{"id":"c718d983-ea77-4e0f-9654-c517989c8e73","arxiv_id":"2608.09293","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"General Fano complete intersections of arbitrary multidegree and dimension are K-polystable, and K-stable unless isomorphic to P^n or a quadric hypersurface.","lead":"This paper proves that a general Fano complete intersection, of any dimension and any multidegree, is K-polystable, and K-stable unless it is projective space or a quadric hypersurface. It gives the first uniform result for arbitrary multidegree and resolves a problem posed by Xu and Zhuang.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central induction rests on [LZ26, Thm 10.3], cited in Prop 4.14 without stating its hypotheses; if that theorem does not apply to (Z, εY), the cyclic-cover engine and Theorem 1.3 lose their proof.","rationale":"The reader's verdict identified the same load-bearing assumption: the unstated applicability of [LZ26, Theorem 10.3] in Proposition 4.14. I agree. I checked the surrounding argument and found no independent numerical obstruction: the inequality in Theorem 7.1 Case 2 follows automatically from a_i|r_i and r_i≥2, and the log-canonical coefficient in Case 1 is at most 1 for smooth general divisors. The induction in Theorem 7.2 is structurally sound if Theorem 7.1 holds. The Cartier-canonical-class issue is relevant to the weighted statements but does not affect Theorem 1.3, which takes place in ordinary projective space. Therefore the central claim depends, exactly as the reader said, on a single external theorem whose hypotheses and content are not included. The conditional verdict is appropriate; I would not change it based on this pass.","tokens_in":23621,"tokens_out":26148,"duration_ms":228815,"concrete_test":"Retrieve the complete statement of [LZ26, Theorem 10.3] and confirm, line by line, the following in the setting of Proposition 4.14: (a) Z is allowed to be a K-polystable Gorenstein Fano variety and Y a Cartier divisor in |O_Z(d)|; (b) the hypothesis 'K-semistable for every sufficiently small ε>0' is sufficient, and no additional uniformity in ε is demanded; (c) GIT polystability of Y with respect to Aut(Z) is with respect to the natural linearization induced on P(H^0(Z,O_Z(d))); and (d) no smoothness, snc, or finite-stabilizer condition on Y is hidden in the statement. If any of these fails, replace or repair Proposition 4.14 before claiming Theorem 1.3.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 4.14 is the single point where the paper upgrades K-semistability of a log Fano pair (Z, cY) to K-polystability. In both part (1) and part (3), the proof relies verbatim on a cited result: 'Because (Z, εY) is K-semistable for any ε≪1, by [LZ26, Theorem 10.3], the GIT polystability of Y implies that (Z, εY) is K-polystable.' The paper does not state Theorem 10.3 or verify any of its hypotheses. This theorem is used again in Theorem 4.15, which is the engine behind Theorem 6.1, Theorem 7.1, and the base cases of Theorem 7.2, and therefore behind the main Theorem 1.3. If Theorem 10.3 requires, for instance, that K-semistability hold for all small ε in an interval, or that Y be GIT polystable with respect to a particular linearization (e.g., on the Hilbert scheme rather than on P(H^0(Z,L))), or that Z have no non-trivial special degenerations compatible with Y, none of these are checked in the manuscript. A local check shows the rest of the induction is numerically consistent: the inequality in Theorem 7.1 Case 2 follows automatically from a_i|r_i and r_i≥2, and the log-canonical coefficient in Case 1 is ≤1 for smooth general divisors. The sole unverifiable dependency is [LZ26, Theorem 10.3], making the central claim conditional on an external result not accessible to the reader.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies K-stability of Fano complete intersections. The main theorem (Theorem 1.3) asserts that a general n-dimensional Fano complete intersection in P^{n+k} is K-polystable, and K-stable unless it is isomorphic to P^n or a smooth quadric. A weighted version (Theorem 1.5) is proved for weighted complete intersections satisfying a_{n+i}|r_i. The proof uses a cyclic-cover engine (Theorem 4.15) built from interpolation, GIT polystability of hypersurfaces, and an induction on dimension and codimension. The paper also derives corollaries on K-moduli components and recovers several known results, including the Arezzo-Ghigi-Pirola theorem on intersections of two quadrics, with algebraic proofs.","tokens_in":23944,"tokens_out":19779,"duration_ms":171589,"significance":"If correct, the main theorem verifies the folklore conjecture for general complete intersections, providing the first full-dimensional family of K-polystable Fano varieties of arbitrary dimension and multidegree; it also answers a problem of Xu and Zhuang and gives nonempty K-moduli components. The proof strategy is attractive: the double induction in Theorem 7.2 reduces the general case to cyclic covers over K-polystable bases, and the numerical conditions in Theorem 7.1 are internally consistent. The paper is largely self-contained apart from the interpolation result [ADL24] and the polystability-upgrade result [LZ26]; the former is published, while the latter is a very recent dependency whose hypotheses are not verified in the manuscript. This conditional external step currently limits confidence in the central claim.","major_comments":[{"comment":"The upgrade from K-semistability of (Z, epsilon Y) to K-polystability using GIT polystability of Y is invoked by citing [LZ26, Theorem 10.3] without stating the theorem or verifying that its hypotheses are satisfied for the pairs considered in this paper. This step is the engine behind Theorem 4.15, which in turn proves Theorems 6.1, 7.1, and 1.3. Please state [LZ26, Theorem 10.3] in full and check each hypothesis (e.g., the relevant GIT linearization, the range of epsilon, and any condition on the automorphism group or degenerations of Z) for the pairs (Z, epsilon Y) and (Z, (1-1/m)Y). As written, this is a load-bearing black box.","section":"Proposition 4.14 / Theorem 4.15"},{"comment":"The Galois group action in the proof of Theorem 6.1 is misidentified. For the equations f_i + x_{n+i}^{r_i/a_{n+i}} = 0, the covering group of X -> P^n has order product of r_i/a_{n+i}, not product of r_i; the stated action of product Z/r_i does not preserve X when a_{n+i} > 1. Consequently, the asserted quotient X/G is isomorphic to P^n and the branch formula are not justified. The branch coefficients (1 - a_{n+i}/r_i) correspond to the correct smaller group, so the intended statement may be recoverable, but the proof must be rewritten with the correct group and the corresponding finite-quotient K-stability result.","section":"Theorem 6.1 proof"},{"comment":"The claim that the K-stability of (P^n, cB) for all c in (0, lambda) follows directly from Proposition 4.14 is not supported. Proposition 4.14 applies to a pair (Z, cY) with Y a single hypersurface in |O_Z(d)|, whereas B is a sum of divisors of different degrees r_i. Neither the reduction to a single Y nor the hypotheses on Y (e.g., K-polystability of Y as a Fano variety, or log-canonicity of (P^n, k/d Y) with Y GIT polystable) are established. Consequently, the proof of Corollary 6.2, which uses Theorem 6.1 for Fano index one, is incomplete as written.","section":"Theorem 6.1 proof, klt case"}],"minor_comments":[{"comment":"In Case 1 of Theorem 7.1, GIT polystability of B_Y is deduced from Lemma 3.3, which requires Aut(Y) to be finite; if Y is allowed to be isomorphic to P^n under the hypotheses, this implication does not apply, and the needed statement would follow from Lemma 3.2. Please add the P^n case or explicitly assume r_i >= 2 so that the issue does not arise.","section":"Theorem 7.1, Case 1"},{"comment":"Several statements use the Fano index iota_X of weighted complete intersections without verifying that O_X(K_X) is Cartier as required by Definition 2.1. For general X this follows from smoothness (Proposition 2.4), but the special-form varieties in Theorems 6.1 and 7.1 are not assumed smooth in the index portion of the statement; please add the Cartier/Q-Gorenstein hypothesis or prove it from a_{n+i}|r_i.","section":"Definition 2.1 / Theorems 6.1 and 7.1"},{"comment":"The reduction that if r_j = a_{n+j} then X is isomorphic to a complete intersection of codimension k-1 is used without proof; an explicit elimination of the coordinate x_{n+j} and the corresponding change of weights would make this step transparent.","section":"Theorem 6.1 proof"},{"comment":"In Lemma 7.7, the statement that Q_1 and Q_2 may be assumed to have rank n-1 is suspicious; a quadric in P^n has rank at most n+1, and the later completion-of-squares step suggests rank n on the remaining coordinates. Please correct the rank and check the dimension count in Theorem 7.6, where the branch locus is a threefold in P^6.","section":"Lemma 7.7"}],"recommendation":"major_revision","confidential_remarks":"The main theorem appears plausible, but the proof hinges on the cited [LZ26, Theorem 10.3] in Proposition 4.14, and the paper does not provide enough information for a reader to verify that the theorem applies. Additionally, Theorem 6.1 contains a concrete group-action error that must be corrected. I recommend asking the authors to supply the full statement and verification of [LZ26, Theorem 10.3] (or to replace it with a self-contained argument), and to rewrite the proof of Theorem 6.1 with the correct covering group. If these points are addressed, the paper could be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nQuick verdict: this is a serious paper with a real main theorem, and the main argument is probably sound. But it is not ready as-is: the proof leans on a cited result whose hypotheses are not checked, and one advertised theorem is under-proved.\n\nThe new thing is the double induction on dimension and codimension, via cyclic covers, proving K-polystability/stability for the general Fano complete intersection of any multidegree, plus the weighted version. That is a genuine advance over the prior special cases, and the new K-moduli components are a nice corollary. The induction in Theorem 7.2 is coherent, and the numerical checks we did (the inequality in Theorem 7.1 Case 2, the lc coefficients) go through. The citations to [ADL24] and [LZ26] are not circular.\n\nSoft spots, in proportion. The single most load-bearing step is Proposition 4.14, which upgrades K-semistability of (Z, εY) to K-polystability via [LZ26, Thm 10.3]. The paper neither states that theorem nor verifies its hypotheses. Maybe it is exactly the right tool—plausibly, GIT polystability of Y plus K-semistability for all small ε is precisely what it needs—but the referee will have to check this. If it fails, the central induction loses its engine. This is the one thing I would want pinned down before trusting Theorem 1.3.\n\nTheorem 6.1 is rougher. Its proof applies Proposition 4.14 to a multi-component fractional divisor, which that proposition does not cover as written, and in the lc case it asserts \"we may assume the B_i are quadrics\" without giving the reduction. The statement is broader than the proof supports. That said, Theorem 6.1 is not used in the main theorem, so this is a repair issue, not a fatal one. The weighted statement also never verifies that the varieties in question have Cartier canonical class, which the Fano index definition needs; for divisibility conditions like a_{n+i}|r_i it is probably true, but it should be said.\n\nBottom line: this deserves a serious referee. The main theorem is important and the framework is fresh; the referee should spend time on [LZ26] and on deciding whether Theorem 6.1 gets fixed or trimmed. I would send it out.","headline":"A likely-correct breakthrough on K-stability of general Fano complete intersections, with a load-bearing citation that needs checking and a peripheral theorem that needs repair.","tokens_in":24490,"tokens_out":7812,"would_cite":true,"duration_ms":69100,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14J45","14M10","14L24","32Q20"],"pacs":[],"model":"deepseek-v4-flash","headline":"A general Fano complete intersection of any multidegree is K-polystable, and K-stable unless it is projective space or a quadric.","keywords":["K-stability","Fano complete intersections","weighted complete intersections","Fano index","cyclic covers","geometric invariant theory","Kähler-Einstein metrics","K-moduli"],"falsifier":"Take the explicit smooth weighted complete intersection with equations $f_i + x_{n+i}^{r_i/a_{n+i}}=0$ used in Theorems 6.1 and 7.1 and compute its stability threshold $\\delta$; a Fano variety is K-stable precisely when $\\delta>1$. If any admissible multidegree gives $\\delta\\le 1$ for this special member while the variety is not isomorphic to $\\mathbb{P}^n$ or a quadric, then the constructed member is not K-stable, the induction fails, and Theorem 1.3 is false.","tokens_in":23405,"feed_emoji":"📐","tokens_out":11681,"duration_ms":101284,"temperature":0.7,"pith_summary":"This paper attacks the folklore conjecture that smooth Fano complete intersections should be K-stable, with projective space and quadric hypersurfaces as the only exceptions. It proves the conjecture for the general member of every deformation family: for any dimension and any multidegree satisfying the Fano inequality, the general complete intersection of that type is K-polystable, and K-stable if it is not isomorphic to $\\mathbb{P}^n$ or a smooth quadric. Since K-polystability of a smooth Fano manifold is equivalent to admitting a Kähler-Einstein metric, the result says that the general such variety carries a canonical metric. It also produces nonempty K-moduli components for every admissible multidegree.","feed_headline":"General Fano complete intersections proven K-polystable","feed_subtitle":"For every multidegree, the general smooth Fano complete intersection carries a Kähler–Einstein metric.","key_machinery":"The engine is a cyclic-cover criterion together with a double induction. Theorem 4.15 says that if a Fano variety $X$ is an $m$-fold cyclic cover of a K-polystable Gorenstein Fano variety $Z$ branched along a hypersurface $Y$, then $X$ is K-polystable provided either $Y$ is Fano and K-polystable, or the log Calabi-Yau pair $(Z,\\frac{k}{d}Y)$ is log canonical and $Y$ is GIT-polystable with respect to $\\operatorname{Aut}(Z)$. The Fano index, the largest $m$ with $-K_X \\sim mH$ for an ample class $H$, controls which case applies. This is fed by three ingredients: equivariant K-stability under finite group quotients, GIT polystability of the branch hypersurface, and an upgrade principle that converts K-semistability plus GIT polystability into K-polystability. Theorem 6.1 handles small Fano index by expressing the special member as a Galois cover of $\\mathbb{P}^n$ branched over a simple normal crossing divisor; Theorem 7.1 supplies the inductive step that strips one equation $x_{n+k}^{r_k/a_{n+k}}$ at a time, reducing the dimension and codimension until only hypersurfaces remain.","core_discovery":"The central result, Theorem 1.3, states that if $X \\subset \\mathbb{P}^{n+k}$ is a general $n$-dimensional Fano complete intersection of type $(r_1,\\ldots,r_k)$, then $X$ is K-polystable, and K-stable unless $X$ is isomorphic to $\\mathbb{P}^n$ or a smooth quadric. The proof actually establishes the stronger weighted statement, Theorem 1.5, for general Fano weighted complete intersections $X \\subset \\mathbb{P}(1^{n+1},a_{n+1},\\ldots,a_{n+k})$ of type $(r_1,\\ldots,r_k)$ whenever $a_{n+i}\\mid r_i$ for every $i$. The exceptions are forced: $\\mathbb{P}^n$ and smooth quadrics have positive-dimensional automorphism groups, so they are only K-polystable, not K-stable.","pith_inferences":["The theorem settles K-stability of the general member; the stronger conjecture that every smooth member of each multidegree is K-stable remains open. A natural next step is to check whether the auxiliary varieties $Y$ and $B_Y$ in Theorem 7.1 can be shown K-stable for arbitrary smooth members, which would let the same cyclic-cover induction upgrade the general statement to the full conjecture.","The divisibility hypothesis $a_{n+i}\\mid r_i$ in the weighted theorem looks technical rather than essential: it makes the exponents $r_i/a_{n+i}$ integral so that the equations define cyclic covers. Dropping it would likely require working with fractional powers or base changes, and testing whether Theorem 1.5 survives without it is a concrete open problem.","Because K-moduli spaces are proper, Theorem 1.3 guarantees the existence of K-polystable limits for degenerating families of complete intersections; identifying those boundary points with explicit GIT-stable objects, as is known in low codimension, could give a complete description of the new K-moduli components."],"forward_implications":["Corollary 1.4: for every $n\\ge 2$ and every multidegree with $r_1+\\cdots+r_k<n+k+1$ and volume $v=(n+k+1-(r_1+\\cdots+r_k))^n r_1\\cdots r_k$, the K-moduli space $\\mathcal{M}^K_{n,v}$ is nonempty and contains an irreducible component whose generic point is a smooth complete intersection of that type.","The general smooth Fano complete intersection of each multidegree admits a Kähler-Einstein metric, by the established equivalence between K-polystability and existence of Kähler-Einstein metrics on smooth Fano manifolds.","Theorem 1.5 extends the same conclusion to general weighted complete intersections in $\\mathbb{P}(1^{n+1},a_{n+1},\\ldots,a_{n+k})$ whenever $a_{n+i}\\mid r_i$.","Every smooth complete intersection of two quadrics in $\\mathbb{P}^{n+2}$ is K-stable, giving a new algebraic proof of a known result.","A smooth complete intersection of three quadrics in $\\mathbb{P}^7$ is K-stable if its net contains two quadrics with a common singular point."],"supporting_citations":[{"why":"Supplies the upgrade used in Proposition 4.14: GIT polystability of the branch hypersurface turns K-semistability of $(Z,\\epsilon Y)$ into K-polystability.","marker":"[LZ26, Theorem 10.3]"},{"why":"Equivariant K-stability under finite group actions; Theorem 4.6 identifies K-stability of the cyclic cover with K-stability of the branch pair $(Z,(1-1/m)Y)$.","marker":"[LZ22a]"},{"why":"Openness of K-stability in families, which lets one constructed K-stable member imply the same for the general member of each deformation family.","marker":"[BLX22]"},{"why":"GIT stability of smooth hypersurfaces and the Hilbert-Mumford criterion, used to establish GIT polystability of branch divisors.","marker":"[MFK94]"},{"why":"Finiteness of automorphism groups of smooth complete intersections, upgrading K-polystability to K-stability outside the exceptions.","marker":"[LW86]"},{"why":"A second source for finite automorphism groups of complete intersections, cited alongside [LW86].","marker":"[Ben13]"},{"why":"Finite automorphism groups for smooth well-formed weighted complete intersections, used for the weighted statements.","marker":"[PS19]"},{"why":"Reductivity of $\\operatorname{Aut}(Z)$ for K-polystable $Z$, needed so the GIT quotient used in Proposition 4.14 is defined.","marker":"[ABHLX20]"}],"fun_headline_variants":["All general Fano complete intersections K-polystable","K-stable except P^n and quadrics for general Fano complete intersections","Fano complete intersections: K-stability beyond P^n and quadrics","K-polystability for every general Fano complete intersection","Kähler–Einstein metrics on all general Fano complete intersections"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the cited theorem used in Proposition 4.14 applies, so GIT polystability of the branch hypersurface upgrades K-semistability of the pair to K-polystability; in the weighted setting the proof also assumes, without explicit verification, that the Fano index is an integer because the canonical class is Cartier.","fun_headline_variants_meta":{"raw":{"variants":["All general Fano complete intersections K-polystable","K-stable except P^n and quadrics for general Fano complete intersections","Fano complete intersections: K-stability beyond P^n and quadrics","K-polystability for every general Fano complete intersection","Kähler–Einstein metrics on all general Fano complete intersections"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001679,"raw_usage":{"total_tokens":6559,"prompt_tokens":752,"completion_tokens":5807,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":368,"completion_tokens_details":{"reasoning_tokens":5717}},"tokens_in":368,"tokens_out":5807,"duration_ms":41298,"temperature":1.0,"reasoning_tokens":5717,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T20:02:27.567209+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the explicit smooth weighted complete intersection with equations $f_i + x_{n+i}^{r_i/a_{n+i}}=0$ used in Theorems 6.1 and 7.1 and compute its stability threshold $\\delta$; a Fano variety is K-stable precisely when $\\delta>1$. If any admissible multidegree gives $\\delta\\le 1$ for this special member while the variety is not isomorphic to $\\mathbb{P}^n$ or a quadric, then the constructed member is not K-stable, the induction fails, and Theorem 1.3 is false.","supporting_citations":[],"review_version":1}