On Calabi-Yau Complete Intersections in Toric Varieties

Lev A. Borisov; Victor V. Batyrev

REVIEW

On Calabi-Yau Complete Intersections in Toric Varieties

Victor V. Batyrev, Lev A. Borisov

alg-geom · Submitted 18 Dec 1994 · 16 references

Pith's verdict: A solid, foundational verification paper: the nef-partition duality is checked against the mirror Hodge-number prediction for the (0,q) and (1,q) rows, with the scope limits stated up front.

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CitationLev A. Borisov, Victor V. Batyrev. "On Calabi-Yau Complete Intersections in Toric Varieties." arXiv:alg-geom/9412017 (1994). https://pith.science/paper/T2UGT5JC

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Editor's note

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Bottom line: this is the paper that turns Borisov's nef-partition construction into a checked statement, at least for the (0,q) and (1,q) rows of the Hodge diamond. If you work on toric complete intersections you already know it; if you don't, this is the cleanest place to see why the duality is believable rather than merely elegant. The construction is Borisov's [6]; what is new here is the verification against the mirror-symmetry prediction, and that is the part worth your time. The abstract says so plainly at abstract:s1, and it does not oversell: abstract:s3 states the restriction to h^{0,q} and h^{1,q} outright, so the reader's worry that the framing is broader than the content does not really land on the text. The (1,q) limitation is likewise acknowledged in Section 1, not buried. The machinery is careful. Theorem 4.16 reduces to irreducible nef-partitions via Theorem 5.8 and a finite-covering argument, and the degeneration there is sound because 2.5 concentrates both E_1 pages in the same degree with the same dimension. Propositions 7.2 and 7.4 are the combinatorial heart of Theorem 7.1, and they are actually proved, not gestured at. The examples in 8.7–8.9 (h^{2,1} = 73, 89, 73, 65 for V_{3,3}, V_{2,4}, V_{2,2,3}, V_{2,2,2,2}) match the standard values, which is independent evidence the formulas are right. Soft spots, in proportion. The load-bearing external input is Proposition 1.1, taken from Batyrev–Dais [4], which was a Bonn preprint at the time. That identification is what licenses reading the result as a statement about the singular V rather than only about V̂ and Ŵ; if it failed, the Hodge-number theorem about the desingularizations would still stand. In fairness, [4] and [6] both appeared in print later, so this is largely a historical artifact of the 1994 timestamp rather than a live gap. Second, a few steps are compressed behind 'it is easy to prove' (6.4, 6.6), which costs a reader an afternoon but nothing more. Who should read it: anyone working on mirror symmetry for toric complete intersections, and anyone who wants the combinatorial side of the story done properly. It deserves a serious referee, and I would bring it to a reading group without hesitation.

Referee report

Significance

This is a foundational contribution to toric mirror symmetry: it extends Batyrev's hypersurface duality to complete intersections via Borisov's nef-partitions and supplies the first systematic verification of the mirror Hodge-number prediction in that setting. The combinatorial core (Theorems 4.16, 5.10, 7.1, 9.6) is internally coherent, and the explicit examples in 8.7–8.9 (h^{2,1} = 73, 89, 73, 65 for V_{3,3}, V_{2,4}, V_{2,2,3}, V_{2,2,2,2}) match the standard literature, which is independent evidence that the formulas are correct. The paper is also honest about its scope: the (1,q) verification is complete only for projective-space complete intersections, and the string-theoretic interpretation is delegated to [4]. The combinatorial lemmas 7.2 and 7.4 are proved in the text rather than quoted.

Major objections (3)

[§1, Proposition 1.1 and abstract (abstract:s3)] The identification h^{p,q}_st(V) = h^{p,q}(V̂) for p = 0, 1 and 0 ≤ q ≤ d−r is cited from [4] (Batyrev–Dais, MPI-94-115, alg-geom/9410001), which is listed as a preprint. Since the paper's headline interpretation — that the nef-partition duality agrees with the mirror-symmetry prediction for the singular V itself — rests on this identification, the authors should state precisely what [4] establishes: whether the identification is a theorem or a conjecture there, and the exact range of q for which it is proved. The paper's own Hodge-number statements about V̂ and Ŵ (Theorems 4.16, 7.1, 9.6) are independent of [4], so this is a concern about the interpretation rather than about the core theorems; a short clarifying paragraph in §1 would resolve it.
[§1 and abstract (abstract:s1)] The abstract's phrasing that the combinatorial duality 'agrees with the duality for Hodge numbers predicted by mirror symmetry' can be read as claiming the full (p,q) relation (2), whereas the paper proves (0,q) duality in general (Theorem 4.16 with Corollary 5.10) and (1,q) duality in general only at the level of Euler characteristics (Theorem 7.1), with full (1,q) duality established only for complete intersections in projective space (Theorem 9.6). The introduction acknowledges this distinction, but the abstract should be sharpened to match, for instance by saying explicitly that the (1,q) verification is complete only for projective-space complete intersections.
[§9, first paragraph] The hypothesis 'd_1 + ⋯ + d_r = d' is inconsistent with the rest of §9 and should read d_1 + ⋯ + d_r = d + 1. The examples 8.7–8.9 confirm the intended condition (3+3 = 6 = 5+1, 2+2+3 = 7 = 6+1, 2+2+2+2 = 8 = 7+1), and the formula in Proposition 9.5, −d + Σ_{i=1}^r (l(∇_i) − 1) = −d + (d+1) = 1, only evaluates to 1 under d_1 + ⋯ + d_r = d + 1. As written, the stated hypothesis contradicts the Calabi–Yau condition and the subsequent computation; the fix is immediate but the hypothesis as printed is wrong.

Minor objections (7)

[§5, proof of Theorem 4.16] The sentence 'Since ''Ẽ^{p,q} degenerates at ''Ẽ^{p,q}_1 (see the above arguments)' appears to refer to degeneration at E_2, which is what the preceding paragraph establishes for irreducible nef-partitions via Corollary 3.5. The conclusion that ''E^{p,q} degenerates at ''E^{p,q}_2 is correct, but the reference to E_1 should be corrected to E_2.
[§6, Proposition 6.1] The claim that the Euler sequence for P̂ can be restricted to V̂ 'without obtaining additional Tor-sheaves' relies on transversality of Ẑ_1, …, Ẑ_r to all toric strata. A sentence of justification, or a citation to the standard toric transversality statement, would make this step self-contained.
[§8, Corollary 8.4] The derivation of the displayed formulas from Corollary 8.2 and Proposition 8.3 involves reorganizing the sum over v ∈ V(∆*) into sums over faces Θ of ∆* of fixed dimension. A sentence explaining this reorganization, and in particular the passage from the pointwise terms h^k(O_{D(v)∩V̂}) to the face-indexed sums, would help the reader follow the computation.
[§9, Proposition 9.5] The equality h^1(Ω^1_V) = 1 used at the end of the proposition is the Lefschetz statement h^{1,1}(V) = 1 for a complete intersection in P^d. This is standard but should be cited or justified in one line, since it is the bridge from the computed value of h^{d−r−1}(Ω^1_Ŵ) to the duality statement of Theorem 9.6.
[§5, Corollary 5.10] The corollary uses the natural isomorphisms H^i(V, O_V) ≅ H^i(V̂, O_V̂) for i = 1, …, d−r. For a MPCP-desingularization this follows from the vanishing of the higher direct images of O_V̂, but the paper does not state or cite this. A short justification or reference would make the step explicit.
[§8 and §9] The formulas in §8 and §9 assume d−r ≥ 3. For d−r ≤ 2, Proposition 4.15 makes V̂ smooth and the Hodge numbers classical; the paper could note in one sentence that these cases are covered by the standard theory and are therefore excluded from the formulas.
[Throughout] There are several typographical errors that should be corrected: 'chahge' in §7, 'cointained' in Definition 5.1, 'pulbacks' in Proposition 8.3, 'analgous' and 'verﬁcation' in the introduction, and 'dege nerate' in §3. The distinction between V and V̂ (and between W and Ŵ) is clear from context but could be emphasized once in the abstract.

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The paper

The paper claims

A combinatorial mirror recipe for toric Calabi–Yau complete intersections is checked against the Hodge-number predictions of mirror symmetry, and it passes in the low-degree cases the authors can reach.

The reading

Mirror symmetry predicts that a Calabi–Yau $n$-fold $V$ and its mirror partner $W$ have Hodge numbers related by $h^{p,q}(V) = h^{n-p,q}(W)$. For Calabi–Yau varieties cut out as complete intersections of $r$ divisors inside a Gorenstein toric Fano variety, Borisov had proposed a purely combinatorial mirror construction: decompose the reflexive polyhedron $P$ of the ambient toric variety into a Minkowski sum $P = P_1 + ... + P_r$ (a so-called nef-partition) and build the mirror from the dual partition. The question this paper takes up is whether that recipe really reproduces the Hodge-number relation, rather than merely looking plausible. The authors prove the $(0,q)$ half of the duality in full, prove the $(1,q)$ half as a sign identity for the Euler characteristic of the sheaf of 1-forms, and obtain the complete $(1,q)$ statement for complete intersections in projective space. What is at stake is whether a finite piece of combinatorics on lattice polyhedra can stand in for the geometry of a mirror family.

What carries the argument

The load-bearing device is the nef-partition and its dual. A reflexive polyhedron $P$ (one whose toric variety is Gorenstein Fano with $0$ as its unique interior lattice point) is written as a Minkowski sum $P = P_1 + ... + P_r$ in such a way that the corresponding piecewise-linear functions take only the values $0$ and $1$ on the primitive normal vectors of the facets; the dual partition is built by taking, for each $i$, the convex hull of $0$ and the normal vectors on which the $i$-th function equals $1$. The Hodge numbers are then extracted by two standard exact sequences run in parallel: the Koszul resolution of the complete intersection, and the toric Euler sequence $0 -> Omega^1 -> mathcal{O}^{oplus d} -> oplus_v mathcal{O}_{D(v)} -> 0$ restricted to the intersection. The sign $(-1)^{d-r}$ in Theorem 7.1 comes from a pairing of lattice points: a nonzero interior lattice point $w$ of a sum $P_i + sum_{j in J} P_j$ corresponds to the origin lying in the relative interior of a dual sum $sum_{j notin J} Q_j(w)$, with the two dimensions adding to $d$ (Proposition 7.2); a second, shifted version of the same pairing handles the divisor-restricted pieces and adds to $d-1$ (Proposition 7.4). The bookkeeping is organised by a reduction theorem (Theorem 5.8) that splits any nef-partition uniquely into irreducible summands, so that the spectral sequences degenerate and the general case follows from the product case by a finite covering argument.

Extended reading notes

Core claim

The paper's central claim is that the nef-partition duality on reflexive lattice polyhedra agrees with the Hodge-theoretic duality demanded by mirror symmetry, for the string-theoretic Hodge numbers $h^{0,q}_{st}$ and $h^{1,q}_{st}$ of a Calabi–Yau complete intersection $V$ of $r$ semi-ample divisors in a $d$-dimensional Gorenstein toric Fano variety, with $0 <= q <= d-r$. Concretely, the authors show that $h^i(mathcal{O}_V) = h^i(mathcal{O}_W)$ for the dual pair (Theorem 4.16), which together with Serre duality gives $h^i(mathcal{O}_{V}) = h^{d-r-i}(mathcal{O}_{W})$, i.e. the $(0,q)$ duality, and they give a closed combinatorial formula for the generating polynomial $E(P,t) = sum_i h^i(mathcal{O}_V) t^i$ as an alternating sum of counts of interior lattice points of Minkowski sums of the pieces of the partition (Corollary 5.9). For the $(1,q)$ numbers they prove that $chi(Omega^1_V) = (-1)^{d-r} chi(Omega^1_W)$ (Theorem 7.1), which is exactly the Euler-characteristic shadow of the expected term-by-term duality $h^{1,q}(V) = h^{1,d-r-q}(W)$, and they upgrade this to the full term-by-term statement for complete intersections in projective space (Theorem 9.6).

Load-bearing premise

The verification is about the usual Hodge numbers of a partial desingularization, so it only counts as a verification for the singular Calabi–Yau variety itself if the string-theoretic Hodge numbers really do coincide with those, as the paper asserts by citing an earlier preprint rather than re-deriving.

Editorial extensions

If this is right

The $(0,q)$ string-theoretic Hodge numbers of a toric Calabi–Yau complete intersection are computable by an explicit lattice-point sum over subsets of the nef-partition, so the mirror partner's Hodge numbers need not be computed independently.
The Euler characteristic of the sheaf of 1-forms changes sign by $(-1)^{d-r}$ under the duality, giving a checkable numerical invariant of the mirror construction in every dimension.
For complete intersections of ample divisors, explicit closed formulas for all $(p,1)$ Hodge numbers follow from the toric Euler sequence and the face structure of the dual polyhedron, yielding the worked values $h^{2,1}(V_{3,3}) = 73$, $h^{2,1}(V_{2,4}) = 89$, $h^{2,1}(V_{2,2,3}) = 73$, and $h^{2,1}(V_{2,2,2,2}) = 65$.
For complete intersections in projective space the $(1,q)$ duality holds term by term, so the full Hodge diamond of the mirror is determined by that of the original.
The vanishing results for intermediate cohomology of $mathcal{O}_V$ give, as a by-product, a criterion for when the complete intersection is nonempty and irreducible, expressed purely in the independence of the lattice polyhedra.

Reading between the lines

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

The gap between what is proved and what mirror symmetry asks for is concentrated in the $(1,q)$ numbers in dimensions $d-r >= 4$: the paper establishes only the alternating sum there, so a natural next test is a four-dimensional example where the individual $h^{1,q}$ can be computed by an independent method and compared against the predicted mirror values.
The lattice-point pairing behind Proposition 7.2 looks reusable: it converts an interior-point count for a Minkowski sum into an interior-point count for a dual sum with complementary dimension, which is the same shape of statement one would want for higher-degree Hodge numbers $h^{p,q}$ with $p >= 2$, where the paper currently has formulas only in the ample case.
The reduction to irreducible nef-partitions suggests that the interesting geometric content sits in the irreducible summands, and that a product of Calabi–Yau varieties should be treated as a single mirror-consistent object rather than as a degenerate case.
Because the argument passes through a finite covering when the partition splits only over a sublattice of finite index, one could ask whether the Hodge-number duality survives on the quotient by the covering group; the paper's method is insensitive to this, which is either a strength or a hidden assumption depending on the example.

Original abstract

We investigate Hodge-theoretic properties of Calabi-Yau complete intersections $V$ of $r$ semi-ample divisors in $d$-dimensional toric Fano varieties having at most Gorenstein singularities. Our main purpose is to show that the combinatorial duality proposed by second author agrees with the duality for Hodge numbers predicted by mirror symmetry. It is expected that the complete verification of mirror symmetry predictions for singular Calabi-Yau varieties $V$ of arbitrary dimension demands considerations of so called {\em string-theoretic Hodge numbers} $h^{p,q}_{\rm st}(V)$. We restrict ourselves to the string-theoretic Hodge numbers $h^{0,q}_{\rm st}(V)$ and $h^{1,q}_{\rm st}(V)$ $(0 \leq q \leq d-r) which coincide with the usual Hodge numbers $h^{0,q}(\widehat{V})$ and $h^{1,q}(\widehat{V})$ of a $MPCP$-desingularization $\widehat{V}$ of $V$.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,958 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.95 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.49 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We investigate Hodge-theoretic properties of Calabi-Yau complete intersections $V$ of $r$ semi-ample divisors in $d$-dimensional toric Fano varieties having at most Gorenstein singularities. Our main purpose is to show that the combinatorial duality proposed by second author agrees with the duality for Hodge numbers predicted by mirror symmetry. It is expected that the complete verification of mirror symmetry predictions for singular Calabi-Yau varieties $V$ of arbitrary dimension demands considerations of so called {\em string-theoretic Hodge numbers} $h^{p,q}_{\rm st}(V)$. We restrict ourselves to the string-theoretic Hodge numbers $h^{0,q}_{\rm st}(V)$ and $h^{1,q}_{\rm st}(V)$ $(0 \leq q \leq d-r) which coincide with the usual Hodge numbers $h^{0,q}(\widehat{V})$ and $h^{1,q}(\widehat{V})$ of a $MPCP$-desingularization $\widehat{V}$ of $V$.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We investigate Hodge-theoretic properties of Calabi-Yau complete intersections $V$ of $r$ semi-ample divisors in $d$-dimensional toric Fano varieties having at most Gorenstein singularities.	contribution 0.94	0.84	0.05	asserted 0.86
s1	Our main purpose is to show that the combinatorial duality proposed by second author agrees with the duality for Hodge numbers predicted by mirror symmetry.	contribution 0.77	0.52	0.55	asserted 0.99
s2	It is expected that the complete verification of mirror symmetry predictions for singular Calabi-Yau varieties $V$ of arbitrary dimension demands considerations of so called {\em string-theoretic Hodge numbers} $h^{p,q}_{\rm st}(V)$.	background 0.97	0.04	0.04	hedged possibility 0.70
s3	We restrict ourselves to the string-theoretic Hodge numbers $h^{0,q}_{\rm st}(V)$ and $h^{1,q}_{\rm st}(V)$ $(0 \leq q \leq d-r) which coincide with the usual Hodge numbers $h^{0,q}(\widehat{V})$ and $h^{1,q}(\widehat{V})$ of a $MPCP$-desingularization $\widehat{V}$ of $V$.	limitation 0.97	0.69	0.07	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.98.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (16)

[6]

L.A. Borisov, Towards the Mirror Symmetry for Calabi-Yau Complete Inter- sections in Gorenstein Toric Fano Varieties , Prepint 1993, alg-geom/9310001

View on Pith Pith review arXiv 1993
[4]

V. V. Batyrev, D.I. Dais, Strong McKay Corrrespondence, String-theoretic Hodge Numbers and Mirror Symmetry , Preprint Max-Plank Inst. Bonn, MPI- 94-115, alg-geom/9410001

View on Pith Pith review arXiv
[1]

V. V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersur - faces in toric varieties , J. Alg. Geom., 3 (1994) 493-535

View on Pith 1994
[2]

V. Batyrev, D. van Straten, Generalized Hypergeometric Functions and Ratio- nal Curves on Calabi-Yau Complete Intersections in Toric Va rieties, Preprint 1993, alg-geom 9307010, to be published in Commun. Math. Phys

View on Pith 1993
[3]

V.V. Batyrev, L.A. Borisov, Dual Cones and Mirror Symmetry for Generalized Calabi-Yau Manifolds , to appear in ”Mirror Symmetry II” (ed. S.-T. Yau), alg-geom/9402002

View on Pith Pith review arXiv
[5]

P. Berglund, T. H¨ ubsch,On a Residue Representation of Deformations, Koszul and Chiral Rings , IASSNS-HEP-94/97, hep-th/9411131

View on Pith Pith review arXiv
[7]

V.I. Danilov, Geometry of toric varieties , Russ. Math. Surv., 33 (1978),97-154

View on Pith 1978
[8]

G. Ellingsrud, S.A. Strømme, Counting twisted cubic curves on general complete intersections, alg-geom/9409006

View on Pith Pith review arXiv
[9]

G. Ellingsrud, S.A. Strømme, Bott’s Formula and Enumerative Geometry , Preprint, November 1994, alg-geom/9411005

View on Pith Pith review arXiv 1994
[10]

P. Griﬃths, J. Harris, Principles of Algebraic Geometry , 1978

View on Pith 1978
[11]

S. Hosono, A. Klemm, S. Theisen and S.T. Yau, Mirror symmetry, mirror map and application to complete intersection Calabi-Yau space s, Preprint HUTMP- 94-02, hep-th/9406055

View on Pith Pith review arXiv
[12]

A. G. Khovansky, Newton polyhedra and toric varieties , Funct. Anal. Appl. 11 (1977), 56-67

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[13]

A. G. Khovansky, Newton polyhedra and the genus of complete intersections , Funct. Anal. Appl. 12 (1978), 38-46

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[14]

A. Klemm, S. Theisen, Mirror Maps and Instanton Sums for Complete Inter- sections in Weighted Projective Space , LMU-TP 93-08, hep-th/9304034

View on Pith Pith review arXiv
[15]

A. Libgober and J. Teitelbaum, Lines on Calabi-Yau complete intersections, mirror symmetry, and Picard-Fuchs equations , Duke Math. J., 69(1): 29-39, Int. Math. Res. Notes (1993). 26

View on Pith 1993
[16]

T. Oda, Convex Bodies and Algebraic Geometry - An Introduction to th e Theory of Toric Varieties, Ergeb. Math. Grenzgeb. (3), vol. 15, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1988. 27

View on Pith 1988

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76 later papers cite it

Evidence: substantial. 76 of the citing papers share no author with it, and 15 of those carry a confident label from Pith's citation classifier.

Pith read 26 of the 127 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

3 later papers by other authors are listed below: 3 used its method or tool.

Uses its method or tool 3 of 12 papers with this label

Used the combinatorial formulas of Batyrev-Borisov from this paper together with an equation.

Modular Calabi-Yau Fourfolds and Connections to M-Theory Fluxes (2023) · 0.99 · paraphrased: the licence does not allow quoting
Used the Batyrev-Borisov construction of this paper for X̃ to obtain the mirror.

Linear Sigma Models With Strongly Coupled Phases -- One Parameter Models (2013) · 0.99 · paraphrased: the licence does not allow quoting
Followed this paper's Batyrev construction, which also requires the dual polytope, to obtain the hypersurface equation for X in 𝒜.

T-Duality and Flavor Symmetries in Little String Theories (2023) · 0.98 · paraphrased: the licence does not allow quoting

How the literature uses it

76 arXiv papers cite this in 127 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 · 74 sentences without a confident label

Builds on it 1 paper

The study we carried out in section [ref] for toric-hypersurface Calabi–Yau fourfolds has a clear generalization to Calabi–Yau fourfolds obtained as complete intersection in toric ambient spaces [ref] .

The Vertical, the Horizontal and the Rest: anatomy of the middle cohomology of Calabi-Yau fourfolds and F-theory applications · Outlook · 0.87

Uses its method or tool 12 papers

Combined with the methods of Batyrev and Borisov [ref] , which we briefly review in sect:Batyrev_Borisov_Review, this allows finding the mirror Calabi-Yau manifold as a subvariety of a suitable toric variety.

Mirror Symmetry for Five-Parameter Hulek-Verrill Manifolds · Toric Geometry and Mirror Symmetry · 0.99
The analysis relies heavily on the machinery of toric geometry and the toric mirror construction by Batyrev and Borisov [ref] .

A pair of Calabi-Yau manifolds from a two parameter non-Abelian gauged linear sigma model · Toric analysis and topological data · 0.98
Using the combinatorial formulas of Batyrev–Borisov [ref] together with eq.

Modular Calabi-Yau Fourfolds and Connections to M-Theory Fluxes · The geometry of the Hulek–Verrill fourfold · 0.99

Compares with it 2 papers

It would also be interesting to compare the result of Theorem [ref] with the previously known formulas for Hodge numbers of complete intersections obtained by Batyrev and Borisov in [ref] .

Closed form expressions for Hodge numbers of complete intersection Calabi-Yau threefolds in toric varieties · Relations with other results · 0.99
Though their reflexive polytope is a Minkowski sum of simplices, they exceed the class of nondegenerate CICYs that can be produced from this sum as in [ref] .

Sch\"on complete intersections · Definition and examples · 0.88

Cites it as background 28 papers

To facilitate this approach to a low-energy phenomenology derived from string theory, mathematicians and physicists have constructed large datasets of Calabi–Yau threefolds [ref] as well as various refined analyses of properties thereof [ref] .

Patterns in Calabi-Yau Distributions · Introduction · 0.80
The theory of mirror symmetry on Calabi-Yau hypersurfaces of toric varieties is very well developed [ref] .[The theory of complete intersections is similar in many aspects [ref] , but we will not be discussing it here.] In this subsection, we will review all aspects required to understand the algorithm for the computation of the genus $0$ Gromov-Witten invariants for this class of Calabi-Yau threefolds.

Determining F-theory matter via Gromov-Witten invariants · Review: an algorithm for computing the Gromov-Witten invariants of anti-canonical hypersurfaces of toric varieties · 1.00
The largest known class of compact CY threefolds is constructed as hypersurfaces in toric varieties: any fine regular star triangulation (FRST) of a four-dimensional reflexive polytope defines a toric Fano variety in which any generic anticanonical hypersurface is a smooth CY threefold [ref] .

Generating Triangulations and Fibrations with Reinforcement Learning · Introduction · 0.97

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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L.A. Borisov, Towards the Mirror Symmetry for Calabi-Yau Complete Inter- sections in Gorenstein Toric Fano Varieties , Prepint 1993, alg-geom/9310001

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V. V. Batyrev, D.I. Dais, Strong McKay Corrrespondence, String-theoretic Hodge Numbers and Mirror Symmetry , Preprint Max-Plank Inst. Bonn, MPI- 94-115, alg-geom/9410001

View on Pith Pith review arXiv

[3] [1]

V. V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersur - faces in toric varieties , J. Alg. Geom., 3 (1994) 493-535

View on Pith 1994

[4] [2]

V. Batyrev, D. van Straten, Generalized Hypergeometric Functions and Ratio- nal Curves on Calabi-Yau Complete Intersections in Toric Va rieties, Preprint 1993, alg-geom 9307010, to be published in Commun. Math. Phys

View on Pith 1993

[5] [3]

V.V. Batyrev, L.A. Borisov, Dual Cones and Mirror Symmetry for Generalized Calabi-Yau Manifolds , to appear in ”Mirror Symmetry II” (ed. S.-T. Yau), alg-geom/9402002

View on Pith Pith review arXiv

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P. Berglund, T. H¨ ubsch,On a Residue Representation of Deformations, Koszul and Chiral Rings , IASSNS-HEP-94/97, hep-th/9411131

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V.I. Danilov, Geometry of toric varieties , Russ. Math. Surv., 33 (1978),97-154

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G. Ellingsrud, S.A. Strømme, Counting twisted cubic curves on general complete intersections, alg-geom/9409006

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G. Ellingsrud, S.A. Strømme, Bott’s Formula and Enumerative Geometry , Preprint, November 1994, alg-geom/9411005

View on Pith Pith review arXiv 1994

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P. Griﬃths, J. Harris, Principles of Algebraic Geometry , 1978

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S. Hosono, A. Klemm, S. Theisen and S.T. Yau, Mirror symmetry, mirror map and application to complete intersection Calabi-Yau space s, Preprint HUTMP- 94-02, hep-th/9406055

View on Pith Pith review arXiv

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A. G. Khovansky, Newton polyhedra and toric varieties , Funct. Anal. Appl. 11 (1977), 56-67

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A. G. Khovansky, Newton polyhedra and the genus of complete intersections , Funct. Anal. Appl. 12 (1978), 38-46

View on Pith 1978

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A. Klemm, S. Theisen, Mirror Maps and Instanton Sums for Complete Inter- sections in Weighted Projective Space , LMU-TP 93-08, hep-th/9304034

View on Pith Pith review arXiv

[15] [15]

A. Libgober and J. Teitelbaum, Lines on Calabi-Yau complete intersections, mirror symmetry, and Picard-Fuchs equations , Duke Math. J., 69(1): 29-39, Int. Math. Res. Notes (1993). 26

View on Pith 1993

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T. Oda, Convex Bodies and Algebraic Geometry - An Introduction to th e Theory of Toric Varieties, Ergeb. Math. Grenzgeb. (3), vol. 15, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 1988. 27

View on Pith 1988