mathcal{F}(Delta)$ equals the Picard number $h^{1,1}$ of the corresponding member of $mathcal{F}(Delta^*)$, and for $n = 4$ the Euler characteristics of the two threefolds are exact negatives. What makes it matter is scope: the recipe is uniform across dimensions and across every reflexive polytope, so it manufactures mirror pairs far beyond the weighted-projective examples physicists had catalogued, and it converts the search for a mirror into finite combinatorics on lattice points.

What carries the argument

Three pieces carry the argument. First, reflexivity: an integral polytope whose facet hyperplanes are all at integral distance one from the origin, a condition that is preserved under taking the polar dual and that guarantees the associated toric variety is Gorenstein Fano with the hypersurface's canonical class trivial. Second, crepant resolution by triangulation: a maximal projective triangulation of $Delta^*$ (or of $Delta$, depending on which side one resolves) produces a maximal projective crepant partial desingularization of both the ambient toric variety and the hypersurface, so singularities can be handled uniformly and the resolved variety is smooth for $n le 4$. Third, the bookkeeping of Hodge-Deligne numbers of affine hypersurfaces computed by Danilov and Khovanskii, combined with a stratification of the resolved hypersurface by strata $Z_{f,Theta} times (mathbb{C}^*)^k$ and additivity of the $(p,q)$-Euler characteristic. The key observation in the second count is that only the open stratum and the codimension-two faces contribute to the relevant Hodge component, which is what makes $h^{1,1}$ on one side computable as a sum over faces of the dual polytope on the other.

Extended reading notes

Core claim

The central claim is that polar duality of reflexive polytopes induces a mirror correspondence between families of Calabi-Yau hypersurfaces in toric varieties. Concretely, let $Delta$ be an $n$-dimensional integral polytope containing the origin, with every facet hyperplane at integral distance one; then $Delta^* = {y : langle x, y rangle ge -1 text{ for all } x in Delta}$ is again such a polytope, and the faces of $Delta$ and $Delta^*$ correspond one-to-one with incidence reversed. The paper proves that for $n ge 4$, the Hodge number $h^{n-2,1}$ of a maximal projective crepant partial desingularization of a $Delta$-regular Calabi-Yau hypersurface in $P_Delta$ equals the Picard number $h^{1,1}$ of the corresponding $Delta^*$-regular hypersurface in $P_{Delta^*}$ (Theorem 4.4.3). For $n = 4$ this yields the physicists' relation $h^{1,1}(V) = h^{2,1}(V')$ and $h^{1,1}(V') = h^{2,1}(V)$, together with $e(hat{Z}_f) = -e(hat{Z}_g)$ for dual polytopes (Corollary 4.5.6). The author states the stronger claim as a conjecture rather than a theorem: that the two conformal field theories attached to the paired families are isomorphic.

Load-bearing premise

The load-bearing premise is the stratum-by-stratum accounting in the Hodge-number computations: that the relevant Hodge component of the resolved hypersurface receives contributions only from the open affine stratum and from codimension-two faces carrying a single $mathbb{C}^*$ factor, with no missed stratum and no Hodge component of type $(n-1,n-3)$ in the boundary cohomology; the author notes that an earlier version of the $h^{1,1}$ formula was wrong and was corrected after others found the error, which marks this bookkeeping as the delicate step.

Editorial extensions

If this is right

Every reflexive polytope in dimension four delivers a Calabi-Yau threefold together with a mirror candidate, so the construction generates mirror pairs well beyond hypersurfaces in weighted projective space.
The Hodge numbers of both members of a pair are read off from lattice-point counts on the two polytopes and their faces, giving closed formulas (Corollary 4.5.1) that need no period or Picard-Fuchs computation.
The Euler characteristic changes sign under duality, so the number of generations $tfrac{1}{2} e(hat{Z}_f)$ of a threefold is minus that of its mirror, matching the physical expectation.
Reflexive simplices reproduce the Greene-Plesser orbifold construction and generalize Roan's verification of the Hodge relation for Fermat-type quotients in weighted projective spaces.
A finite abelian group attached to a reflexive pair controls how one family arises as a quotient of another, giving a Galois-style correspondence that is compatible with mirror duality.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because only finitely many reflexive polytopes exist in each dimension up to unimodular equivalence (Theorem 4.1.12), the mirror-search problem becomes a finite enumeration: in principle one can list all four-dimensional reflexive polytopes and read off every toric mirror pair at once, which is a concrete algorithmic programme the paper sets up but does not carry out.
The formulas for $h^{1,1}$ and $h^{2,1}$ include correction sums over codimension-two faces that vanish when the ambient toric variety is smooth; testing the duality on polytopes where these corrections are genuinely nonzero would probe the parts of the bookkeeping that the smooth examples cannot reach.
Matching Hodge numbers and Euler characteristics is necessary but not sufficient for the conjectured isomorphism of conformal field theories; a sharper test would compare Yukawa couplings or mirror maps computed from the two families, which the paper does not attempt.
The Galois correspondence for finite morphisms of reflexive pairs suggests that orbifold mirror pairs should be organized by the fundamental groups of the polytopes, a structural prediction that could be checked against known orbifold mirror constructions.

Original abstract

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, polyhedron $\Delta^*$ in the dual space defines another family ${\cal F}(\Delta^*)$ of Calabi-Yau varieties, so that we obtain the remarkable duality between two {\em different families} of Calabi-Yau varieties. It is shown that the properties of this duality coincide with the properties of {\em Mirror Symmetry} discovered by physicists for Calabi-Yau $3$-folds. Our method allows to construct many new examples of Calabi-Yau $3$-folds and new candidats for their mirrors which were previously unknown for physicists. We conjecture that there exists an isomorphism between two conformal field theories corresponding to Calabi-Yau varieties from two families ${\cal F}(\Delta)$ and ${\cal F}(\Delta^*)$.

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The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

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, polyhedron $\Delta^*$ in the dual space defines another family ${\cal F}(\Delta^*)$ of Calabi-Yau varieties, so that we obtain the remarkable duality between two {\em different families} of Calabi-Yau varieties. It is shown that the properties of this duality coincide with the properties of {\em Mirror Symmetry} discovered by physicists for Calabi-Yau $3$-folds. Our method allows to construct many new examples of Calabi-Yau $3$-folds and new candidats for their mirrors which were previously unknown for physicists. We conjecture that there exists an isomorphism between two conformal field theories corresponding to Calabi-Yau varieties from two families ${\cal F}(\Delta)$ and ${\cal F}(\Delta^*)$.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
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s0	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 ${\bf T} =({\bf C}^*)^n$.	background 0.80	0.28	0.03	asserted 0.83
s1	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, polyhedron $\Delta^$ in the dual space defines another family ${\cal F}(\Delta^)$ of Calabi-Yau varieties, so that we obtain the remarkable duality between two {\em different families} of Calabi-Yau varieties.	result 0.84	0.51	0.55	asserted 0.99
s2	It is shown that the properties of this duality coincide with the properties of {\em Mirror Symmetry} discovered by physicists for Calabi-Yau $3$-folds.	interpretation 0.55	0.66	0.46	asserted 1.00
s3	Our method allows to construct many new examples of Calabi-Yau $3$-folds and new candidats for their mirrors which were previously unknown for physicists.	contribution 0.90	0.89	0.16	asserted 0.99
s4	We conjecture that there exists an isomorphism between two conformal field theories corresponding to Calabi-Yau varieties from two families ${\cal F}(\Delta)$ and ${\cal F}(\Delta^*)$.	future work 0.60	0.21	0.19	speculative 1.00

What kind of paper the classifier reports this is: theoretical result 0.75 · evidence from proof or derivation 0.94.

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References

Works cited (45)

[1]

P.S. Aspinwall and C.A. L¨ utken, and G.G. Ross, Construction and couplings of mirror manifolds, Phys. Lett. B 241, n.3, (1990), 373-380

View on Pith 1990
[2]

P.S. Aspinwall and C.A. L¨ utken, Geometry of mirror manifolds , Nuclear Physics B, 353 (1991), 427-461

View on Pith 1991
[3]

P.S. Aspinwall and C.A. L¨ utken, Quantum algebraic geometry of superstring compact- iﬁcations, Nuclear Physics B, 355 (1991), 482-510

View on Pith 1991
[4]

P.S. Aspinwall, Multiple Mirror Manifolds and Topology Change in String The ory, Preprint IASSNS-HEP-93/4

View on Pith
[5]

V. V. Batyrev, Boundedness of the degree of higher-dimensional toric Fano varieties, Moscow Univ. Math. Bull. 37, n.1, (1982), 28-33

View on Pith 1982
[6]

V. V. Batyrev, Variations of the Mixed Hodge Structure of Aﬃne Hypersurfac es in Algebraic Tori, Duke Math. J. 69, (1993), 349-409

View on Pith 1993
[7]

V. V. Batyrev, Quantum Cohomology Rings of Toric Manifolds , Preprint MSRI, (1993)

View on Pith 1993
[8]

V.V. Batyrev, D. A. Cox, On the Hodge Structure of Projective Hypersurfaces in Toric Varieties, Preprint 1993

View on Pith 1993
[9]

V.V. Batyrev, D. van Straten, Generalized Hypergeometric Functions and Rational Curves on Calabi-Yau Complete Intersections in Toric Varie ties, Preprint 1993

View on Pith 1993
[10]

P. Berglund, T. H¨ ubsch,A Generalized Construction of Mirror Manifolds , in Essays on Mirror Manifolds (ed. S.-T. Yau), International Press Co., Hong Kong, 1992, 388-407. 41

View on Pith 1992
[11]

A.A. Borisov, L.A. Borisov, Singular toric Fano varieties , Mat. Sbornik 183, n.2, (1992), 134-141 (in Russian)

View on Pith 1992
[12]

P. Candelas, M. Lynker, R. Schimmrigk, Calabi-Yau Manifolds in Weighted P4, Nucl. Phys., B 341 (1990), 383-402

View on Pith 1990
[13]

P. Candelas, X.C. de la Ossa, P.S. Green, and L. Parkes, A pair of Calabi-Yau mani- folds as an exactly soluble superconformal theory , Nuclear Phys. B 359 (1991), 21-74

View on Pith 1991
[14]

V.I. Danilov, The geometry of toric varieities . Russian Math. Survey, 33, n.2, (1978), 97-154

View on Pith 1978
[15]

V.I. Danilov, A.G. Khovanski ˆi, Newton polyhedra and an algorithm for computing Hodge-Deligne numbers, Math. USSR Izv. 29 (1987) 279-298

View on Pith 1987
[16]

P. Deligne, Th´ eory de Hodge, II, III, Publ. Math. I.H.E.S. 40 (1971) 5-58; 44 (1975) 5-77

View on Pith 1971
[17]

L. Dixon, in Superstrings. Uniﬁed Theories and Cosmology 1987 , G. Furlan et al., eds., World Scientiﬁc 1988

View on Pith 1987
[18]

G. Ellingsrud and S.A. Strømme, The number of twisted cubic curves on the general quintic threefold, in Essays on Mirror Manifolds (ed. S.-T. Yau), Internation al Press Co., Hong Kong (1992), 181-240

View on Pith 1992
[19]

A. Font, Periods and duality symmetries in Calabi-Yau com pactiﬁcat ions, Nucl. Phys. B 389 (1993) 153

View on Pith 1993
[20]

I. M. Gelfand, A.V. Zelevinsky, and M.M. Kapranov, Equations of hypergeometric type and toric varieties , Func. Anal. Appl. 28 (1989), n. 2, 12-26; English trans. 94-106

View on Pith 1989
[21]

M.B. Green, J.H. Schwarz, E. Witten, Superstring theory I, II, Cambrdge Univ. Press, Cambridge, New York, New Rochelle, Melbourne, Sydney, 1988

View on Pith 1988
[22]

B.R. Greene and M.R. Plesser, Duality in Calabi-Yau moduli space , Nuclear Phys. B 338 (1990), 15-37

View on Pith 1990
[23]

B.R. Greene and M.R. Plesser, An Introduction to Mirror Manifolds , in Essays on Mirror Manifolds (ed. S.-T. Yau), International Press Co., Hong Kong, 1992, 1-30

View on Pith 1992
[24]

R. Hartshorn, Algebraic Geometry, Springer-Verlag, B erline-Heidelberg-New York, 1977

View on Pith 1977
[25]

D. Hensley, Lattice vertex polytopes with interior lattice points , Paciﬁc J. Math. , 105, n.1, 1983, pp. 183-191

View on Pith 1983
[26]

S. Katz, On the ﬁnitness of rational curves on quintic threefolds, Compositio Math. 60 (1986) 151-162. 42

View on Pith 1986
[27]

S. Katz, Rational curves on Calabi-Yau 3-folds, in Essays on Mirror Manifolds (Ed. S.-T. Yau), Int. Press Co., Hong Kong, (1992) 168-180

View on Pith 1992
[28]

S. Katz, Rational curves on Calabi-Yau manifolds: verifying of pred ictions of Mirror Symmetry, Preprint, alg-geom/9301006 (1993)

View on Pith Pith review arXiv 1993
[29]

A.G. Khovansi ˆi, Newton polyhedra and the genus of full intersections , Funk. Anal Priloz, 12, n.1, (1978), 51-61

View on Pith 1978
[30]

A.G. Khovanski ˆi, Newton polyhedra (resolution of singularities ), Sovremen. Probl. Mat. 22, Moscow (1983) (in Russian). Engl. trans.: J. of Soviet Math ., v.27, 1984, 2811-2830

View on Pith 1983
[31]

A. Klemm, S. Theisen, Consideration of One Modulus Calabi-Yau Compactiﬁcation : Picard-Fuchs Equation, K¨ ahler Potentials and Mirror Maps, Nucl. Phys. B 389 (1993) 153

View on Pith 1993
[32]

A. Klemm. S. Theisen, Mirror Maps and Instanton Sums for Complete Intersections in Weighted Projective Spaces , LMU-TPW 93-08, Preprint April 1993

View on Pith 1993
[33]

A.G. Kouchnirenko, Poly´ edres de Newton et nombres de Milnor , Invent. Math., 32, n.1 (1976)

View on Pith 1976
[34]

M. Kreuzer, R. Schimmrigk, H. Skarke, Abelian Landau-Ginzburg Orbifolds and Mirror Symmetry, Nucl. Hys. B 372 (1992) 61-86

View on Pith 1992
[35]

W. Lerche, C. Vafa and N. Warner, Nucl. Phys. B324 (1989) 427

View on Pith 1989
[36]

A. Libgober, and J. Teitelbaum, Lines on Calabi-Yau complete intersections, mirror symmetry, and Picard-Fuchs equations, Duke Math. J., Int. Math. Res Notices 1 (1993) 29

View on Pith 1993
[37]

M. Lynker, R. Schimmrigk, Landau-Ginzburg theories as orbifolds , Phys. Lett. B, 249, n.2 (1990) 237-242

View on Pith 1990
[38]

D.G. Markushevich, Resolution of singularities (toric method) , appendix to: D.G. Markushevich, M.A. Olshanetsky, and A.M. Perelomov, Description of a class of su- perstring compactiﬁcation related to semi-simple Lie alge bras, Comm. Math. Phys. 111 (1987), 247-274

View on Pith 1987
[39]

D. Morrison, Mirror symmetry and rational curves on quintic threefolds: a guide for mathematicians, J. Amer. Math. Soc. 6 (1993), 223-247

View on Pith 1993
[40]

D. Morrison, Picard-Fuchs equations and mirror maps for hypersurfaces , in Essays on Mirror Manifolds (ed. S.-T. Yau), International Press Co., Hong Kong, 1992, 241-264

View on Pith 1992
[41]

T. Oda, Convex Bodies and Algebraic Geometry - An Introduction to th e Theory of Toric Varieties, Ergebnisse der Math. (3)15, Springer-Verlag, Berlin, Hei ldelberg,New York, London, Paris, Tokyo,1988. 43

View on Pith 1988
[42]

M. Reid, Decomposition of toric morphisms , in Arithmetic and Geometry, papers ded- icated to I.R. Shafarevich on the occasion of his 60th birthd ay (M. Artin and J. Tate, eds.), vol.II, Geometry, Progress in Math. 36, Birkhauser, Boston, Basel, Stuttgar t, 1983, 395-418

View on Pith 1983
[43]

S.-S. Roan, S.-T. Yau On Ricci Flat 3-Folds, Acta Mathematica Sinica, New Series 3 (1987), 256-288

View on Pith 1987
[44]

S.-S. Roan, The Mirror of Calabi-Yau Orbifold , Int. J. of Math., vol. 2, n. 4 (1991) 439-455

View on Pith 1991
[45]

R. Schimmrigk, The construction of mirror symmetry , preprint HD-THEP-92-17. 44

View on Pith

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How later papers used it

177 later papers cite it

Evidence: substantial. 176 of the citing papers share no author with it, and 54 of those carry a confident label from Pith's citation classifier.

Pith read 22 of the 424 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

5 later papers by other authors are listed below: 1 built on it and 4 used its method or tool.

Builds on it 1 of 6 papers with this label

Established the cohomology correspondence between the mirror pair previously found in this paper, as an extension of the mirror correspondence in another work.

Toric Representation and Positive Cone of Picard Group and Deformation Space in Mirror Symmetry of Calabi-Yau Hypersurfaces in Toric Varieties (2011) · 0.98 · paraphrased: the licence does not allow quoting

Uses its method or tool 4 of 40 papers with this label

Used the correspondence proposed by Batyrev between Calabi-Yau families and reflexive polyhedra.

Coincidences between Calabi-Yau manifolds of Berglund-Hubsch type and Batyrev polytopes (2020) · 0.97 · paraphrased: the licence does not allow quoting
Calculated the Hodge numbers h^2,1 and h^1,1 following Batyrev's formulae.

A Generalized Construction of Calabi-Yau Models and Mirror Symmetry (2016) · 0.98 · paraphrased: the licence does not allow quoting
Followed Batyrev's approach to build the mirror of the orbifold as a hypersurface in a toric variety.

On the equivalence of Batyrev and BHK Mirror symmetry constructions (2020) · 0.91 · paraphrased: the licence does not allow quoting
Made these manifolds explicitly via Batyrev's construction by triangulating polytopes in the Kreuzer-Skarke database using CYTools.

QCD Axion Dark Matter in String Theory: Haloscopes and Helioscopes as Probes of the Landscape (2024) · 0.99 · paraphrased: the licence does not allow quoting

How the literature uses it

177 arXiv papers cite this in 423 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 230 sentences without a confident label

Contests it 1 paper

Equivalently, $_Σ_0$ is isomorphic to an abelian quotient of weighted projective space [ref] [Some misprints occur in RIMS journal paper [ref] page 832, where $d_j:= d/n_j$ and $dia.[ e^2πi/d_1, …, e^2πi/ d_5]$ (on line 14, 15) should be $d_j:= d/g.c.d(d, n_j)$, $dia.[ e^2πi n_1/d, …, e^2πi n_5/ d]$ respectively.]: $_Σ_0 ≃_(w_i)^n/G$, with weights $w_i ∈_>0 (1≤i ≤n+1)$, and $G$ a subgroup of the diagonal finite group $SD (:={dia[ t_1, …, t_n+1] | t_i^d_i = ∏_it_i=1 ( 1 ≤i ≤n+1)})$, containing the element $dia [ e^2πi w_1/d, …, e^2πiw_n+1/d] ∈G$, where $d= ∑_i=1^n+1 w_i$ and $d_i = d/g.c.d(d, w_i)$.

Toric Representation and Positive Cone of Picard Group and Deformation Space in Mirror Symmetry of Calabi-Yau Hypersurfaces in Toric Varieties · Kahler cone of a toric variety · contest 0.82

Builds on it 6 papers

By identifying the toric representatives of the representation, we establish the cohomology correspondence between the mirror pair previously found in [ref] , as an extension of the mirror correspondence in [ref] .

Toric Representation and Positive Cone of Picard Group and Deformation Space in Mirror Symmetry of Calabi-Yau Hypersurfaces in Toric Varieties · Concluding Remarks · 0.98
In light of this, we propose the following conjecture generalizing Batyrev's construction [ref] [The toric construction of mirror pairs of Calabi-Yau manifolds restricted to Fermat hypersurfaces in weighted projective space was first observed by Roan [ref] .]:

A Generalized Construction of Calabi-Yau Models and Mirror Symmetry · Combinatorial calculations · 0.91
This generalizes the transposition prescription [ref] and [ref] , in Cox variables [ref] , to transpolar pairs of VEX polytopes [ref] .

Hirzebruch Surfaces, Tyurin Degenerations and Toric Mirrors: Bridging Generalized Calabi-Yau Constructions · Mirror Pairs · 0.98

Uses its method or tool 40 papers

For this, we use the correspondence proposed by Batyrev between the CY families and reflexive polyhedra [ref] .

Coincidences between Calabi-Yau manifolds of Berglund-Hubsch type and Batyrev polytopes · Introduction · 0.97
On the other hand, following Batyrev [ref] , we can build a mirror of the original orbifold as follows.

On the equivalence of Batyrev and BHK Mirror symmetry constructions · Introduction · 0.85
To do this, we use Batyrev's approach to the mirror symmetry [ref] in the same way as in [ref] to construct GLSM for Fermat hypersurface cases.

GLSM for Calabi-Yau Manifolds of Berglund-Hubsch Type · Localization and Mirror symmetry · 0.95

Uses its data or values 3 papers

These searches cannot be done exhaustively: the largest-known collection contains up to $10^296$ [ref] CYs[More pointedly, the currently best algorithm for enumerating these CYs [ref] would require iterating over at least $10^276$ items to generate this entire collection [ref] .], each constructed from a certain fine, regular, `star' triangulation (denoted FRST) of a $4$D polytope [ref] .

Sampling Triangulations and Calabi-Yau Threefolds with Autoregressive GNNs · Introduction · 0.85
However, at most 14 of the 20 relations can be linearly independent, which can be seen from Batyrev's calculation of $b_4$ of the quintic mirror [ref] .

Enumerative geometry of the mirror quintic · The mirror quintic, its curves, and its divisors · 0.83
The mirror[According to [ref] $χ=2160$ moreover from Thm. 4.3.7 of [ref] ($n$ in [ref] is the dimension of $ℙ_Δ$) one has $h_31(M_4)=426$, $h_11(M_4)=1$.

Landscaping with fluxes and the E8 Yukawa Point in F-theory · Integral basis for the primitive horizontal homology of fourfolds · 0.95

Compares with it 5 papers

In Section [ref], we found that the technique suggested by Hori and Vafa [ref] for determining the Picard-Fuchs equations satisfied by $Π_M_G$ when the Calabi-Yau condition ([ref]) holds yields results which agree with those obtained in [ref] working in the Batyrev-Borisov [ref] framework.

Hori-Vafa mirror periods, Picard-Fuchs equations, and Berglund-H\"{u}bsch-Krawitz duality · Discussion · 0.95
Not only do the numerical (and $m$-independent) values of the so-defined contributions $ĉ_1$ and $ĉ_2$ equal the total sums [ref] , respectively, but they match term-by-term the contributions [ref] in Batyrev's formula for the Euler characteristic [ref] (denoting $X'=X_2,3,4$ and $X”=X_5,6$): ĉ_-1 =c_2(p^*) c_3(X') c_1(X”) c_1^-1() =(J_1^ 2)(J_1^ 2)(2J_2)(4J_1^ -1+(2-m)J_2^ -1) =1·1·2·4 =8. ĉ_0 =[c_1(p^*) c_3(X') c_1(X”) +c_2(p^*) c_2(X') c_1(X”)](c_1^ 0()=1) =[1·1·2 + 1·3·2]·1 =8. ĉ_1 =[c_1(p^*) c_3(X') c_0(X”) +c_1(p^*) c_2(X') c_1(X”) +c_2(p^*) c_1(X') c_1(X”) +c_2(p^*) c_2(X') c_0(X”)]c_1(), =[1·1·1·(2-m)+m·1·1·4]_22+3m +[1·3·2·4] +[1·3·2·4]_2×6·4 +[1·3·1·(2-m)]_3(21-m) =56. ĉ_2 =[c_1(p^*) c_2(X') c_0(X”) +c_1(p^*) c_1(X') c_1(X”) +c_2(p^*) c_0(X') c_1(X”) +c_2(p^*) c_1(X') c_0(X”)]c_1^ 2(), =[1·3·1·8(2-m)+m·3·1·16] _3·8(22+m) +[1·3·2·16] +[1·1·2·16]_4·2·16 +[1·3·1·8(2-m)]_3·8(21-m)

Chern Characteristics and Todd-Hirzebruch Identities for Transpolar Pairs of Toric Spaces · Four Dimensions · 0.98
In contrast to the standard works for Calabi-Yau hypersurfaces in Fano toric varieties, e.g. in [ref] for threefolds and in [ref] for the sextic fourfold, where the anti-canonical divisor is ample, we have to deal with the case where the anti-canonical divisor is only semiample and hence compatible with the resolution of singularities.

Three-form periods on Calabi-Yau fourfolds: Toric hypersurfaces and F-theory applications · Calabi-Yau fourfold hypersurfaces · 0.83

Cites it as support 2 papers

The result then follows from Proposition 3.1.3 of [ref] .

Resolution of degenerate mirror families via toric morphisms · Singular locus in $𝒳^*_S$ · 0.98
These components are smooth and irreducible by [ref] .

Statistics of Limit Root Bundles Relevant for Exact Matter Spectra of F-Theory MSSMs · Dual graph · 0.98

Cites it as background 82 papers

The Batyrev–Borisov construction of mirror Calabi–Yau manifolds [ref] relies on swapping the roles played by the reflexive (convex) polytopes in a standard polar pair [ref] .

Chern Characteristics and Todd-Hirzebruch Identities for Transpolar Pairs of Toric Spaces · The Mirror-Symmetric Framework · 0.92
In [ref] , Batyrev identified that the toric varieties induced by fine, regular, star triangulations (FRSTs) of four-dimensional reflexive polytopes admit smooth CY threefold hypersurfaces.

Calabi-Yau Threefolds from Vex Triangulations · Introduction · 0.84
In what follows, we will focus on the so called stringy topological data as originally introduced by Batyrev in the context of reflexive polytopes, $$, and the associated toric varieties, $X_$ [ref] .[Related work on calculating the stringy Chern classes, $c_i(X)$, in terms of the combinatorial data for more general polytopes $Δ$ can be found in [ref] .] The main idea is that certain topological invariants, including the Hodge numbers and in particular the Euler number, are independent of the choice of the so called crepant desingularization of $X_$ [ref] .

Machine Learning Kreuzer--Skarke Calabi--Yau Threefolds · Analytic Formulæ · 0.87

Discussion

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[1] [1]

P.S. Aspinwall and C.A. L¨ utken, and G.G. Ross, Construction and couplings of mirror manifolds, Phys. Lett. B 241, n.3, (1990), 373-380

View on Pith 1990

[2] [2]

P.S. Aspinwall and C.A. L¨ utken, Geometry of mirror manifolds , Nuclear Physics B, 353 (1991), 427-461

View on Pith 1991

[3] [3]

P.S. Aspinwall and C.A. L¨ utken, Quantum algebraic geometry of superstring compact- iﬁcations, Nuclear Physics B, 355 (1991), 482-510

View on Pith 1991

[4] [4]

P.S. Aspinwall, Multiple Mirror Manifolds and Topology Change in String The ory, Preprint IASSNS-HEP-93/4

View on Pith

[5] [5]

V. V. Batyrev, Boundedness of the degree of higher-dimensional toric Fano varieties, Moscow Univ. Math. Bull. 37, n.1, (1982), 28-33

View on Pith 1982

[6] [6]

V. V. Batyrev, Variations of the Mixed Hodge Structure of Aﬃne Hypersurfac es in Algebraic Tori, Duke Math. J. 69, (1993), 349-409

View on Pith 1993

[7] [7]

V. V. Batyrev, Quantum Cohomology Rings of Toric Manifolds , Preprint MSRI, (1993)

View on Pith 1993

[8] [8]

V.V. Batyrev, D. A. Cox, On the Hodge Structure of Projective Hypersurfaces in Toric Varieties, Preprint 1993

View on Pith 1993

[9] [9]

V.V. Batyrev, D. van Straten, Generalized Hypergeometric Functions and Rational Curves on Calabi-Yau Complete Intersections in Toric Varie ties, Preprint 1993

View on Pith 1993

[10] [10]

P. Berglund, T. H¨ ubsch,A Generalized Construction of Mirror Manifolds , in Essays on Mirror Manifolds (ed. S.-T. Yau), International Press Co., Hong Kong, 1992, 388-407. 41

View on Pith 1992

[11] [11]

A.A. Borisov, L.A. Borisov, Singular toric Fano varieties , Mat. Sbornik 183, n.2, (1992), 134-141 (in Russian)

View on Pith 1992

[12] [12]

P. Candelas, M. Lynker, R. Schimmrigk, Calabi-Yau Manifolds in Weighted P4, Nucl. Phys., B 341 (1990), 383-402

View on Pith 1990

[13] [13]

P. Candelas, X.C. de la Ossa, P.S. Green, and L. Parkes, A pair of Calabi-Yau mani- folds as an exactly soluble superconformal theory , Nuclear Phys. B 359 (1991), 21-74

View on Pith 1991

[14] [14]

V.I. Danilov, The geometry of toric varieities . Russian Math. Survey, 33, n.2, (1978), 97-154

View on Pith 1978

[15] [15]

V.I. Danilov, A.G. Khovanski ˆi, Newton polyhedra and an algorithm for computing Hodge-Deligne numbers, Math. USSR Izv. 29 (1987) 279-298

View on Pith 1987

[16] [16]

P. Deligne, Th´ eory de Hodge, II, III, Publ. Math. I.H.E.S. 40 (1971) 5-58; 44 (1975) 5-77

View on Pith 1971

[17] [17]

L. Dixon, in Superstrings. Uniﬁed Theories and Cosmology 1987 , G. Furlan et al., eds., World Scientiﬁc 1988

View on Pith 1987

[18] [18]

G. Ellingsrud and S.A. Strømme, The number of twisted cubic curves on the general quintic threefold, in Essays on Mirror Manifolds (ed. S.-T. Yau), Internation al Press Co., Hong Kong (1992), 181-240

View on Pith 1992

[19] [19]

A. Font, Periods and duality symmetries in Calabi-Yau com pactiﬁcat ions, Nucl. Phys. B 389 (1993) 153

View on Pith 1993

[20] [20]

I. M. Gelfand, A.V. Zelevinsky, and M.M. Kapranov, Equations of hypergeometric type and toric varieties , Func. Anal. Appl. 28 (1989), n. 2, 12-26; English trans. 94-106

View on Pith 1989

[21] [21]

M.B. Green, J.H. Schwarz, E. Witten, Superstring theory I, II, Cambrdge Univ. Press, Cambridge, New York, New Rochelle, Melbourne, Sydney, 1988

View on Pith 1988

[22] [22]

B.R. Greene and M.R. Plesser, Duality in Calabi-Yau moduli space , Nuclear Phys. B 338 (1990), 15-37

View on Pith 1990

[23] [23]

B.R. Greene and M.R. Plesser, An Introduction to Mirror Manifolds , in Essays on Mirror Manifolds (ed. S.-T. Yau), International Press Co., Hong Kong, 1992, 1-30

View on Pith 1992

[24] [24]

R. Hartshorn, Algebraic Geometry, Springer-Verlag, B erline-Heidelberg-New York, 1977

View on Pith 1977

[25] [25]

D. Hensley, Lattice vertex polytopes with interior lattice points , Paciﬁc J. Math. , 105, n.1, 1983, pp. 183-191

View on Pith 1983

[26] [26]

S. Katz, On the ﬁnitness of rational curves on quintic threefolds, Compositio Math. 60 (1986) 151-162. 42

View on Pith 1986

[27] [27]

S. Katz, Rational curves on Calabi-Yau 3-folds, in Essays on Mirror Manifolds (Ed. S.-T. Yau), Int. Press Co., Hong Kong, (1992) 168-180

View on Pith 1992

[28] [28]

S. Katz, Rational curves on Calabi-Yau manifolds: verifying of pred ictions of Mirror Symmetry, Preprint, alg-geom/9301006 (1993)

View on Pith Pith review arXiv 1993

[29] [29]

A.G. Khovansi ˆi, Newton polyhedra and the genus of full intersections , Funk. Anal Priloz, 12, n.1, (1978), 51-61

View on Pith 1978

[30] [30]

A.G. Khovanski ˆi, Newton polyhedra (resolution of singularities ), Sovremen. Probl. Mat. 22, Moscow (1983) (in Russian). Engl. trans.: J. of Soviet Math ., v.27, 1984, 2811-2830

View on Pith 1983

[31] [31]

A. Klemm, S. Theisen, Consideration of One Modulus Calabi-Yau Compactiﬁcation : Picard-Fuchs Equation, K¨ ahler Potentials and Mirror Maps, Nucl. Phys. B 389 (1993) 153

View on Pith 1993

[32] [32]

A. Klemm. S. Theisen, Mirror Maps and Instanton Sums for Complete Intersections in Weighted Projective Spaces , LMU-TPW 93-08, Preprint April 1993

View on Pith 1993

[33] [33]

A.G. Kouchnirenko, Poly´ edres de Newton et nombres de Milnor , Invent. Math., 32, n.1 (1976)

View on Pith 1976

[34] [34]

M. Kreuzer, R. Schimmrigk, H. Skarke, Abelian Landau-Ginzburg Orbifolds and Mirror Symmetry, Nucl. Hys. B 372 (1992) 61-86

View on Pith 1992

[35] [35]

W. Lerche, C. Vafa and N. Warner, Nucl. Phys. B324 (1989) 427

View on Pith 1989

[36] [36]

A. Libgober, and J. Teitelbaum, Lines on Calabi-Yau complete intersections, mirror symmetry, and Picard-Fuchs equations, Duke Math. J., Int. Math. Res Notices 1 (1993) 29

View on Pith 1993

[37] [37]

M. Lynker, R. Schimmrigk, Landau-Ginzburg theories as orbifolds , Phys. Lett. B, 249, n.2 (1990) 237-242

View on Pith 1990

[38] [38]

D.G. Markushevich, Resolution of singularities (toric method) , appendix to: D.G. Markushevich, M.A. Olshanetsky, and A.M. Perelomov, Description of a class of su- perstring compactiﬁcation related to semi-simple Lie alge bras, Comm. Math. Phys. 111 (1987), 247-274

View on Pith 1987

[39] [39]

D. Morrison, Mirror symmetry and rational curves on quintic threefolds: a guide for mathematicians, J. Amer. Math. Soc. 6 (1993), 223-247

View on Pith 1993

[40] [40]

D. Morrison, Picard-Fuchs equations and mirror maps for hypersurfaces , in Essays on Mirror Manifolds (ed. S.-T. Yau), International Press Co., Hong Kong, 1992, 241-264

View on Pith 1992

[41] [41]

T. Oda, Convex Bodies and Algebraic Geometry - An Introduction to th e Theory of Toric Varieties, Ergebnisse der Math. (3)15, Springer-Verlag, Berlin, Hei ldelberg,New York, London, Paris, Tokyo,1988. 43

View on Pith 1988

[42] [42]

M. Reid, Decomposition of toric morphisms , in Arithmetic and Geometry, papers ded- icated to I.R. Shafarevich on the occasion of his 60th birthd ay (M. Artin and J. Tate, eds.), vol.II, Geometry, Progress in Math. 36, Birkhauser, Boston, Basel, Stuttgar t, 1983, 395-418

View on Pith 1983

[43] [43]

S.-S. Roan, S.-T. Yau On Ricci Flat 3-Folds, Acta Mathematica Sinica, New Series 3 (1987), 256-288

View on Pith 1987

[44] [44]

S.-S. Roan, The Mirror of Calabi-Yau Orbifold , Int. J. of Math., vol. 2, n. 4 (1991) 439-455

View on Pith 1991

[45] [45]

R. Schimmrigk, The construction of mirror symmetry , preprint HD-THEP-92-17. 44

View on Pith