Topology and Euler characteristics of tropical varieties

Botong Wang; Chenxi Wu; Connor Simpson; Scott Hiatt

REVIEW 3 major objections 4 minor 13 references

Topology and Euler characteristics of tropical varieties

T0 review · 3 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read H-regular tropical subvarieties obey the Green–Lazarsfeld sign law: (−1)ᵈ χ(X) ≥ 0.

desk verdict A serious tropical-geometry paper proving a Green–Lazarsfeld analogue for H-regular subvarieties; the main theorem is plausible and the architecture is sound, but Proposition 6.1 needs a small fix and the Section 8 sphere count has a typo. read the letter →

arxiv 2606.10817 v2 pith:IABW5NEL submitted 2026-06-09 math.AG math.CO

Scott Hiatt , Connor Simpson , Botong Wang , Chenxi Wu This is my paper · ORCID

classification math.AGmath.CO MSC 14T0514C4014F08

keywords tropicalabelianvarietyEulercharacteristicH-regularfanGreen–LazarsfeldtheoremhomologyLefschetz-typeBergmanMorsetheory

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 tropical analogue of a classical theorem of Green and Lazarsfeld: on a complex abelian variety, the signed Euler characteristic of a smooth subvariety X satisfies (−1)^{dim X} χ(X, O_X) ≥ 0. Here, for a `d`-dimensional tropical subvariety X of a tropical abelian variety, the paper establishes (−1)ᵈ χ(X) ≥ 0 under an `H-regularity` condition that locally models X on tropicalizations of well-behaved very affine varieties. The sign law is combinatorial: χ(X) is computed from tropical Hodge numbers, and the paper's local vanishing theorem also yields a Lefschetz-type statement for affine H-regular tropical varieties. The authors further show the sign inequality fails for general tropical subvarieties and give an explicit non-H-regular tropical fan whose link is not a bouquet of spheres.

What carries the argument

H-regularity: a closed irreducible subvariety U of the complex affine torus (C*)ⁿ is H-regular if its closure in some smooth projective toric variety is smooth, meets every torus orbit transversely, and each nonempty orbit intersection is connected. A tropical fan is H-regular if it is the tropicalization of such a U. This condition is the load-bearing geometric input that guarantees the existence of a toric compactification whose boundary has rationally smooth strata, allowing the authors to compare the tropical link to a dual complex and apply mixed-Hodge-module vanishing results.

What would settle it

A direct combinatorial check: for any complete simplicial fan Δ and linear functional ℓ, try to exhibit a rational simplicial cone σ_S contained in {ℓ>0}∪{0} whose relative interior contains every ℓ-positive ray of Δ, while also being wide enough for the deformation retraction argument. If no such cone can exist (e.g., for a fan with two opposing ℓ-positive rays nearly spanning the complement of a thin cone), Proposition 6.1 fails.

Watch

Extended reading notes

Core claim

Theorem 1.1: If A is a tropical abelian variety (a real torus with positive definite quadratic form) and X is an H-regular, pure-dimensional tropical subvariety of dimension d, then (−1)ᵈ χ(X) ≥ 0. This is proved via a Morse-theoretic argument built on a local statement: for an H-regular tropical fan T of dimension d and a linear function ℓ that does not vanish on any ray, the reduced homology of the link of T ∩ {ℓ ≤ 0} vanishes in all degrees except possibly d−1 (Theorem 2.8). The local vanishing is established by relating the link to the dual complex of a toric compactification, converting the problem into a vanishing of top-weight cohomology for a quasi-projective variety that admits a pr

Load-bearing premise

The existence of a sufficiently fine simplicial cone σ_S in the positive half-space of ℓ that contains all ℓ-positive rays of the ambient fan—used to build the retraction to the link of T∩{ℓ≤0}—is asserted by Proposition 6.1; if this cone cannot be constructed, the local vanishing theorem has no proof.

Editorial extensions

If this is right

If the main theorem holds, every H-regular d-dimensional tropical subvariety of a tropical abelian variety has signed Euler characteristic of the predicted sign, determined solely by dimension.
The signed Euler characteristic of such a subvariety equals the signed alternating sum of tropical Hodge numbers, linking the topological statement to tropical Hodge theory.
The local vanishing theorem (Theorem 2.8) yields a Lefschetz-type result: for a general affine hyperplane H, the relative homology H_k(X, X∩H) vanishes for all k≠d.
The sign law fails without H-regularity: the paper constructs a 2-dimensional tropical cycle in a quotient of a product of Jacobians with Euler characteristic −4.
The local vanishing gives a new proof of a slight weakening of a result of Adiprasito–Björner for Bergman fans of matroids realizable over C, and answers a question of Mikhalkin–Ziegler in that case.

Reading between the lines

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

A natural testable extension is whether the signed inequality generalizes to larger classes of tropical varieties, e.g., those locally connected in codimension 1 but not H-regular; the authors suspect it does not, and their Example 2.11 gives a starting point.
The dependency of Proposition 6.1 on a 2024 preprint about common stellar subdivisions of fans means that if that result were incomplete, the local vanishing theorem would lack a fully established proof; a reader might check whether the cone construction can be simplified to avoid that citation.
The paper's explicit 3-dimensional tropical fan whose link is not a bouquet of spheres shows that H-regularity is not merely a technical condition but controls fundamental topological obstructions; it would be interesting to see whether such non-bouquet links obstruct any combinatorial Hodge-theoretic positivity.
The authors raise the question of a tropical Chern-class refinement; if a signed Chern-class positivity holds for H-regular tropical varieties, it would give a stronger structural constraint than the Euler characteristic inequality alone.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Thanks for the report. I read the paper with your notes. My take: this is a real result, not a dressed-up special case. The main novelty is Theorem 1.1 plus the local vanishing Theorem 2.8, and the proof route through MHM, toric compactifications, and Morse theory is coherent. I checked the weight arithmetic in Prop 4.1 and the spectral sequence in Prop 5.1; those hold. The counterexamples matter—Example 2.11 shows H-regularity is needed, and Section 8 addresses Hacking's question. I agree with you on the load-bearing issue: Proposition 6.1 is under-specified. The stress-test note is right that your "wide cone cannot be contained in ℓ>0" is not accurate—a dual cone can be wide and still lie inside ℓ>0 if its generators surround v. But the gap is real: the proof never states that v must lie in the interior of cone(v_i). Without it, the chosen σ_S can have rays with ℓ<0. Since n vectors can form a simplicial cone around v, the fix is easy: choose v_i so that v ∈ int(cone(v_i)) and all are close to v. Add that sentence and the argument goes through.

Two smaller things. First, the "moreover" in Prop 6.1 uses [AP24] (common stellar subdivisions). That is a 2024 preprint resolving a conjecture; the paper cites it without comment. The rest of the paper does not depend on that specific theorem, but it should be flagged. Second, Lemma 8.1 has a typo/arithmetic slip: the 2-skeleton of a 6-simplex has H2 = Z^20, not Z^10, and the displayed H3=Z^10 should be H2=Z^10 (or Z^20). The example's conclusion is unaffected because H1=Z is what violates the bouquet-of-2-spheres property, but the count should be corrected.

I also noted two standard facts used without citation (purity of cohomology of rationally smooth projective varieties; the fiber-bundle claim in Theorem 7.1). These are minor and fixable.

Net: the main claims are believable, the architecture is sound, and the flaws are repairable. The paper deserves a serious referee; I'd send it to someone comfortable with mixed Hodge modules. I'd cite it once the Prop 6.1 fix is in.

Referee Report

3 major / 4 minor

Summary. The paper proves a tropical analogue of the Green–Lazarsfeld theorem: for an H-regular d-dimensional tropical subvariety X of a tropical abelian variety, the signed Euler characteristic satisfies (−1)^d χ(X) ≥ 0. The proof is based on a local vanishing theorem (Theorem 2.8) for H-regular tropical fans, proved via mixed-Hodge-module weight vanishing (Prop. 4.1), a comparison between dual complexes and weight-graded cohomology (Prop. 5.1), and a Morse-theoretic argument (Thm. 7.1). The paper also gives a Lefschetz-type theorem for affine H-regular tropical varieties, discusses the necessity of H-regularity through a quotient construction, and constructs a 3-dimensional tropical fan whose link has nontrivial homology in an unexpected degree, addressing a question of Hacking.

Significance. If the main theorem is correct, it establishes a clean combinatorial sign law for a natural class of tropical subvarieties of tropical abelian varieties, exactly parallel to the classical Green–Lazarsfeld inequality. The proof strategy is well structured and uses sophisticated tools (Saito's mixed Hodge modules, decomposition theorem, toric transversality) in a way that is likely influential. The paper also gives a new proof of a version of the Adiprasito–Björner vanishing for realizable matroids and supplies a counterexample to a proposed generalization. The writing is careful and the modular architecture (Prop 4.1 → Prop 5.1 → Thm 2.8 → Thm 7.1 → Thm 1.1) is coherent. However, the main local vanishing theorem rests on Proposition 6.1, whose proof as written has a real gap, and on an external 2024 preprint ([AP24]). These issues are load-bearing and must be resolved before the central claim can be considered fully established.

major comments (3)

[§6, Proposition 6.1 (page 15)] The construction of the cone σ_S is incomplete. The proof chooses unit vectors v_1,…,v_n “sufficiently close to v” and sets σ_S = {u : (u,v_i) ≥ 0}. For the subsequent claims (σ_S ⊂ {ℓ>0}∪{0} and every ℓ-positive ray of Δ lying in the relative interior of σ_S) one needs v ∈ int(cone(v_i)); “sufficiently close” does not force this, because the v_i could all lie in a half-space through v. This is not a mere technicality: if σ_S contains a ray with ℓ<0, the subsequent link retraction argument can fail. Please add the surrounding condition explicitly and prove that such v_i exist. In addition, the bullet property asserting that the homotopies H^S_σ and H^Σ_σ stay outside σ_S is asserted without proof; since the retraction moves points along straight segments, this requires verification. A complete proof of Prop. 6.1 is necessary for Theorem 2.8.
[§6, Proposition 6.1, “moreover” statement (page 16)] The existence of the common stellar refinement Δ′ is imported from [AP24, Theorem 1.1], an arXiv preprint from 2024 that resolves a classical conjecture. This places a substantial external result at the base of the main theorem. If the result has since been accepted for publication, please cite the published version; if not, either prove the special case needed here (where one of the fans is a subdivision containing the cone σ_S) or clearly flag the dependence. As it stands, the unsupported preprint makes the proof of Theorem 2.8 conditional.
[§6, application of Proposition 5.1; §5, Proposition 5.1] Proposition 5.1 requires each D_I to be connected and rationally smooth. Corollary 3.8 establishes rational smoothness of intersections with toric subvarieties, but connectedness of D_I = \bar{U} ∩ (intersection of boundary divisors) is not proved. The H-regularity assumption only guarantees that U∩O is connected for each torus orbit O; it does not, as written, imply connectedness of the intersection of \bar{U} with an orbit closure, which may meet several orbits. This gap affects the identification H^0(D_I,Q)^∨ ≅ Q used in the proof of Prop. 5.1. Please supply a proof or a precise argument showing that the hypotheses of Prop. 5.1 hold in the setting of Section 6.

minor comments (4)

[§8, Lemma 8.1] The stated homology is internally inconsistent: a wedge of ten 2-spheres has H_2 ≅ Z^{10}, not H_3 ≅ Z^{10}. Moreover, the homology of the 2-skeleton of a 6-simplex has H_2 ≅ Z^{20} (Euler characteristic 21 − 1 = 20), not 10. Please correct the Betti numbers and the wedge count. The main point of the example (non-vanishing H_1) is unaffected.
[Abstract and Introduction] The abstract says the link is “not homotopy equivalent to a bouquet of 2-spheres,” while the introduction says “not homotopy equivalent to a bouquet of spheres.” The stronger statement with arbitrary spheres is false for the constructed example, which is a bouquet of S^1 and 2-spheres. Please make the wording consistent and accurate.
[§2, Remark 2.5] “Sch¨ on” should be “Schön.”
[§2, Example 2.11] The proof refers to Figure 1, but the figure is not reproduced in the text. If this is an arXiv rendering issue, please ensure the figure appears in the final version.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation chain is anchored to external Hodge-theoretic and tropical-geometric results, and no prediction reduces to a fit or to its own definition.

full rationale

The central implication Theorem 1.1 is derived from Theorem 2.8 through a fully written circle-valued Morse argument (Theorem 7.1 and Lemma 7.3), while Theorem 2.8 is derived from mixed Hodge module vanishing (Proposition 4.1), a dual-complex dictionary (Proposition 5.1), and an explicit toric compactification construction (Section 6). H-regularity is an explicit hypothesis taken from Hacking [Hac08], not a conclusion manufactured by the paper. The identity chi(X) = sum_q (-1)^q h^{0,q}(X) is imported from [Ite+19], and the Green-Lazarsfeld comparison is explicitly described as an analogue, not used as an input. The self-citations [LMW21] and [BW15] are used only as methodological templates or sources of an example, and the corresponding proofs are written out in the paper, so they are not load-bearing. The main external risk is Proposition 6.1: its 'moreover' statement invokes [AP24, Theorem 1.1], a recent preprint resolving a classical conjecture, and the cone construction in its proof is subtle. If that theorem or the cone construction fails, the proof of Theorem 2.8 would be incomplete. That is a correctness or verification concern, not circularity: Proposition 6.1 is not assumed as an input and no fitted parameter is later relabeled as a prediction. Likewise, the paper's claims are checked against external benchmarks (GL87, Hac08, Pay13, MS15, Saito's MHM, BBD82), giving independent content to the central result.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No fitted parameters and no invented entities: this is a pure mathematics paper whose inputs are standard Hodge-theoretic theorems, the H-regularity definition, and cited results ([AP24], [Ite+19], [ST08], [MS15], [CLS11]). The only free choices are generic linear functions in the Morse arguments, which are existential choices, not fitted values. The axioms above list the unproved background and the domain hypotheses the central claim rests on; the riskiest item is [AP24, Theorem 1.1].

assumptions (6)

standard math Saito's theory of mixed Hodge modules, the decomposition theorem, and Artin vanishing for perverse sheaves ([Sai89], [Sai91], [BBD82])
Prop. 4.1 is proved inside this framework; deep external input.
standard math Purity of intersection cohomology of projective varieties (hence purity of H^q(Z,Q) for rationally smooth projective Z)
Used in Prop. 5.1 to justify that H^q(D_I,Q) is pure of weight q; not cited by the paper.
domain assumption [AP24, Theorem 1.1]: any two complete rational simplicial fans have a common rational simplicial refinement obtained by stellar subdivisions
Load-bearing for the 'moreover' part of Prop. 6.1 (existence of Δ' and Δ_+), hence for Thm 2.8 and Thm 1.1; recent preprint resolving a classical conjecture.
domain assumption Tropical Euler characteristic satisfies χ(X) = Σ_q (−1)^q h^{0,q}(X) with h^{0,q} the tropical homology Hodge numbers ([Ite+19, Example 17])
This is the definition/correspondence converting the topological χ into the object the proof controls; external input from Itenberg–Katzarkov–Mikhalkin–Zharkov.
domain assumption H-regularity axioms (Defs 2.1, 2.3): the variety is locally modeled on trop(U) for a very affine U whose closure in some smooth projective toric variety is smooth, transverse to all torus orbits, with connected or empty intersections (strengthened beyond Hacking, Remark 2.2)
The hypothesis of the central theorem; a strong technical condition the paper proves is necessary (Example 2.11).
standard math Tropicalization dictionary: |Σ| = trop(U) for the orbit-meeting subfan ([ST08, Prop 3.9]); trop(π(U)) = trop(π)(trop(U)) ([MS15, Cor 3.2.13])
Used in §6 and §8 to identify the fans under compactification and image.

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Pith. "Pith review of Topology and Euler characteristics of tropical varieties." pith.science (2026). https://pith.science/paper/IABW5NEL

@misc{pith2026260610817,
  author       = {Pith},
  title        = {Pith review of: Topology and Euler characteristics of tropical varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IABW5NEL}},
  note         = {Machine review of arXiv:2606.10817}
}

read the original abstract

We study Euler characteristics of tropical subvarieties of tropical abelian varieties. We prove that every H-regular subvariety, locally modeled on tropicalizations of sufficiently well-behaved very affine varieties, has nonnegative signed Euler characteristic. This gives a tropical analogue of a theorem of Green-Lazarsfeld for subvarieties of complex abelian varieties. The main input is a local vanishing theorem for H-regular tropical fans, which also yields a Lefschetz-type theorem for affine H-regular tropical varieties. We further show that the signed Euler characteristic inequality fails for general tropical subvarieties of tropical abelian varieties, and we construct a 3-dimensional tropical fan whose link is not homotopy equivalent to a bouquet of 2-spheres.

Figures

Figures reproduced from arXiv: 2606.10817 by the authors.

**Figure 1.** A genus two curve embedded in its Jacobian. 3. Transversality in toric varieties Let σ ⊂ R n be a strongly convex rational polyhedral cone of codimension m = n − dim σ. Then Xσ is an affine normal toric variety with dense torus T = (C ∗ ) n . The minimal-dimensional T-orbit of Xσ, denoted O, is isomorphic to (C ∗ ) m. After choosing a noncanonical splitting of the ambient lattice, we may write Xσ ∼= (C ∗ ) m × Xσ′, … view at source ↗

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Works this paper leans on

13 extracted references · 6 linked inside Pith

[2]

Elimination theory for trop- ical varieties

Proc. Sympos. Pure Math. Amer. Math. Soc., Providence, RI, 1991, pp. 509–517.isbn: 0-8218-1490-7 (cit. on p. 12). [ST08] Bernd Sturmfels and Jenia Tevelev. “Elimination theory for trop- ical varieties”. In:Math. Res. Lett.15.3 (2008), pp. 543–562. issn: 1073-2780 (cit. on pp. 16, 17). University of Wisconsin-Madison Email address:shiatt@wisc.edu Sydney Ma...

1991
[7]

arXiv:2405.12012 [math.AG](cit. on p. 25). [DW25] Ya Deng and Botong Wang.L 2-vanishing theorem and a con- jecture of Koll´ ar

arXiv
[8]

On the signed Euler characteristic property for subvarieties of abelian varieties

arXiv:2409.11399 [math.AG](cit. on p. 25). [EGM18] Eva Elduque, Christian Geske, and Laurentiu Maxim. “On the signed Euler characteristic property for subvarieties of abelian varieties”. In:J. Singul.17 (2018), pp. 368–387.issn: 1949-2006 (cit. on p. 18). [FK00] Joseph Franecki and Mikhail Kapranov. “The Gauss map and a noncompact Riemann-Roch formula for...

arXiv 2018
[9]

The homology of tropical varieties

arXiv:2503.12835 [math.AG](cit. on p. 24). [Hac08] Paul Hacking. “The homology of tropical varieties”. In:Collect. Math.59.3 (2008), pp. 263–273.issn: 0010-0757,2038-4815 (cit. on pp. 2, 3, 13, 22, 23). [Ite+19] Ilia Itenberg et al. “Tropical homology”. In:Math. Ann.374.1-2 (2019), pp. 963–1006.issn: 0025-5831,1432-1807 (cit. on pp. 1, 24). [LMW21] Yongqi...

arXiv 2008
[10]

arXiv:2309.00229 [math.CO] (cit. on p. 25). [MS15] Diane Maclagan and Bernd Sturmfels.Introduction to tropical geometry. Vol

arXiv
[100]

The signed Euler characteris- tic of very affine varieties

Ast´ erisque. Soc. Math. France, Paris, 1982, pp. 5– 171 (cit. on p. 12). [BW15] Nero Budur and Botong Wang. “The signed Euler characteris- tic of very affine varieties”. In:Int. Math. Res. Not. IMRN14 (2015), pp. 5710–5714.issn: 1073-7928,1687-0247 (cit. on p. 5). [Cha+26] Melody Chan et al. “Piecewise-exponential functions and Ehrhart fans”. In:J. Lond....

1982
[124]

Th´ eorie de Hodge. II

Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2011, pp. xxiv+841.isbn: 978-0-8218-4819-7 (cit. on p. 17). [Del71] Pierre Deligne. “Th´ eorie de Hodge. II”. In:Inst. Hautes ´Etudes Sci. Publ. Math.40 (1971), pp. 5–57.issn: 0073-8301,1618-1913 (cit. on pp. 12, 13). [DP14] Alexandru Dimca and Stefan Papadima. “Non-abelian co...

2011
[161]

Tropical curves, their Ja- cobians and theta functions

Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2015, pp. xii+363.isbn: 978-0-8218-5198-2 (cit. on pp. 3, 5, 23). [MZ08a] Grigory Mikhalkin and Ilia Zharkov. “Tropical curves, their Ja- cobians and theta functions”. In:Curves and abelian varieties. Vol

2015

Show all 13 references

[465]

Connectivity of weighted rank 3 matroids

Contemp. Math. Amer. Math. Soc., Providence, RI, 2008, pp. 203–230.isbn: 978-0-8218-4334-5 (cit. on p. 5). [MZ08b] Grigory Mikhalkin and G¨ unter M. Ziegler. “Connectivity of weighted rank 3 matroids”. In:Oberwolfach Reports5.4 (2008). Problem Session edited by J. Pach, pp. 24...

2008
[2021]

Lagrangian geometry of matroids

arXiv:1401.7301 [math.AG](cit. on p. 2). [ADH23] Federico Ardila, Graham Denham, and June Huh. “Lagrangian geometry of matroids”. In:J. Amer. Math. Soc.36.3 (2023), pp. 727–794.issn: 0894-0347,1088-6834 (cit. on p. 24). [AHL25] Omid Amini, June Huh, and Matt Larson.A decomposi...

2023 arXiv
[2023]

arXiv:2310.15367 [math.CO](cit. on p. 23). [AP24] Karim Adiprasito and Igor Pak.All triangulations have a com- mon stellar subdivision

arXiv
[2024]

Perverse sheaves on varieties with large fundamental groups

arXiv:2404 . 05930 [math.CO] (cit. on p. 16). [A W25] Donu Arapura and Botong Wang. “Perverse sheaves on varieties with large fundamental groups”. In:J. Differential Geom.129.1 (2025), pp. 1–15.issn: 0022-040X,1945-743X (cit. on p. 25). [BBD82] A. A. Beilinson, J. Bernstein, a...

2025
[2025]

arXiv:2511.02026 [math.AG] (cit. on p. 24). 26 REFERENCES [AP23] Omid Amini and Matthieu Piquerez.Hodge theory for tropical fans

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [2]

Elimination theory for trop- ical varieties

Proc. Sympos. Pure Math. Amer. Math. Soc., Providence, RI, 1991, pp. 509–517.isbn: 0-8218-1490-7 (cit. on p. 12). [ST08] Bernd Sturmfels and Jenia Tevelev. “Elimination theory for trop- ical varieties”. In:Math. Res. Lett.15.3 (2008), pp. 543–562. issn: 1073-2780 (cit. on pp. 16, 17). University of Wisconsin-Madison Email address:shiatt@wisc.edu Sydney Ma...

1991

[2] [7]

arXiv:2405.12012 [math.AG](cit. on p. 25). [DW25] Ya Deng and Botong Wang.L 2-vanishing theorem and a con- jecture of Koll´ ar

arXiv

[3] [8]

On the signed Euler characteristic property for subvarieties of abelian varieties

arXiv:2409.11399 [math.AG](cit. on p. 25). [EGM18] Eva Elduque, Christian Geske, and Laurentiu Maxim. “On the signed Euler characteristic property for subvarieties of abelian varieties”. In:J. Singul.17 (2018), pp. 368–387.issn: 1949-2006 (cit. on p. 18). [FK00] Joseph Franecki and Mikhail Kapranov. “The Gauss map and a noncompact Riemann-Roch formula for...

arXiv 2018

[4] [9]

The homology of tropical varieties

arXiv:2503.12835 [math.AG](cit. on p. 24). [Hac08] Paul Hacking. “The homology of tropical varieties”. In:Collect. Math.59.3 (2008), pp. 263–273.issn: 0010-0757,2038-4815 (cit. on pp. 2, 3, 13, 22, 23). [Ite+19] Ilia Itenberg et al. “Tropical homology”. In:Math. Ann.374.1-2 (2019), pp. 963–1006.issn: 0025-5831,1432-1807 (cit. on pp. 1, 24). [LMW21] Yongqi...

arXiv 2008

[5] [10]

arXiv:2309.00229 [math.CO] (cit. on p. 25). [MS15] Diane Maclagan and Bernd Sturmfels.Introduction to tropical geometry. Vol

arXiv

[6] [100]

The signed Euler characteris- tic of very affine varieties

Ast´ erisque. Soc. Math. France, Paris, 1982, pp. 5– 171 (cit. on p. 12). [BW15] Nero Budur and Botong Wang. “The signed Euler characteris- tic of very affine varieties”. In:Int. Math. Res. Not. IMRN14 (2015), pp. 5710–5714.issn: 1073-7928,1687-0247 (cit. on p. 5). [Cha+26] Melody Chan et al. “Piecewise-exponential functions and Ehrhart fans”. In:J. Lond....

1982

[7] [124]

Th´ eorie de Hodge. II

Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2011, pp. xxiv+841.isbn: 978-0-8218-4819-7 (cit. on p. 17). [Del71] Pierre Deligne. “Th´ eorie de Hodge. II”. In:Inst. Hautes ´Etudes Sci. Publ. Math.40 (1971), pp. 5–57.issn: 0073-8301,1618-1913 (cit. on pp. 12, 13). [DP14] Alexandru Dimca and Stefan Papadima. “Non-abelian co...

2011

[8] [161]

Tropical curves, their Ja- cobians and theta functions

Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2015, pp. xii+363.isbn: 978-0-8218-5198-2 (cit. on pp. 3, 5, 23). [MZ08a] Grigory Mikhalkin and Ilia Zharkov. “Tropical curves, their Ja- cobians and theta functions”. In:Curves and abelian varieties. Vol

2015