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Toric Representation and Positive Cone of Picard Group and Deformation Space in Mirror Symmetry of Calabi-Yau Hypersurfaces in Toric Varieties
Algebraic Geometry · Submitted 12 Jun 2011
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We further identify the Kahler cone of the toric variety and degeneration cone of CY hypersurfaces, by which the Kahler cone and degeneration cone for a mirror CY pair are interchangeable under mirror symmetry.
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We derive the combinatorial representations of Picard group and deformation space of anti-canonical hypersurfaces of a toric variety using techniques in toric geometry. The mirror cohomology correspondence in the context of mirror symmetry is established for a pair of Calabi-Yau (CY) ${\sf n}$-spaces in toric varieties defined by reflexive polytopes for an arbitrary dimension ${\sf n}$. We further identify the Kahler cone of the toric variety and degeneration cone of CY hypersurfaces, by which the Kahler cone and degeneration cone for a mirror CY pair are interchangeable under mirror symmetry. In particular, different degeneration cones of a CY 3-fold are corresponding to flops of its mirror 3-fold.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We derive the combinatorial representations of Picard group and deformation space of anti-canonical hypersurfaces of a toric variety using techniques in toric geometry. | contribution 0.89 | 0.96 | 0.16 | asserted 1.00 |
| s1 | The mirror cohomology correspondence in the context of mirror symmetry is established for a pair of Calabi-Yau (CY) ${\sf n}$-spaces in toric varieties defined by reflexive polytopes for an arbitrary dimension ${\sf n}$. | result 0.60 | 0.79 | 0.33 | asserted 1.00 |
| s2 | We further identify the Kahler cone of the toric variety and degeneration cone of CY hypersurfaces, by which the Kahler cone and degeneration cone for a mirror CY pair are interchangeable under mirror symmetry. | result 0.85 | 0.90 | 0.39 | asserted 1.00 |
| s3 | In particular, different degeneration cones of a CY 3-fold are corresponding to flops of its mirror 3-fold. | result 0.85 | 0.79 | 0.16 | asserted 1.00 |
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The paper
Original abstract
We derive the combinatorial representations of Picard group and deformation space of anti-canonical hypersurfaces of a toric variety using techniques in toric geometry. The mirror cohomology correspondence in the context of mirror symmetry is established for a pair of Calabi-Yau (CY) ${\sf n}$-spaces in toric varieties defined by reflexive polytopes for an arbitrary dimension ${\sf n}$. We further identify the Kahler cone of the toric variety and degeneration cone of CY hypersurfaces, by which the Kahler cone and degeneration cone for a mirror CY pair are interchangeable under mirror symmetry. In particular, different degeneration cones of a CY 3-fold are corresponding to flops of its mirror 3-fold.
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What it cites · 18 sentences without a confident label
Contests 2 papers
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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 $d_i = d/g.c.d(d, w_i)$.
Picard Groups of Hypersurfaces in Toric Varieties · Kahler cone of a toric variety · contest 0.82
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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 $d_i = d/g.c.d(d, w_i)$.
Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties · Kahler cone of a toric variety · contest 0.82
Builds on 2 papers
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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] .
Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties · Concluding Remarks · 0.98
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This paper can be regarded as a continuation of the work in [ref] about Picard group of hypersurfaces in toric varieties.
Picard Groups of Hypersurfaces in Toric Varieties · Concluding Remarks · 0.83
Cites as background 2 papers
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In this Appendix, we summarize results in [ref] and some basic facts in toric geometry used in this paper for easy reference.
Picard Groups of Hypersurfaces in Toric Varieties · Appendix: Some Basic Facts in Toric Geometry · 0.98
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Definition. $( , L )$ is a reflexive polytope iff both $(, L)$ and $(^*, L^*)$ are integral [ref] .
Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties · Introduction · 0.95
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