{"id":"fadcb371-d7b0-405e-a3ea-6f6517d3f20f","arxiv_id":"2411.17873","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"On smooth projective toric varieties, every coherent sheaf has a minimal line bundle resolution of length at most the dimension, and for toric subvarieties the Betti numbers are compactly supported cohomology groups of mirror-torus strata.","lead":"This paper uses mirror symmetry to construct minimal resolutions of coherent sheaves on toric varieties by line bundles, and it gives a formula for the Betti numbers in terms of the topology of a mirror torus. For toric subvarieties, it recovers and improves the resolutions of Hanlon, Hicks, and Lazarev, producing minimal versions with explicit term counts.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central claim is only as secure as the imported non-equivariant CCC and its functoriality for finite toric morphisms; Proposition 4.9 is the least-checked bridge and should be verified on a nontrivial finite map.","rationale":"I read the paper in good faith and found no internal inconsistency: the constructive parts are coherent, the examples are consistent with the stated formulas, and the Macaulay2 verification of the cubic is independent evidence for at least that case. The strongest_claim is exactly Theorem 4.13, and the reader's weakest_assumption identifies the same soft spot that I see: the chain from topology to algebra goes through external results whose hypotheses and convention bookkeeping are not reproved. The paper's own exposition flags the dependence on Zhou's CCC, Favero-Huang exodromy, and Treumann's functoriality. My stress-test does not turn up a concrete contradiction, so I would not reject or lower the verdict on the evidence available. The most useful next step is a targeted computational check of Proposition 4.9 on a case with an independently known answer, such as Frobenius or the quadric map. If that check fails, Theorem 4.13 and the Betti formula would need revision; if it passes, the remaining concern is only the usual reliance on cited theorems in a subject where those theorems are currently the accepted state of the art.","tokens_in":27036,"tokens_out":20008,"duration_ms":199853,"concrete_test":"Run the full chain on the Frobenius morphism F_k:P^1→P^1, where the answer is known independently by Thomsen's theorem: compute the module M from Proposition 4.11 using the stated exit-space formula, take its minimal projective resolution over A_Ent, tensor with T, and compare the resulting summands with the Thomsen decomposition of F_k_*O. Also perform the same check on the quadric map P^1→P^2 of Example 4.17, where the resolution is O(-2)→O. Any shift, sign, or v^* versus v! error in Proposition 4.9 changes the multiplicities, so either check would settle whether the functoriality bridge actually lands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim, Theorem 4.13, converts a topological computation on the mirror torus into a minimal line-bundle resolution of u_*O_X1. The conversion passes through four imported equivalences: Zhou's non-equivariant CCC (Theorem 3.4), the exodromy identification of Sh(T^n, Λ_c) with mod-A_Ent (Proposition 3.8), the wrapping adjunction (Theorem 3.14), and the functoriality statement for finite toric morphisms (Proposition 4.9). The least locally justified step is Proposition 4.9: it asserts that κ_2 ∘ u_* = v! ∘ κ_1 in the stated conventions, proved by bootstrapping from Treumann's functoriality for κ' after two duality twists. The proof does not display the sign and shift bookkeeping, and an error here would shift or permute the Betti numbers in Theorem 4.13 rather than merely lengthen the resolution. The paper also relies on Corollary 3.15, the full faithfulness of RHom(T,-), which rests on the same bridge. This is not an internal inconsistency, but it is the single load-bearing assumption: if Proposition 4.9 fails in the needed generality, the topological formula for β_{i,-a} does not transfer to XΣ. The examples and the Macaulay2 check for the cubic support internal consistency, but they do not remove the dependence on the black-box functoriality.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies minimal resolutions of coherent sheaves on smooth projective toric varieties by sums of line bundles from Bondal's finite collection Φ. The main results are Theorem 4.4, which gives a minimal resolution of length at most n for any coherent sheaf, and Theorem 4.13, which for a finite toric morphism u:XΣ1→XΣ2 gives a minimal resolution of u_*O_{XΣ1} of length k=dim XΣ2−dim XΣ1 with multiplicities β_{i,-a}=dim H^i_c(S^c_[a]∩V), where V is the zero fiber of the dual map on tori. The proof passes through Zhou's non-equivariant Coherent-Constructible Correspondence, the Favero–Huang exodromy identification with entrance path algebras, and wrapping functors between singular-support categories. A secondary result recovers the cellular resolutions of Hanlon–Hicks–Lazarev from a non-minimal injective resolution of the constant sheaf on V.","tokens_in":27296,"tokens_out":23015,"duration_ms":215757,"significance":"If correct, the paper gives a striking and useful bridge from topology to explicit minimal resolutions: the Betti numbers of the resolution of u_*O_{XΣ1} are computed by compactly supported cohomology of the cube stratification intersected with a subtorus. This generalizes and improves the Hanlon–Hicks–Lazarev resolutions, and Theorem 4.4 strengthens known bounds via the pdim bound of Favero–Huang. The main formulas are concrete and falsifiable, and the examples, including the Macaulay2 check for the normalization of the cubic, support the claimed terms. The chief weaknesses are the heavy reliance on imported equivalences and the terse, notationally inconsistent treatment of the functoriality of the mirror functor, which is load-bearing for the transfer of the topological Betti numbers.","major_comments":[{"comment":"The functoriality statement for finite toric morphisms is not written in a mathematically well-formed way. The map v is defined as v:M2,R/M2→M1,R/M1, so v_! is a pushforward from T^{n2} to T^{n1}. The expression v_!j_{0*}C_0 applies v_! to a sheaf on T^{n1}, which is not a valid composition. The proof also asserts that the right adjoint of v_! is ρR∘v_!, which is not a standard adjunction and is not justified. The intended statement is presumably that the mirror of u_*O_{XΣ1} is the pullback (up to the appropriate shift) of the skyscraper at 0 along v, i.e., the constant sheaf on V. Because this step transfers the topological computation into the algebraic resolution, the direction and the sign/shift bookkeeping must be corrected and proved explicitly.","section":"§4.3, Proposition 4.9 and equation (13)"},{"comment":"The purity argument for MΣ1 uses the equality t! = t* 'since t is proper by Proposition 2.21', but Proposition 2.21 only asserts that exit spaces are closed and contractible; it does not state compactness or properness. For the cube stratification the exit spaces are in fact compact polytopes under the strong convexity assumption, but this needs to be stated and proved. If an exit space were unbounded, the compactly supported cohomology computation that proves concentration in degree 0 would fail. This is a load-bearing step because it is how the paper concludes that MΣ1 is quasi-isomorphic to a pure module.","section":"§4.3, proof of Proposition 4.11"},{"comment":"The Betti-number computation conflates Hom in the derived category with the graded Hom complex. Equation (17) correctly asks for Ext^i(MΣ1,S[a]), but the following chain computes Hom_{D(mod-A)}(MΣ1,S[a]) and then concludes β_{i,-a}=dim H^i_c(S^c_[a]∩V). To make this valid, the chain should be an isomorphism of graded vector spaces, with the Serre-functor step accompanied by the appropriate shift. The final formula may be correct, but the homological-degree bookkeeping needs to be written out precisely; as it stands, the displayed equalities do not directly imply the claimed Ext computation.","section":"§4.3, proof of Theorem 4.13, equations (17)-(18)"},{"comment":"The proof of acyclicity of the cellular injective complex relies on the assertion that every (k−1)-cell in the induced CW structure on V is contained in the closure of exactly two k-cells. This is true for a regular CW decomposition of a closed manifold, but the induced stratification of a subtorus is only introduced in §2.3; Proposition 2.8 establishes the CW properties for the torus T^n, not for the induced stratification of V. Since Example 2.11 shows that strata of a subtorus can have multiple connected components, the frontier axiom and regularity for S^V_cw should be proved before this assertion is used. This affects both the length bound in Theorem 4.13 and the recovery of the Hanlon–Hicks–Lazarev resolutions.","section":"§4.5, Proposition 4.25"}],"minor_comments":[{"comment":"There are many typos and minor grammatical errors: 'Strumfels' should be 'Sturmfels', 'Hirzerbruch' should be 'Hirzebruch', 'verticies' should be 'vertices', 'morpshism' should be 'morphism', and 'startification' should be 'stratification'. A careful proofreading pass is needed.","section":"§1 and throughout"},{"comment":"The sentence about 'the pioneering work of Bayer, Strumfels, and Peeva' does not cite the correct reference; the citation [BHS21] is the Bruce–Heller–Sayrafi paper, not the monomial-ideal resolution work. The appropriate reference should be added.","section":"§1, paragraph on Bayer–Peeva–Sturmfels"},{"comment":"The phrase 'the zero-dimensional stratum has a trivial intersection' is ambiguous: if the intersection is a single point, the compactly supported cohomology is indeed one-dimensional in degree 0, but 'trivial' normally means empty. Please rephrase.","section":"§4.3, Example 4.17"},{"comment":"The notation MΣ1[a] is used before the module value at a vertex is defined; a short sentence explaining that this is the stalk at the vertex [a] would improve readability.","section":"§4.3, Proposition 4.11"}],"recommendation":"major_revision","confidential_remarks":"The central idea is attractive and the computational examples are convincing, but the proof of the functoriality transfer in §4.3 needs substantial rewriting before the main theorem can be considered established. In particular, the direction of the mirror functoriality and the shift/bookkeeping in Proposition 4.9 and Theorem 4.13 are load-bearing and should not be left as terse claims. The paper also depends on the authors' own prior work (FH22, FH23) and on 'forthcoming work' for Corollary 3.15; the editor may wish to ensure that the results are not duplicating unpublished material and that the dependence is clearly stated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Punchline: this is a solid paper with genuine new results, but the topological Betti formula depends on a chain of imported equivalences where the weakest link is Proposition 4.9. I'd send it to a careful referee.\n\nThe genuinely new content is not Theorem 1.1—the paper itself says that length-at-most-n resolutions were already accessible from Hanlon–Hicks–Lazarev. The new content is minimality, the exact formula β_{i,-a} = dim H^i_c(S^c_[a] ∩ V), the uniqueness when the skeleton equality holds, and the recovery of the HHL resolutions as a cellular refinement. They also give worked examples, including an independent Macaulay2 check for the cubic, which is real evidence. The writing is clear and honest about what is imported.\n\nThe soft spot is the bridge. The main theorem converts a topological computation on the mirror torus into an algebraic resolution via four imported equivalences: Zhou's non-equivariant CCC, the exodromy identification with modules over the entrance path algebra, the wrapping adjunction, and the functoriality statement for finite toric morphisms (Proposition 4.9). The paper does not reprove these; it uses them as black boxes. The least locally justified step is Proposition 4.9, which asserts κ₂ ∘ u_* = v_! ∘ κ₁ and is proven by bootstrapping from Treumann's functoriality after two duality twists. The proof does not display the sign and shift bookkeeping; an error there would shift or permute the Betti numbers rather than merely lengthen the resolution. That is a load-bearing assumption, and the examples do not remove it. The cellular resolution of §4.5 also has a somewhat sketchy argument about the number of adjacent cells, though the examples check out.\n\nNone of this is an internal inconsistency. The paper is not circular: the topological formula is derived from established equivalences, not assumed. The self-citations to Favero–Huang are to published, independently proven theorems, so that alone is not a concern.\n\nWho is this for? Researchers in toric geometry, syzygies, and homological mirror symmetry. It gives an explicit formula for Betti numbers that was not available before. I would bring it to a reading group and would cite it if I worked in the area.\n\nRecommendation: send it to peer review. Ask a referee with real expertise in stratified sheaves and toric geometry to check Proposition 4.9, and ask the authors to expand the sign/shift bookkeeping there. The paper deserves referee time.","headline":"A genuinely useful toric geometry paper: minimal resolutions with explicit topological Betti numbers, but the proof rests on a chain of imported equivalences where Proposition 4.9 deserves close scrutiny.","tokens_in":27866,"tokens_out":3165,"would_cite":true,"duration_ms":27258,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14M25","14F08","13D02","16G20","55N30"],"pacs":[],"model":"deepseek-v4-flash","headline":"A finite toric morphism's pushforward of the structure sheaf admits a minimal line bundle resolution of length equal to the codimension, with every Betti number given by compactly supported cohomology of a cube stratum intersected with…","keywords":["toric varieties","line bundle resolutions","Coherent-Constructible Correspondence","homological mirror symmetry","constructible sheaves","entrance path algebras","Betti numbers","finite toric morphisms"],"falsifier":"Compute the minimal resolution of the normalization module $N=\\mathbb{C}[x_0,x_1,x_2,t]/(tx_1-x_0x_2,\\,tx_0-x_1^2,\\,t^2-x_1x_2)$ as a module over the Cox ring of $\\mathbb{P}^2$, arising from the finite toric morphism $u:\\mathbb{P}^1\\to\\mathbb{P}^2$ with lattice map $(a)\\mapsto(2a,3a)$. Since $\\dim\\mathbb{P}^2-\\dim\\mathbb{P}^1=1$, the theorem predicts exactly the two-term complex $O(-2)^{\\oplus 2}\\to O(-1)\\oplus O\\to u_*O_{\\mathbb{P}^1}$; any extra term, any missing summand, or a minimal resolution of length different from $1$ would refute Theorem 4.13.","tokens_in":26808,"feed_emoji":"📐","tokens_out":13037,"duration_ms":106912,"temperature":0.7,"pith_summary":"This paper establishes that minimal resolutions of coherent sheaves on smooth projective toric varieties can be read off from the topology of a mirror real torus. For any coherent sheaf, it produces a minimal resolution by sums of line bundles from a finite collection introduced by Bondal, up to a common twist, with length at most the dimension of the variety. For a structure sheaf pushed forward along a finite toric morphism of codimension $k$, the resolution has length exactly $k$, and each Betti number is the dimension of a compactly supported cohomology group of one cube-stratum of the mirror torus intersected with the zero fiber of the dual map on tori. The upshot is that syzygies of toric subvarieties become topological counts, and the earlier cellular resolutions of [HHL23] appear as the non-minimal cellular version of the same construction.","feed_headline":"Syzygies of toric subvarieties are counts on a mirror torus","feed_subtitle":"Every syzygy count in a minimal resolution equals the cohomology of a stratum sliced by a subtorus.","key_machinery":"The load-bearing device is the commuting triangle of Theorem 3.14, which identifies the derived category of coherent sheaves on $X_\\Sigma$, the category of constructible sheaves on the mirror torus with singular support in the cube skeleton $\\Lambda^c$, and the derived category of modules over the entrance path algebra $A_{\\mathrm{Ent}}$ of the cube stratification. Under this equivalence, line bundles $O(-a)$ correspond to indecomposable projectives $P_{[a]}$, the left and right wrapping functors move between the smaller singular-support category of the Coherent–Constructible Correspondence and the larger stratified category, and right wrapping preserves co-probe sheaves. For pushforwards, the mirror of $u_*O$ is $v_!\\mathbb{C}_{\\{0\\}}$; Proposition 4.11 shows this object is represented, after a Serre twist and exodromy, by a pure module over $A_{\\mathrm{Ent}}$, so its minimal projective resolution exists and transfers back to a minimal line bundle resolution. The Betti numbers are then Ext groups, computed as compactly supported cohomology of the strata intersections.","core_discovery":"The paper's central discovery is a mirror-symmetry formula for minimal resolutions. Theorem 4.13 states that for a finite toric morphism $u:X_{\\Sigma_1}\\to X_{\\Sigma_2}$ of smooth projective toric varieties, the pushforward $u_*O_{X_{\\Sigma_1}}$ admits a minimal resolution by sums of line bundles of length $k=\\dim X_{\\Sigma_2}-\\dim X_{\\Sigma_1}$, and the multiplicity of $O(-a)$ in the $i$-th term is $\\beta_{i,-a}=\\dim H^i_c(S^c_{[a]}\\cap V)$, where $V$ is the zero fiber of the dual map on tori and $S^c_{[a]}$ are the strata of the cube stratification of the mirror torus. The same mechanism gives, for an arbitrary coherent sheaf, a minimal line bundle resolution of length at most $n$ after a twist by a sufficiently high power of an ample line bundle. The proof transfers the sheaf across the non-equivariant Coherent–Constructible Correspondence, views it as a module over the entrance path algebra of the stratified torus, and resolves that module; minimality and the length bound come from the finite-dimensional algebra's homological dimension.","pith_inferences":["The Betti formula suggests a practical algorithm that avoids constructing differentials first: triangulate the stratified torus, intersect with $V$, compute $H^i_c$ of each stratum intersection, and assemble the terms of the minimal resolution; this could be benchmarked against direct computer algebra computations of syzygies.","Because a generically finite toric morphism factors as a birational map followed by a finite one, the same mirror mechanism may yield minimal resolutions for birational pushforwards, with a length bound that adds a correction to the codimension; the paper does not work out this case.","The uniqueness that holds when $\\Lambda_\\Sigma=\\Lambda^c$ suggests that on projective space and products of projective spaces, minimal line bundle resolutions are canonical invariants of coherent sheaves; comparing them across toric embeddings may yield a toric analogue of multigraded Betti tables."],"forward_implications":["Any coherent sheaf on a smooth projective $n$-dimensional toric variety admits a minimal resolution by sums of line bundles from the finite collection, after a sufficiently high twist, with length at most $n$.","For a finite toric morphism of smooth projective toric varieties, the pushforward of the structure sheaf admits a minimal line bundle resolution of length exactly the codimension $k$.","The multiplicities in that resolution are topological invariants: $\\beta_{i,-a}=\\dim H^i_c(S^c_{[a]}\\cap V)$, so Betti numbers can be computed from compactly supported cohomology on the mirror torus.","Whenever the singular-support skeleton equals the cube skeleton, for example on projective space and products of projective spaces, the minimal resolution is unique and hence irreducible.","The cellular resolutions of [HHL23] are recovered as the non-minimal CW version of this construction, and Frobenius pushforwards decompose as direct sums of line bundles."],"supporting_citations":[{"why":"Supplies the non-equivariant Coherent–Constructible Correspondence (Theorem 3.4) and the mirror of the structure sheaf used in Proposition 4.10.","marker":"[Zho17]"},{"why":"Provides the exodromy equivalence identifying constructible sheaves on the stratified torus with modules over the entrance path algebra, and the ring isomorphism to line bundle endomorphisms.","marker":"[FH22]"},{"why":"Bounds the projective dimension of the entrance path algebra by the dimension of the torus, giving the length bounds for the resolutions.","marker":"[FH23]"},{"why":"Supplies functoriality of the mirror map for finite toric morphisms, used to compute the mirror of the pushforward in Proposition 4.9.","marker":"[Tre10]"},{"why":"The cellular line bundle resolutions recovered in Section 4.5 serve as the comparison point for the minimal resolutions.","marker":"[HHL23]"},{"why":"Gives the algorithmic minimal projective resolutions of quiver algebras used to make the resolutions explicit in Section 4.4.","marker":"[GS07]"},{"why":"The Frobenius pushforward decomposition recovered in Example 4.20 provides a consistency check for the Betti formula.","marker":"[Tho00]"}],"fun_headline_variants":["Syzygies counted by mirror torus strata","Mirror symmetry turns syzygies into torus counts","Cohomology of mirror strata gives syzygies","Line bundle resolutions mirror torus topology","Minimal resolutions from mirror torus strata"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the mirror-symmetry bridge is lossless: the non-equivariant Coherent–Constructible Correspondence and the exodromy identification with entrance-path-algebra modules are imported as black boxes, so if either equivalence fails for the toric varieties considered, the topological Betti formula would not transfer back to resolutions.","fun_headline_variants_meta":{"raw":{"variants":["Syzygies counted by mirror torus strata","Mirror symmetry turns syzygies into torus counts","Cohomology of mirror strata gives syzygies","Line bundle resolutions mirror torus topology","Minimal resolutions from mirror torus strata"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00044,"raw_usage":{"total_tokens":2253,"prompt_tokens":984,"completion_tokens":1269,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":600,"completion_tokens_details":{"reasoning_tokens":1197}},"tokens_in":600,"tokens_out":1269,"duration_ms":10031,"temperature":1.0,"reasoning_tokens":1197,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:46:56.676343+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the minimal resolution of the normalization module $N=\\mathbb{C}[x_0,x_1,x_2,t]/(tx_1-x_0x_2,\\,tx_0-x_1^2,\\,t^2-x_1x_2)$ as a module over the Cox ring of $\\mathbb{P}^2$, arising from the finite toric morphism $u:\\mathbb{P}^1\\to\\mathbb{P}^2$ with lattice map $(a)\\mapsto(2a,3a)$. Since $\\dim\\mathbb{P}^2-\\dim\\mathbb{P}^1=1$, the theorem predicts exactly the two-term complex $O(-2)^{\\oplus 2}\\to O(-1)\\oplus O\\to u_*O_{\\mathbb{P}^1}$; any extra term, any missing summand, or a minimal resolution of length different from $1$ would refute Theorem 4.13.","supporting_citations":[],"review_version":1}