Pith. sign in

REVIEW 3 major objections 3 minor 68 references

Categorical mirror symmetry on cohomology for a complex genus 2 curve

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A genus 2 curve is shown to satisfy mirror symmetry on cohomology: line bundles on the curve correspond to Lagrangians in a symplectic fibration, and products are computed by holomorphic triangle counts.

desk verdict A serious, detailed construction of the first non-exact SYZ mirror for a curve of general type, whose main theorem is conditional on unpublished localization machinery for the target category. read the letter →

arxiv 1908.04227 v3 pith:47FBIMCE submitted 2019-08-12 math.SG

Catherine Cannizzo This is my paper · ORCID
classification math.SG MSC 53D3714F0853D4014K25
keywords homologicalmirrorsymmetrygenus2curveFukayacategoryderivedofcoherentsheavesLandau-GinzburgmodelsymplecticfibrationthetafunctionsSYZ
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves a cohomological form of homological mirror symmetry for a complex genus 2 curve $\Sigma_2$, realized as a $\theta$ divisor $H=s^{-1}(0)$ inside the abelian surface $V=(\mathbb{C}^*)^2/\Gamma_B$ with $\Gamma_B=\mathbb{Z}\langle\gamma',\gamma''\rangle$, $\gamma'=(2,1)$, $\gamma''=(1,2)$. It constructs the mirror as a Landau-Ginzburg model $(Y,v_0)$, a $\Gamma_B$-quotient of an infinite-type toric variety with superpotential $v_0=xyz$, equipped with the structure of a symplectic fibration. The main theorem states that the natural restriction diagram commutes and that the vertical maps are fully faithful embeddings: $D^b_{\mathrm{LCoh}}(V)\hookrightarrow H^0\mathrm{Fuk}(V^\vee)$ and $D^b_{\mathrm{LCoh}}(H)\hookrightarrow H^0\mathrm{FS}(Y,v_0)$. A sympathetic reader should care because the product in the mirror category is computed by counting holomorphic triangles, so the canonical ring of $\Sigma_2$ becomes a symplectic invariant: multiplication of sections of powers of the canonical bundle is reproduced by triangle counts in the mirror. The paper is also one of the first to define and use a Fukaya category for a non-exact, non-Lefschetz symplectic fibration with compact torus fibers.

What carries the argument

The central object is the family of linear Lagrangians $\ell_k$ in the SYZ dual abelian variety $V^\vee=T_B\times T_F$, defined in action-angle coordinates by $\theta\equiv-k\lambda\xi\pmod{\mathbb{Z}^2}$ with $\lambda=\begin{pmatrix}2&1\\1&2\end{pmatrix}^{-1}$; these are Lagrangian graphs whose intersections with $\ell_j$ and $\ell_i$ number $(j-i)^2$, matching the dimension of $H^0(V,L^{j-i})$. The same Lagrangians, parallel transported over U-shaped curves in the base of the superpotential $v_0=xyz$, generate the Fukaya-Seidel-type category on $(Y,v_0)$. The symplectic fibration itself is built from a Kähler potential patched from the toric potentials of $\mathbb{C}^3$ charts and of the singular fiber $\mathbb{CP}^2(3)/\Gamma_B$, with the symplectic form chosen so that $v_0$ is a symplectic fibration. The computational engine is a Leibniz rule reducing the differential on all Lagrangians to the differential for the moment-map fiber Lagrangian $t_x$, followed by a cobordism argument that identifies the desired count with $J_0$-holomorphic discs; that disc count equals the $\theta$ function defining the line bundle, while disc-with-sphere configurations are included through an open mirror theorem for infinite-type toric Calabi-Yau manifolds.

What would settle it

Compute the morphism space between $L_k$ and its upward pushoff in the localized category of Definition 4.6 by both the directed bigon count and by roofs; if the two answers differ for some regular almost complex structure, Lemma 4.8 fails and the right vertical embedding has no well-defined target.

Watch

Extended reading notes

Core claim

On its own terms, the discovery is Theorem 1.2: choose $L\to V$ to be the ample line bundle whose factor of automorphy is $s(\gamma,x)=x^{\lambda(\gamma)}\tau^{\kappa(\gamma)}$, with $\lambda=\begin{pmatrix}2&1\\1&2\end{pmatrix}^{-1}$; then $H=s^{-1}(0)$ is a complex genus 2 curve. The paper proves that the diagram with $D^b_{\mathrm{LCoh}}(V)\to D^b_{\mathrm{LCoh}}(H)$ on the complex side and $H^0\mathrm{Fuk}(V^\vee)\to H^0\mathrm{FS}(Y,v_0)$ on the symplectic side commutes, with fully faithful vertical embeddings. The objects on the symplectic side are explicit: $L^k$ maps to the linear Lagrangian $\ell_k$ in $V^\vee$, and $L^k|_H$ maps to the fibered Lagrangian obtained by parallel transporting $\ell_k$ around a U-shaped curve in the base of $v_0$. The core numerical fact is that the $\mu_2$ product in the torus, which counts holomorphic triangles, matches the product of $\theta$ functions computing the multiplication of sections; since $L|_H$ is the canonical bundle of $\Sigma_2$, the ring structure on $\bigoplus_{i\ge0}H^0(\Sigma_2,L^i)$ is determined by those triangle counts. The proof of the right vertical arrow uses a computation of the differential as the $\theta$ function, with disc counts contributing in the leading order and sphere-bubble corrections handled separately.

Load-bearing premise

The load-bearing premise is that the unpublished categorical-localization technology for defining morphisms in a Fukaya-Seidel category works for this non-exact symplectic fibration with compact torus fibers and sphere bubbles; if it does not, the category $H^0\mathrm{FS}(Y,v_0)$ used in the main theorem is not defined.

Editorial extensions

If this is right

  • If the embeddings of Theorem 1.2 hold, the canonical ring $\bigoplus_{i\ge0}H^0(\Sigma_2,L^i)$ is isomorphic to the cohomological endomorphism ring of the mirror Lagrangians, so the projective embeddings of $\Sigma_2$ are determined by Lagrangian intersection data.
  • The product in that ring is computable, in principle, by counting holomorphic triangles in $(Y,v_0)$; the paper writes the $\mu_2$ coefficient as a sum of weights $\tau^{-(l/l'l'')\kappa(\cdots)}$ over lattice elements $\gamma_A$.
  • For the abelian surface $V$, the embedding $D^b_{\mathrm{LCoh}}(V)\hookrightarrow H^0\mathrm{Fuk}(V^\vee)$ gives a direct verification in this example that multiplication of sections of a line bundle matches the triangle count in the mirror torus.
  • Because the line bundles and their shifts generate the subcategories considered, an $A_\infty$ enhancement would extend these cohomological embeddings to the whole derived category $D^b\mathrm{Coh}(H)$, a step the paper identifies as the natural next one.

Reading between the lines

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

  • Editorial inference: the same pattern -- linear Lagrangians mirroring powers of an ample line bundle, with products counted by triangles -- should extend to any smooth curve in an abelian variety; the honeycomb tropicalization of the corresponding theta function would determine the mirror quotient and the triangle weights.
  • Editorial inference: the equality between theta-function multiplication and the $\mu_2$ count gives a testable numerical prediction: for low powers $L$, $L^2$, $L^3$ on $V$, the coefficients of the product of theta functions should reproduce the weighted triangle sums term by term as $\tau\to0$.
  • Editorial inference: if the unpublished localization results used for Definition 4.6 are replaced by an explicit construction, the cohomological Fukaya-Seidel category of $(Y,v_0)$ may admit a description as a wrapped-type Fukaya category on the smooth part of the fibration, which would make the right vertical embedding computable by standard wrapped-Floer methods.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper proposes a cohomological mirror symmetry statement for a complex genus 2 curve realized as a theta divisor H in an abelian surface V. It constructs the SYZ mirror as a Landau-Ginzburg model (Y,v0), where Y is a Gamma_B-quotient of an infinite-type toric variety and v0 = xyz, equipped with a non-exact symplectic fibration structure. Theorem 1.2 asserts two fully faithful embeddings: D^b_LCoh(V) into H^0Fuk(V^vee) (left vertical arrow) and D^b_LCoh(H) into H^0FS(Y,v0) (right vertical arrow). The left arrow is established by explicit computation of morphisms between linear Lagrangians and a triangle count whose weights match theta-function multiplication in Lemma 2.20. The right arrow rests on a definition of a DFS-type Fukaya-Seidel category via categorical localization (Definition 4.6), a monodromy computation (Lemma 4.20), and a computation of the differential as proportional to the theta function after incorporating disc and sphere counts. The main structural gap is that the target category for the right arrow is not constructed self-containedly: its localized morphism and quasi-invariance lemmas are postponed to unpublished work.

Significance. If the category foundations can be supplied, the paper would be a significant contribution: a concrete, computable HMS statement for a general-type curve with the curve on the complex side, and one of the first Fukaya-Seidel categories for a non-exact, non-monotone symplectic fibration with compact fibers. The explicit triangle count in Lemma 2.5 and its matching with theta multiplication in Lemma 2.20 are strong internal evidence for the left arrow; the monodromy computation is detailed; and the differential computation is ambitious and, conditional on the standard tools, internally consistent. The paper also gives a clean description of how the mirror is forced by the tropicalization of the defining theta function. The manuscript is not self-contained in a load-bearing way; however, the gaps are of the missing-proof or unpublished-reference type rather than obvious contradictions, so the result is credible as a research announcement and worthy of a major revision.

major comments (3)
  1. [Section 4.1, Definition 4.6; Lemmas 4.8, 4.9, 4.54] The category H^0FS(Y,v0) in Theorem 1.2 is the target of the main embedding, but its definition is not self-contained: Definition 4.6 defines morphisms by categorical localization, and Lemmas 4.8 and 4.9, which identify the localized morphisms and compositions with directed ones, are assigned to unpublished work of Abouzaid-Seidel and Abouzaid-Auroux; Lemma 4.54 (quasi-invariance on regular choices) is tagged "References for proof." Because Remark 4.11 explicitly notes that the fibration is not exact (compact torus fibers) and Remark 4.13 notes it is not monotone, the standard localization and quasi-invariance arguments do not automatically apply, and no replacement argument is supplied. This is load-bearing: without these lemmas the right vertical arrow is not defined.
  2. [Section 4.6, Lemma 4.54] Quasi-invariance of H^0FS on regular choices is needed for the computation in Section 5 to be an invariant of the category: the paper computes the differential at J0 with an admissible perturbation and uses a cobordism from a generic J, but the cobordism and the independence of the count from the perturbation must be established within the non-exact setting. The proof reference to unpublished work is not enough for a foundational lemma of this kind, since the category used in the main theorem is defined only after this invariance is known.
  3. [Section 5.3, Theorem 5.8] The sphere-bubble contributions to the differential are imported from [KL19] for the infinite-type toric cover ~Y, while the target Y is the quotient by Gamma_B with compact fibers. The paper does not prove that the relevant counts, homology classes, and weights descend to the quotient, or that the open Gromov-Witten invariants used in Corollary 5.3 are Gamma_B-invariant. Without such a descent statement, the proportionality of the differential to the theta function is not established on Y itself.
minor comments (3)
  1. [Highlights] There is a typo in the Highlights: "sympectic side" should be "symplectic side."
  2. [Theorem 1.2 and Section 2] The paper uses H^0Fuk(V^vee) and H^0FS(Y,v0) for cohomological categories, but the Maslov grading conventions for the linear Lagrangians are not spelled out; since the grading determines which part of Floer cohomology is taken, a sentence specifying the grading would help the reader.
  3. [Definition 3.28] The bump functions alpha3,...,alpha6 are constrained only by inequalities in the text; their precise domains, boundary conditions, and derivative bounds should be collected in one place, since Appendix A is invoked repeatedly and the reader cannot easily check the claimed smallness of derivative contributions.

Circularity Check

1 steps flagged · score 6.0 of 10

The mirror is built from Trop(s) and then 'recovers' the same theta function s as the computed differential; the recovery reduces to unpacking the construction, though the categorical embedding retains independent content.

  1. self definitional [Section 3.3, Eq. (3.10); Section 5.2, Theorem 5.4]
    "∆˜Y := {(ξ1, ξ2, η) ∈ R^3 | η ≥ Trop(s)(ξ)} ... Trop(s)(ξ) := max_γ κ(γ) + ⟨ξ, λ(γ)⟩ ... Theorem 5.4. ... Then the disc count equals the defining theta function s(x) = Σ_{n∈Z^2} x_1^{-n1} x_2^{-n2} τ^{1/2 n^t ((2,1),(1,2)) n} up to a coordinate change."

    The mirror (Y, v0) is built in Section 3.3 from the tropicalization of the same theta function s that Section 5.2 presents as a computed output: ∆_Y = {η ≥ Trop(s)} with Trop(s) = max_γ κ(γ)+⟨ξ,λ(γ)⟩. The proof of Theorem 5.4 obtains disc areas from the facet data of this same polytope, ν(F_{m1,m2})=(−m1,−m2,1)^t and α(F_{m1,m2})=m1^2+m1m2+m2^2, then compares the [CO06] series with the theta series and finds identical exponents and coefficients. The 'recovery' of s as the differential is therefore not an independent symplectic prediction; it is the input tropical function repackaged through the construction of (Y, v0), up to the standard local-system coordinate change. The embedding statements retain independent content, but this derivation of s is by construction.

full rationale

The main circular step is the theta-function input-output loop: the mirror's SYZ polytope is defined by η ≥ Trop(s)(ξ), and Theorem 5.4 then announces that the disc count equals s(x) 'up to a coordinate change.' Because the facet equations used in the count are read off from the same tropical function, the differential computation reduces to unpacking the definition of (Y, v0) rather than to an independent verification of the canonical ring of Σ2. This is partial circularity: the fully faithful embedding of D^b_LCoh(H) into H^0FS(Y, v0) is a category-theoretic claim that does not automatically follow from the construction of the polytope, and the left-arrow computation for abelian varieties is an independent flat-torus triangle count matched with theta multiplication. I did not count the reliance on unpublished Abouzaid-Seidel localization lemmas (Lemmas 4.8, 4.9, 4.54) as circularity: those are external cited tools, not self-citations, and the concern about whether they apply in this non-exact, compact-fiber setting is a correctness or rigor issue rather than a circularity. No other load-bearing step was found to reduce to its own inputs. Score 6 reflects one central 'prediction' that is forced by the construction while the overall embedding program retains independent content.

Assumptions & free parameters 4 free parameters · 7 assumptions · 2 invented entities

The central claim rests on: (i) the AAK generalized SYZ construction (imported, not re-proven); (ii) the unpublished Abouzaid-Seidel localization used to define the target category; (iii) external counting theorems ([CO06], [Cha11], [KL19], Givental) for the disc and sphere contributions; and (iv) a sequence of hand-made choices: the Kahler potential exponents (a,b,c) = (1,1,2) and rotation exponents (alpha,beta) = (-2,1) (Claim 3.15), the axial P1 areas normalized to 1 (Claim 3.21), and the bump functions alpha3-alpha6 (Definition 3.28) with the smallness parameters tau and T. The free choices are what make the construction glue; the external theorems carry the analytic weight. The genuine contribution is the synthesis: the monodromy computation (Lemma 4.20), the cobordism (Lemma 5.1), and the identification of the differential with the theta function (Theorem 5.4).

free parameters (4)
  • Kahler potential exponents (a,b,c) and rotation exponents (alpha,beta) = (a,b,c) = (1,1,2), (alpha,beta) = (-2,1).
    In Claim 3.15, the equivariance equations have multiple solutions; the paper chooses a = b = 1, which fixes alpha = -2, beta = 1, c = 2. The Gamma_B-invariance of omega and the monodromy computation depend on this choice.
  • Symplectic area normalization of the three axial P1s = Area = 1 for each axis (Claim 3.21).
    The polytope is normalized so each boundary P1 has symplectic area 1; this aligns the A-side area exponents with the B-side quadratic form kappa and hence with the theta-function exponents in Theorem 5.4.
  • Bump functions alpha3, alpha4, alpha5, alpha6 = Ranges: 2/3 <= alpha3 <= 1, -1/2 <= alpha4 <= 1/2, 0 <= alpha5 <= 1, 0 <= alpha6 <= 1, monotone with specified…
    The symplectic form in Definition 3.28 is defined via these interpolating bump functions; non-degeneracy (Lemma 3.29) requires derivative estimates deferred to Appendices A and B.
  • Family parameters tau and T = tau in R_+ << 1, T << 1, with |v0| <= T^l.
    tau is the complex structure/Novikov parameter of the genus 2 curve and T of the mirror; they are genuine family parameters, but the smallness conditions and the cutoff |v0| < T^l are choices on which proper discontinuity of the Gamma_B-action (Section 3.3) depends.
assumptions (7)
  • domain assumption Generalized SYZ mirror construction of Abouzaid-Auroux-Katzarkov: the blow-up Bl_{H x 0}(V x C) admits a Lagrangian torus fibration, with Fuk(X) ~ Fuk(H) and D^bCoh(X) = <D^bCoh(V x C), D^bCoh(H)>.
    Theorem 3.4 is imported from [AAK16, Section 4] and is the starting point of the entire mirror construction (Sections 3.1-3.3).
  • ad hoc to paper The localized Fukaya-Seidel category of the non-exact, non-Lefschetz symplectic fibration (Y,v0) is well-defined, and morphisms and compositions in it agree with those in the directed category.
    Definition 4.6 defines objects and morphisms by categorical localization; Lemmas 4.8 and 4.9 cite unpublished Abouzaid-Seidel work. This is the load-bearing premise for the target category of the main theorem.
  • standard math Seidel's Fukaya category framework for Lefschetz fibrations [Sei08] (moduli spaces, regularity, strip-like ends) adapts to this non-Lefschetz setting.
    Sections 4.3-4.5 model all definitions and the regularity proof on [Sei08] and [MS12], adapting from Lefschetz thimbles to U-shaped parallel transports.
  • domain assumption The open mirror theorem for toric Calabi-Yau manifolds of infinite type [KL19, Theorem 3.10], together with [Cha11] and Givental's mirror theorem, computes the disc plus sphere contributions as exp(g_I(q)).
    Section 5.3 imports this theorem to count configurations that are not regular for J0; the series exp(g_I) is taken as given and not re-derived in the paper.
  • domain assumption Quasi-invariance of the Fukaya category under change of regular almost complex structure holds for this fibration.
    Lemma 4.54 is stated with References for proof; the continuation-map argument is delegated to [Sei08, Section 10c] and a Sard-Smale argument.
  • domain assumption For the U-shaped linear Lagrangians, there is no disc bubbling, no strip-breaking, and no Maslov-zero disc in a fiber; the only possible degenerations are sphere bubbles, whose moduli have negative dimension for regular J (Example 4.34, Corollary 4.53).
    These exclusions use that linear Lagrangians in tori bound no discs and the projection argument to the base; Corollary 4.53 then uses [Laz11] and [Laz00]. They are needed to make the counts well-defined.
  • standard math Standard background: action-angle coordinates and quasi-Hamiltonian moment maps; Delzant's theorem; classification of line bundles on abelian varieties (Appell-Humbert); h0(V,L^l) = l^2; Riemann-Roch; the theta divisor of the Jacobian of a genus 2 curve is a genus 2 curve with L|_H = K_H.
    Section 2.2 assembles the complex side from [Pol03], [BL04], and [Huy05], and Remark 1.3 identifies L|_H with the canonical bundle; these are standard textbook facts.
invented entities (2)
  • DFS-type Fukaya-Seidel category of linear Lagrangians in (Y,v0)
    purpose: Target category of the HMS embedding; replaces the not-yet-defined full FS(Y,v0).
    Its definition (Definition 4.6) uses categorical localization owed to unpublished Abouzaid-Seidel work, and Lemmas 4.8 and 4.9 are referenced to unpublished results; whether the category exists for non-exact, non-Lefschetz fibrations with sphere-bubble corrections is not verifiable within the paper.
  • The quotient toric variety (Y,v0) with the chosen Gamma_B-complex structure (Definition 3.14) independent evidence
    purpose: The mirror Landau-Ginzburg model for H = Sigma_2.
    It is an explicit, checkable geometric object: the critical locus of v0 is claimed to be the banana manifold, matching Seidel's mirror [Sei11], and the equivariance of the moment map is checked in Lemma 3.19.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Categorical mirror symmetry on cohomology for a complex genus 2 curve." pith.science (2026). https://pith.science/paper/47FBIMCE

@misc{pith2026190804227,
  author       = {Pith},
  title        = {Pith review of: Categorical mirror symmetry on cohomology for a complex genus 2 curve},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/47FBIMCE}},
  note         = {Machine review of arXiv:1908.04227}
}
abstract

Motivated by observations in physics, mirror symmetry is the concept that certain manifolds come in pairs $X$ and $Y$ such that the complex geometry on $X$ mirrors the symplectic geometry on $Y$. It allows one to deduce symplectic information about $Y$ from known complex properties of $X$. Strominger-Yau-Zaslow arXiv:hep-th/9606040 described how such pairs arise geometrically as torus fibrations with the same base and related fibers, known as SYZ mirror symmetry. Kontsevich arXiv:alg-geom/9411018 conjectured that a complex invariant on $X$ (the bounded derived category of coherent sheaves) should be equivalent to a symplectic invariant of $Y$ (the Fukaya category, see references in article abstract). This is known as homological mirror symmetry. In this project, we first use the construction of "generalized SYZ mirrors" for hypersurfaces in toric varieties following Abouzaid-Auroux-Katzarkov arXiv:1205.0053v4, in order to obtain $X$ and $Y$ as manifolds. The complex manifold is the genus 2 curve $\Sigma_2$ (so of general type $c_1<0$) as a hypersurface in its Jacobian torus. Its generalized SYZ mirror is a Landau-Ginzburg model $(Y,v_0)$ equipped with a holomorphic function $v_0:Y \to \mathbb{C}$ which we put the structure of a symplectic fibration on. We then describe an embedding of a full subcategory of $D^bCoh(\Sigma_2)$ into a cohomological Fukaya-Seidel category of $Y$ as a symplectic fibration. While our fibration is one of the first nonexact, non-Lefschetz fibrations to be equipped with a Fukaya category, the main geometric idea in defining it is the same as in Seidel's construction for Fukaya categories of Lefschetz fibrations and in Abouzaid-Seidel.

Figures

Figures reproduced from arXiv: 1908.04227 by the authors.

Figure 1
Figure 1. A triangle in V ∨ contributing to µ 2 , viewed in ξ1, ξ2 plane in the universal cover R 4 of the three vectors around the triangle must be zero. Let the ξ coordinates of these vectors be ξ, ξ0 , ξ00 respectively. Then setting their sum, and the sum of θ-coordinates equal to zero: (2.6) ξ + ξ 0 + ξ 00 = 0 iξ + jξ0 − k(ξ + ξ 0 ) = 0 ∴ ξ 0 = − l l 00 ξ ξ 00 = − l 0 l 00 ξ Now we apply the constraint that p1 ∈ `i ∩ `j a… view at source ↗
Figure 2
Figure 2. CP2 example Dually, the toric polytope for CP2 is a triangle, suppose delineated by the two coor￾dinate axes (say m1 = 0, m2 = 0) and m1 + m2 = −1. The line bundle described by this polytope is O({z0 = 0}) with sections given by the integral vertices of the polytope χ (0,0) = 1, χ(−1,0), χ(0,−1). Thus ω = i 2π ∂∂ log(1 + |z1/z0| 2 + |z2/z0| 2 ), which recovers the Fubini-Study form. 3.1. Finding a Lagrangian torus f… view at source ↗
Figure 3
Figure 3. Moment map gives Lagrangian torus fibration: CP2 example The above Lagrangian torus fibration arose from a moment map, so is called toric. If a Lagrangian torus fibration is not from a moment map, it’s called non-toric. Note that CP2 blown up at [1 : 0 : 0] corresponds to removing a small triangle on the base, which gives a quadrilateral. This can still be the base of a Lagrangian torus fibration by taking T 2 ’s ab… view at source ↗
Figures from the paper (22 more)
Figure 4
Figure 4. Figure 4: Base of exterior and interior blow-up on CP2 These describe Lagrangian torus fibrations over a base with a symplectic affine structure. Taking the Legendre transform, we obtain a Lagrangian torus fibration over a base with a complex affine structure, namely log | · |. …
Figure 5
Figure 5. Figure 5: L) Trop(1 + x1 + x2) = 0 R) Moment polytope in R 3 Tropicalizing the infinite series given by the theta function at first sight seems hard. In fact, it satisfies a periodicity property which allows us to see the tropicalization as a honeycomb shape when projected to (ξ…
Figure 6
Figure 6. Figure 6: The (0,0) tile delimited by the tropical curve Now pick any (ξ1, ξ2) ∈ R 2 . Choose γ such that (ξ1 − γ1, ξ2 − γ2) ∈ F0,0, which we can do since the hexagon is the same size as the fundamental domain for the ΓB-action by subtraction. Let λ(γ) =: (m1, m2) t . Then again…
Figure 7
Figure 7. Figure 7: The (m1, m2) tile delimited by the tropical curve [PITH_FULL_IMAGE:figures/full_fig_p027_7.png]
Figure 8
Figure 8. Figure 8: Moment polytope for central fiber of (Y, v0) when H = Σ2. restrict to a symplectic form away from the zero fiber. In a neighborhood of the zero fiber, this corresponds to a neighborhood of the facets in ∆Y˜ . Away from a vertex, we take the limit of the symplectic form…
Figure 9
Figure 9. Figure 9: In one dimension lower, the boundary of ∆Y˜ is the moment map image of a string of P 1 ’s. In the polytope, |v0| increases in the (0, 1) direction. In the fibration v0, |v0| is the radius of the circle in the base. and the toric K¨ahler form for CP2 (3 points), the blo…
Figure 10
Figure 10. Figure 10: Depiction of 3D ∆Y˜ . Coordinates respect ΓB-action, see be￾low; magenta parallelogram = fundamental domain. Vertices = C 3 charts. Coordinate transitions, see Lemma 3.16. Expressions in the center of tiles indicate e.g. η ≥ ϕ(ξ) = −ξ1 − 1 over that tile, so tile(0,0)…
Figure 11
Figure 11. Figure 11: L) Regions near a vertex, R) Number of regions interpolated between where Region VII is where α1 = α2 = 1/3 and F = 1 3 (gxy + gxz + gyz) ≈ 2 3 ((T rx) 2 + (T ry) 2 + (T rz) 2 ) via the log approximation and in the other regions α1, α2 interpolate between the three K¨…
Figure 12
Figure 12. Figure 12: How the three angular directions vary for rxryrz constant on a fiber We define functions φx, φy, φz for expressions we will use often in the coming definitions [PITH_FULL_IMAGE:figures/full_fig_p039_12.png]
Figure 13
Figure 13. Figure 13: Delineating regions in coordinates (dI , θI ) and (dII , θII ) will not have contributions from the αi making it degenerate. This is proven in Appendix A. Now we define the symplectic form in terms of rx, ry, rz in these regions. Definition 3.28 (Definition of symplec…
Figure 14
Figure 14. Figure 14: Delineated fundamental domain are still small so we still have an approximation for the K¨ahler potential. The calculation for the negligible terms is given in Appendix B. Dominant terms in the remainder of C 3 patch. See [PITH_FULL_IMAGE:figures/full_fig_p046_14.png]
Figure 15
Figure 15. Figure 15: Li = parallel transport `i around U-shaped curve in base of v0 a compact set by rays in the right half plane, i.e. lines reiθ0 for a fixed θ0 ∈ [−π, π), as in [PITH_FULL_IMAGE:figures/full_fig_p049_15.png]
Figure 16
Figure 16. Figure 16: Monodromy in fiber, thought of as a section over the parallelo￾gram (ξ1, ξ2) 7→ (f1(ξ1, ξ2), f2(ξ1, ξ2)). thus b = 1 and a must be zero as η depends on |v0|. So the horizontal lift is of the form: Xhor = ∂/∂θη + f1∂/∂θ1 + f2∂/∂θ2 Also we saw above in Claim 4.5 that fo…
Figure 17
Figure 17. Figure 17: Example of strip-like end Next we find the 2-homology of Y . This is where the pseudo-holomorphic discs map to in the target. Lemma 4.28 (Homology of Y ). H2(Y ) ∼= H2(CP2 (3)/ΓB) ∼= Z 4 where all homology classes will be over Z. Proof. Note that Y deformation retract…
Figure 18
Figure 18. Figure 18: The homotopy between ∂ (left) and the count of discs we compute (right) maps on the torus fiber. We take two steps to reduce the calculation of M1 to something that is computable: first in Lemma 5.1 we construct a cobordism between M1 : CF(`i+1, tx) → CF(`i , tx) and …
Figure 19
Figure 19. Figure 19: Gromov compactification Lemma 5.1. Choose β0 and let βr = φr∗ (β0) where φr : (Y, S γ1 `) ∼=−→ (Y, S γr `) is a dif￾feomorpism inducing an isomorphism φr∗ on homology for 0 < r ≤ 1. Then for a suitable family Jr described in the proof, [ r∈(0,1] Mˆ ((Y,[ γr `); βr; Jr…
Figure 20
Figure 20. Figure 20: Gromov-Witten theory background for mirror symmetry of toric varieties Gromov-Witten invariants. He proved a relation between these two functions, i.e. a mirror theorem. Closed mirror theorem: [CCIT15]. The closed mirror theorem relates the I and J function (defined i…
Figure 21
Figure 21. Figure 21: Leibniz rule The reason we use the Leibniz rule is because we would like to use S circle tx as the La￾grangian boundary condition and not S circle `j , as this will allow us to count discs with boundary in the preimage of a moment map, as in [CO06]. Note that M1 : CF(…
Figure 22
Figure 22. Figure 22: Diagram illustrating Leibniz rule (5.9) homleft(`j , tx) ⊗ homright(`i+1, `j ) 3 p∞,j ⊗ p k i+1,j M2 ✲ hommiddle(`i+1, tx) homleft(`j , tx) ⊗ homleft(`i , `j ) M1 ⊗ 1 + 1 ⊗ M1 ❄ M2 = µ 2 left ✲ homleft(`i , tx) M1 ❄ [PITH_FULL_IMAGE:figures/full_fig_p082_22.png]
Figure 23
Figure 23. Figure 23: Simplified diagram on fibers Differentiating the product M2 , the Leibniz rule implies (where l = j−i and so p k i,j indexed over k can instead be written as pe,l indexed over e) (5.10) M1 (M2 (p∞,j , pe,l−1)) = M2 (M1 (p∞,j ), pe,l−1) + M2 (p∞,j , M1 (pe,l−1)) = µ 2 …
Figure 24
Figure 24. Figure 24: A triangle in V ∨ contributing to µ 2 , viewed in ξ1, ξ2 plane in the universal cover R 4 The three vertices of the triangle are on lifts of `i ∩ tx 3 p∞,i, `j ∩ tx 3 p∞,j , and `i ∩ `j 3 pe,l ˜ . Translate pe,l ˜ so that it lies in the fundamental domain for the ΓB-a…
Figure 25
Figure 25. Figure 25: Proof of Main Theorem Ext groups here are computed from injective resolutions: (0 ✲ L −1 ✲ OV ✲ OH ✲ 0) ⊗ Lj−i 0 ✲ L j−i−1 ✲ L j−i ✲ L j−i H ✲ 0 so taking the cohomology long exact sequence we obtain: (6.1) 0 → H 0 (L j−i−1 ) → H 0 (L j−i ) → H 0 (L j−i |H) → H 1 (L j…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

68 extracted references · 46 canonical work pages

  1. [1]

    Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces

    Mohammed Abouzaid, Denis Auroux, and Ludmil Katzarkov. Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces. Publ. Math. Inst. Hautes \'Etudes Sci. , 123:199--282, 2016

    work page 2016

  2. [2]

    Morse homology, tropical geometry, and homological mirror symmetry for toric varieties

    Mohammed Abouzaid. Morse homology, tropical geometry, and homological mirror symmetry for toric varieties. Selecta Math. (N.S.) , 15(2):189--270, 2009. ://doi.org/10.1007/s00029-009-0492-2

    doi:10.1007/s00029-009-0492-2 2009

  3. [3]

    Mirror symmetry for weighted projective planes and their noncommutative deformations

    Denis Auroux, Ludmil Katzarkov, and Dmitri Orlov. Mirror symmetry for weighted projective planes and their noncommutative deformations. Ann. of Math. (2) , 167(3):867--943, 2008. doi:10.4007/annals.2008.167.867

    work page doi:10.4007/annals.2008.167.867 2008

  4. [4]

    Summer graduate school, D erived C ategories

    Nicolas Addington, Alexander Polishchuk, and Ed Segal. Summer graduate school, D erived C ategories. https://www.msri.org/summer_schools/821

    work page

  5. [5]

    L efschetz fibration methods in wrapped F loer cohomology

    Mohammed Abouzaid and Paul Seidel. L efschetz fibration methods in wrapped F loer cohomology. in preparation

    work page

  6. [6]

    Homological mirror symmetry for the 4-torus

    Mohammed Abouzaid and Ivan Smith. Homological mirror symmetry for the 4-torus. Duke Math. J. , 152(3):373--440, 2010

    work page 2010

  7. [7]

    Khovanov homology from F loer cohomology

    Mohammed Abouzaid and Ivan Smith. Khovanov homology from F loer cohomology. J. Amer. Math. Soc. , 32(1):1--79, 2019. doi:10.1090/jams/902

    doi:10.1090/jams/902 2019

  8. [8]

    Mirror symmetry and T -duality in the complement of an anticanonical divisor

    Denis Auroux. Mirror symmetry and T -duality in the complement of an anticanonical divisor. J. G\"okova Geom. Topol. GGT , 1:51--91, 2007

    work page 2007

Show all 68 references
  1. [9]

    A beginner's introduction to F ukaya categories

    Denis Auroux. A beginner's introduction to F ukaya categories. In Contact and symplectic topology , volume 26 of Bolyai Soc. Math. Stud. , pages 85--136. J\' a nos Bolyai Math. Soc., Budapest, 2014. ://doi.org/10.1007/978-3-319-02036-5_3

    2014 doi

  2. [10]

    Complex abelian varieties , volume 302 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]

    Christina Birkenhake and Herbert Lange. Complex abelian varieties , volume 302 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Springer-Verlag, Berlin, second edition, 2004

    2004

  3. [11]

    Homological mirror symmetry for the genus 2 curve in an abelian variety and its generalized S trominger- Y au- Z aslow mirror

    Catherine Cannizzo. Homological mirror symmetry for the genus 2 curve in an abelian variety and its generalized S trominger- Y au- Z aslow mirror . PhD thesis, U niversity of C alifornia, Berkeley, 2019. https://arxiv.org/abs/1908.04227

    2019 arXiv

  4. [12]

    A mirror theorem for toric stacks

    Tom Coates, Alessio Corti, Hiroshi Iritani, and Hsian-Hua Tseng. A mirror theorem for toric stacks. Compos. Math. , 151(10):1878--1912, 2015. ://doi.org/10.1112/S0010437X15007356

    1912 doi

  5. [13]

    Gross fibrations, SYZ mirror symmetry, and open G romov- W itten invariants for toric C alabi- Y au orbifolds

    Kwokwai Chan, Cheol-Hyun Cho, Siu-Cheong Lau, and Hsian-Hua Tseng. Gross fibrations, SYZ mirror symmetry, and open G romov- W itten invariants for toric C alabi- Y au orbifolds. J. Differential Geom. , 103(2):207--288, 2016. ://projecteuclid.org/euclid.jdg/1463404118

    2016

  6. [14]

    Symplectic T oric M anifolds

    Ana Cannas da Silva. Symplectic T oric M anifolds. https://people.math.ethz.ch/ acannas/Papers/toric.pdf

  7. [15]

    http://www.polyfolds.org/index.php?title=Polyfold_constructions_for_Fukaya_categories

    Polyfold constructions for F ukaya categories. http://www.polyfolds.org/index.php?title=Polyfold_constructions_for_Fukaya_categories

  8. [16]

    A formula equating open and closed G romov- W itten invariants and its applications to mirror symmetry

    Kwokwai Chan. A formula equating open and closed G romov- W itten invariants and its applications to mirror symmetry. Pacific J. Math. , 254(2):275--293, 2011. ://doi.org/10.2140/pjm.2011.254.275

    2011 doi

  9. [17]

    Cox and Sheldon Katz

    David A. Cox and Sheldon Katz. Mirror symmetry and algebraic geometry , volume 68 of Mathematical Surveys and Monographs . American Mathematical Society, Providence, RI, 1999. ://doi.org/10.1090/surv/068

    1999 doi

  10. [18]

    S YZ mirror symmetry for toric C alabi- Y au manifolds

    Kwokwai Chan, Siu-Cheong Lau, and Naichung Conan Leung. S YZ mirror symmetry for toric C alabi- Y au manifolds. J. Differential Geom. , 90(2):177--250, 2012. ://projecteuclid.org/euclid.jdg/1335230845

    2012

  11. [19]

    Cox, John B

    David A. Cox, John B. Little, and Henry K. Schenck. Toric varieties , volume 124 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, 2011. doi:10.1090/gsm/124

    2011 doi

  12. [20]

    Floer cohomology and disc instantons of L agrangian torus fibers in F ano toric manifolds

    Cheol-Hyun Cho and Yong-Geun Oh. Floer cohomology and disc instantons of L agrangian torus fibers in F ano toric manifolds. Asian J. Math. , 10(4):773--814, 2006

    2006

  13. [21]

    Symplectic homology and the E ilenberg- S teenrod axioms

    Kai Cieliebak and Alexandru Oancea. Symplectic homology and the E ilenberg- S teenrod axioms. Algebr. Geom. Topol. , 18(4):1953--2130, 2018. doi:10.2140/agt.2018.18.1953. Appendix written jointly with Peter Albers

    1953 doi

  14. [22]

    Stacks for everybody

    Barbara Fantechi. Stacks for everybody. In European C ongress of M athematics, V ol. I ( B arcelona, 2000) , volume 201 of Progr. Math. , pages 349--359. Birkh\" a user, Basel, 2001

    2000

  15. [23]

    The coherent-constructible correspondence and homological mirror symmetry for toric varieties

    Bohan Fang, Chiu-Chu Melissa Liu, David Treumann, and Eric Zaslow. The coherent-constructible correspondence and homological mirror symmetry for toric varieties. In Geometry and analysis. N o. 2 , volume 18 of Adv. Lect. Math. (ALM) , pages 3--37. Int. Press, Somerville, MA, 2011

    2011

  16. [24]

    Lagrangian intersection F loer theory: anomaly and obstruction

    Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono. Lagrangian intersection F loer theory: anomaly and obstruction. P art I , volume 46 of AMS/IP Studies in Advanced Mathematics . American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009

    2009

  17. [25]

    Lagrangian intersection F loer theory: anomaly and obstruction

    Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono. Lagrangian intersection F loer theory: anomaly and obstruction. P art II , volume 46 of AMS/IP Studies in Advanced Mathematics . American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009

    2009

  18. [26]

    Kuranishi Structures and Virtual Fundamental Chains

    Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono. Kuranishi Structures and Virtual Fundamental Chains . Springer Monographs in Mathematics. Springer Singapore, 2020. doi:10.1007/978-981-15-5562-6

    2020 doi

  19. [27]

    Gromov convergence of pseudoholomorphic disks

    Urs Frauenfelder. Gromov convergence of pseudoholomorphic disks. J. Fixed Point Theory Appl. , 3(2):215--271, 2008. doi:10.1007/s11784-008-0078-1

    2008 doi

  20. [28]

    M ath 465, S pring 2010: T opology of manifolds

    J Francis. M ath 465, S pring 2010: T opology of manifolds. http://math.northwestern.edu/ jnkf/classes/mflds/, 2010

    2010

  21. [29]

    Mirror symmetry of abelian varieties and multi-theta functions

    Kenji Fukaya. Mirror symmetry of abelian varieties and multi-theta functions. J. Algebraic Geom. , 11(3):393--512, 2002

    2002

  22. [30]

    Mat 667 advanced topics in topology: Global symplectic geometry via pseudo-holomorphic curves

    Kenji Fukaya. Mat 667 advanced topics in topology: Global symplectic geometry via pseudo-holomorphic curves. http://scgp.stonybrook.edu/video_portal/video.php?id=4438, 2020. https://onedrive.live.com/view.aspx?resid=2232CE7612EBBB59 onenote:https://d.docs.live.net/2232ce7612eb...

    2020

  23. [31]

    Introduction to toric varieties , volume 131 of Annals of Mathematics Studies

    William Fulton. Introduction to toric varieties , volume 131 of Annals of Mathematics Studies . Princeton University Press, Princeton, NJ, 1993. ://doi.org/10.1515/9781400882526. The William H. Roever Lectures in Geometry

    1993 doi

  24. [32]

    A polyfold proof of the arnold conjecture, 2018, 1810.06180 http://arxiv.org/abs/1810.06180

    Benjamin Filippenko and Katrin Wehrheim. A polyfold proof of the arnold conjecture, 2018, 1810.06180 http://arxiv.org/abs/1810.06180

    2018 arXiv

  25. [33]

    Math 257b: Topics in symplectic geometry -- aspects of F ukaya categories

    Sheel Ganatra. Math 257b: Topics in symplectic geometry -- aspects of F ukaya categories. https://drive.google.com/file/d/1-anqMQbQCxgu01AKCK3rUqmOF95vkhvK/view, 2016

    2016

  26. [34]

    A mirror theorem for toric complete intersections

    Alexander Givental. A mirror theorem for toric complete intersections. In Topological field theory, primitive forms and related topics ( K yoto, 1996) , volume 160 of Progr. Math. , pages 141--175. Birkh\" a user Boston, Boston, MA, 1998

    1996

  27. [35]

    Topological mirror symmetry

    Mark Gross. Topological mirror symmetry. Invent. Math. , 144(1):75--137, 2001. doi:10.1007/s002220000119

    2001 doi

  28. [36]

    Mirror symmetry and the S trominger- Y au- Z aslow conjecture

    Mark Gross. Mirror symmetry and the S trominger- Y au- Z aslow conjecture. In Current developments in mathematics 2012 , pages 133--191. Int. Press, Somerville, MA, 2013

    2012

  29. [37]

    Lagrangian torus fibration for symplectic toric degenerations

    Roberta Guadagni. Lagrangian torus fibration for symplectic toric degenerations . PhD thesis, University of T exas at A ustin, 2017. https://repositories.lib.utexas.edu/handle/2152/63642

    2017

  30. [38]

    Kaehler structures on toric varieties

    Victor Guillemin. Kaehler structures on toric varieties. J. Differential Geom. , 40(2):285--309, 1994. ://projecteuclid.org/euclid.jdg/1214455538

    1994

  31. [39]

    Mirror symmetry , volume 1 of Clay Mathematics Monographs

    Kentaro Hori, Sheldon Katz, Albrecht Klemm, Rahul Pandharipande, Richard Thomas, Cumrun Vafa, Ravi Vakil, and Eric Zaslow. Mirror symmetry , volume 1 of Clay Mathematics Monographs . American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2003...

    2003

  32. [40]

    Complex geometry

    Daniel Huybrechts. Complex geometry . Universitext. Springer-Verlag, Berlin, 2005. An introduction

    2005

  33. [41]

    Polyfold and Fredholm Theory

    Helmut Hofer , Krzysztof Wysocki , and Eduard Zehnder . Polyfold and Fredholm Theory . arXiv e-prints , page arXiv:1707.08941, Jul 2017, 1707.08941 http://arxiv.org/abs/1707.08941

    2017 arXiv

  34. [42]

    Local C alabi- Y au manifolds of type A via SYZ mirror symmetry

    Atsushi Kanazawa and Siu-Cheong Lau. Local C alabi- Y au manifolds of type A via SYZ mirror symmetry. J. Geom. Phys. , 139:103--138, 2019. ://doi.org/10.1016/j.geomphys.2018.12.015

    2019 doi

  35. [43]

    urich, 1994) , pages 120--139. Birkh\

    Maxim Kontsevich. Homological algebra of mirror symmetry. In Proceedings of the I nternational C ongress of M athematicians, V ol.\ 1, 2 ( Z \"urich, 1994) , pages 120--139. Birkh\"auser, Basel, 1995

    1994

  36. [44]

    Lazzarini

    L. Lazzarini. Existence of a somewhere injective pseudo-holomorphic disc. Geom. Funct. Anal. , 10(4):829--862, 2000. ://doi.org/10.1007/PL00001640

    2000 doi

  37. [45]

    Relative frames on J -holomorphic curves

    Laurent Lazzarini. Relative frames on J -holomorphic curves. J. Fixed Point Theory Appl. , 9(2):213--256, 2011. ://doi.org/10.1007/s11784-010-0004-1

    2011 doi

  38. [46]

    Chiu-Chu Melissa Liu. Moduli of J -holomorphic curves with L agrangian boundary conditions and open G romov- W itten invariants for an S^1 -equivariant pair, 2002, math/0210257 http://arxiv.org/abs/math/0210257

    2002 arXiv

  39. [47]

    Lectures on the h -cobordism theorem

    John Milnor. Lectures on the h -cobordism theorem . Notes by L. Siebenmann and J. Sondow. Princeton University Press, Princeton, N.J., 1965

    1965

  40. [48]

    J -holomorphic curves and symplectic topology , volume 52 of American Mathematical Society Colloquium Publications

    Dusa McDuff and Dietmar Salamon. J -holomorphic curves and symplectic topology , volume 52 of American Mathematical Society Colloquium Publications . American Mathematical Society, Providence, RI, second edition, 2012

    2012

  41. [49]

    Introduction to Symplectic Topology

    Dusa McDuff and Dietmar Salamon. Introduction to Symplectic Topology . Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford, third edition, 2017. ://doi.org/10.1093/oso/9780198794899.001.0001

    2017

  42. [50]

    Virtual Fundamental Cycles in Symplectic Topology

    Dusa McDuff, Mohammad Tehrani, Kenji Fukaya, and Dominic Joyce. Virtual Fundamental Cycles in Symplectic Topology . American Mathematical Society, 2019

    2019

  43. [51]

    Floer cohomology of L agrangian intersections and pseudo-holomorphic disks

    Yong-Geun Oh. Floer cohomology of L agrangian intersections and pseudo-holomorphic disks. I . Comm. Pure Appl. Math. , 46(7):949--993, 1993. ://doi.org/10.1002/cpa.3160460702

    1993 doi

  44. [52]

    M 392c: L agrangian F loer homology, hw 1

    James Pascaleff. M 392c: L agrangian F loer homology, hw 1. https://faculty.math.illinois.edu/ jpascale/courses/2014/m392c/notes/homework1.pdf , 2014

    2014

  45. [53]

    Abelian varieties, theta functions and the F ourier transform , volume 153 of Cambridge Tracts in Mathematics

    Alexander Polishchuk. Abelian varieties, theta functions and the F ourier transform , volume 153 of Cambridge Tracts in Mathematics . Cambridge University Press, Cambridge, 2003. doi:10.1017/CBO9780511546532

    2003 doi

  46. [54]

    Categorical mirror symmetry: The Elliptic curve

    Alexander Polishchuk and Eric Zaslow. Categorical mirror symmetry: The Elliptic curve . Adv. Theor. Math. Phys. , 2:443--470, 1998

    1998

  47. [55]

    Fukaya categories and P icard- L efschetz theory

    Paul Seidel. Fukaya categories and P icard- L efschetz theory . Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Z\"urich, 2008

    2008

  48. [56]

    Homological mirror symmetry for the genus two curve

    Paul Seidel. Homological mirror symmetry for the genus two curve. J. Algebraic Geom. , 20(4):727--769, 2011

    2011

  49. [57]

    Some speculations on pairs-of-pants decompositions and F ukaya categories

    Paul Seidel. Some speculations on pairs-of-pants decompositions and F ukaya categories. In Surveys in differential geometry. V ol. XVII , volume 17 of Surv. Differ. Geom. , pages 411--425. Int. Press, Boston, MA, 2012. doi:10.4310/SDG.2012.v17.n1.a9

    2012 doi

  50. [58]

    Homological mirror symmetry for the quartic surface

    Paul Seidel. Homological mirror symmetry for the quartic surface. Mem. Amer. Math. Soc. , 236(1116):vi+129, 2015. doi:10.1090/memo/1116

    2015 doi

  51. [59]

    Homological mirror symmetry for C alabi- Y au hypersurfaces in projective space

    Nick Sheridan. Homological mirror symmetry for C alabi- Y au hypersurfaces in projective space. Invent. Math. , 199(1):1--186, 2015. doi:10.1007/s00222-014-0507-2

    2015 doi

  52. [60]

    On the F ukaya category of a F ano hypersurface in projective space

    Nick Sheridan. On the F ukaya category of a F ano hypersurface in projective space. Publ. Math. Inst. Hautes \' E tudes Sci. , 124:165--317, 2016. doi:10.1007/s10240-016-0082-8

    2016 doi

  53. [61]

    Four dimensions from two in symplectic topology

    Margaret Symington. Four dimensions from two in symplectic topology. In Topology and geometry of manifolds ( A thens, GA , 2001) , volume 71 of Proc. Sympos. Pure Math. , pages 153--208. Amer. Math. Soc., Providence, RI, 2003. doi:10.1090/pspum/071/2024634

    2001 doi

  54. [62]

    Mirror symmetry is T -duality

    Andrew Strominger, Shing-Tung Yau, and Eric Zaslow. Mirror symmetry is T -duality. Nuclear Phys. B , 479(1-2):243--259, 1996

    1996

  55. [63]

    Homological mirror symmetry for toric del P ezzo surfaces

    Kazushi Ueda. Homological mirror symmetry for toric del P ezzo surfaces. Comm. Math. Phys. , 264(1):71--85, 2006. doi:10.1007/s00220-005-1509-0

    2006 doi

  56. [64]

    Berkeley math 278: Analysis of pseudoholomorphic curves

    Katrin Wehrheim. Berkeley math 278: Analysis of pseudoholomorphic curves. https://piazza.com/berkeley/fall2013/berkeleymath278/resources, 2013

    2013

  57. [65]

    Regularization of moduli spaces of pseudoholomorphic curves

    Katrin Wehrheim. Regularization of moduli spaces of pseudoholomorphic curves. https://math.berkeley.edu/ katrin/teach/regularization/lectures.shtml, 2014

    2014

  58. [66]

    Lectures on Symplectic Field Theory

    Chris Wendl . Lectures on Symplectic Field Theory . arXiv e-prints , page arXiv:1612.01009, Dec 2016, 1612.01009 http://arxiv.org/abs/1612.01009

    2016 arXiv

  59. [67]

    A _ -structures from M orse trees with pseudoholomorphic disks

    Katrin Wehrheim and Jiayong Li. A _ -structures from M orse trees with pseudoholomorphic disks. https://math.berkeley.edu/ katrin/papers/disktrees.pdf

  60. [68]

    write newline

    " write newline "" before.all 'output.state := FUNCTION fin.entry add.period write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence '...

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.