claims depot shelf
The Hodge Conjecture
Formal claims (Lean)
Stated claims
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The main theorem (Theorem 5.3) asserts that for the threefold X=(Y×E)/G, where Y is a Dwork quartic with Picard rank 19 and G=(Z/4)^2, the 2-torsion Brauer class α with B-field b/2 has per(α)=2 and ind(α)=8. Thus the unramified period-index conjecture, which would require ind|per^{dim−1}=4, fails. The proof establishes that no integral Hodge classes c,d satisfy b^2+bc+d≡0 mod 2, the condition that a Hodge-theoretic bound would impose if the index divided 4; combined with Matzri's bound ind|8, the index must be exactly 8.
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For every three-dimensional isolated cluster variety X_{a,b} the perverse filtration associated with the real-analytic map h(x,x',y,y',z)=(|x|^2-|x'|^2,|y|^2-|y'|^2,log|z|) (upper-middle perversity) equals the weight filtration: P_k H^*(X,Q)=W_{2k} H^*(X,Q)=W_{2k+1} H^*(X,Q), and the same identity holds for intersection cohomology. The equality is proved by matching both filtrations, graded piece by graded piece, with the kernels of restriction maps to the open set {|z|≠1}.
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At its center, the paper claims that a configuration of curves is governed by the node contribution ψ=½Σ nᵢ(dᵢ−2) (equivalently Σ(k_x−1) over singular points). This one integer reappears in the region count f=ψ+2 (all closed) or f=ψ+κ+1 (κ open curves), in the homology of the incidence complex H₁ ≅ Z^{ψ+κ+1}, in the deletion-restriction recurrence ψ(C)=ψ(C′)+v₀, and in the algebraic and Hodge setting as b₂=ψ for line arrangements and dim Gr^W_4 H²=ψ. The paper also claims that the absence of triple points is sufficient (but not necessary) for the OS-type algebra to factor through cohomology, and that for line arrangements the difference between the simplified model and H² is exactly Σ_{k_x≥4
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The central theorem states that G_dRB(A) = MT(A) for the four listed families. The proof avoids the classical algorithm for computing Mumford-Tate groups, which relies on the Deligne torus and is not available for dRB groups. Instead it uses a prior construction: G_dRB(A) is isogenous to G_m × G^h_dRB(A), where G^h_dRB(A) is a connected reductive subgroup of the Hodge group Hdg(A). The task becomes showing G^h_dRB(A) = Hdg(A). For simple CM fourfolds, the paper classifies the two-dimensional subtori of the unitary group U_E and rules them out using a Galois-theoretic criterion on character groups, the explicit comparison matrix for the associated Weil dRB structures, and the algebraic indepe
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The central new result, Corollary 7.8, states: if every primitive component in the Lefschetz decomposition of every class in H^{2i-1}_alg(X, Q) is itself algebraic — the condition the author names (Λ, X, i) — then for codimension-i cycles algebraically equivalent to zero, incidence equivalence coincides with Abel-Jacobi equivalence after tensoring with Q. The condition is shown to hold for codimensions 1, 2, and d, for abelian varieties because the Λ-operator is algebraic there, and for odd-dimensional complete intersections whose odd cohomology is concentrated in the middle degree. In the codimension-2 case the paper repairs a gap in an earlier published proof.
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The paper proves that, for each commutative, associative, or Lie mixed Hodge diagram A, there exist successively defined classes θ_k in PH^{k, 2−k}_{DB}(H^*(A)) for k ≥ 3 such that if all these classes are zero, then A is mixed Hodge formal (Theorem 4.17). For a simply connected compact Kähler manifold X, the first obstruction θ₃ maps, under a well-defined comparison map, to the invariant u(π₃) that measures the splitting of the extension 0 → H³(X) → π₃(X) → Ker(µ) → 0 of mixed Hodge structures (Corollary 5.7). This shows that non-vanishing of u(π₃) enforces non-mixed-Hodge formality even though X is classically formal. The paper also establishes that mixed Hodge formality does not descend a
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The central result is Theorem 1.3: if X is an n-dimensional compact normal analytic variety in Fujiki's class with klt singularities and nef anti-canonical divisor, then for any nef and big classes α1,...,αn−2 in Bott–Chern cohomology, the orbifold second Chern class satisfies ĉ2(X)·α1·...·αn−2 ≥ 0. For the canonical-divisor side, Theorem 1.1 gives Miyaoka's inequality (3ĉ2(X) − ĉ1(X)^2)·α1·...·αn−2 ≥ 0 when K_X is nef and X has canonical singularities (or quotient singularities in codimension 2 with rational singularities). The paper also proves a version for klt Kähler varieties with a small perturbing polarization, and derives Miyaoka–Yau type inequalities from these results.
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For a split spin group G=Spin(n) with n≥7, let Φ_n carry algebraic cycles on the classifying space BG to the W-invariant part of the Chow ring of a maximal torus. The q_i are the recursive generators in a standard presentation of that invariant ring, and the main theorem is a complete classification: q_i lies in Im Φ_n if and only if (n,i)=(10,3). The same statement for the special Clifford group Γ⁺(n) says f_i lies in the image of its characteristic map exactly when (n,i)=(7,2). Over C, naturality of cycle-class maps identifies membership in the Chow image with algebraicity modulo torsion of the corresponding class u_i in H^{2^{i+1}}(BG;Z)/torsion, so the classification transfers verbatim.
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The central discovery is that the second Chern class of a K3[3] hyperkähler sixfold is universal for Hodge-Weil classes: for a very general abelian fourfold B' of Weil type with discriminant one, there is an embedding of the Kummer variety K(B) of an isogenous B into a projective K3[3] manifold X, and the pull-back of c2(T_X) is a non-zero Hodge-Weil class in HW(B',K). The proof uses the Beauville-Bogomolov-Fujiki form: its associated class q∨_X in H^4(X) is proportional to c2(T_X), and its restriction defines a polarization class on a K3-type sub-Hodge structure T_1 ⊂ H^2(B'), which Proposition 1.11 identifies as a Hodge-Weil class. Since the K-action spans HW(B',K) from any non-trivial ele
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The central theorem is Theorem 12.1: for a projective variety X of dimension n ≥ 3 with isolated singularities of klt type, there are natural isomorphisms K_{-n+2}(X)_Q ≃ KH_{-n+2}(X)_Q ≃ H^{n-2}(Pic(X•))_Q ≃ gr^W_2 H^n(X,Q). In words, the next-to-bottom negative K-group is rationally controlled by the weight-two piece of middle cohomology, and can be computed as a quotient and kernel of restriction maps among Picard groups of the surface and curve strata in a log resolution. The paper also formulates systematic conjectures B, C, E, and G: weight bounds on mixed Hodge structure imply vanishing of rational homotopy K-theory; pre-m-rational singularities imply the same for ordinary K-theory; a
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If X is a projective hyperkähler manifold of K3^[n] type and its transcendental lattice T(X) embeds rationally into U^{3}, then there exists an abelian variety A together with an algebraic correspondence that induces a Hodge isometry T(X)⊗Q ≅ T(A)⊗Q.
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The generalized cross-ratio of four complementary linear subspaces in projective n-space serves as a single object that simultaneously encodes the Archimedean height pairing (over C), the non-Archimedean local height (via valuation), and the intersection degree (over a DVR). The key mechanism is that the cross-ratio, defined as an alternating product of determinantal pairings between subspaces and their annihilators, reduces to the classical cross-ratio when n=1, and the authors prove it coincides with the extension class of a mixed Hodge structure via an incidence correspondence between the Grassmannian and projective space. This identification, combined with the valuation-intersection dier
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The central object is the microlocal Bernstein–Sato polynomial eb_{(X,f)}(s), defined by applying Saito's partial microlocalization to the graph embedding of the intersection cohomology Hodge module IC_X^H along f. The core discovery is that this polynomial strictly refines the reduced b-function on singular ambient varieties—it can divide b_{(X,f)}(s)/(s+1) properly—and that its largest root, the minimal exponent eα(X,f), provides a sharp criterion for rational singularities: if X has rational singularities, then D = X ∩ V(f) has rational singularities if and only if eα(X,f) > 1. The paper also shows that eb_{(X,f)}(s) having no integer roots characterizes purity of the local cohomology H^1
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The central claim is Theorem 1.3: for λ ∈ S^1 \ {±1} and U ⊂ S a complex analytic subvariety, the restriction σ_Dol|_U is holomorphic if and only if λ·σ_Dol|_U is an isomonodromic deformation. The mechanism is first-order. Along a first-order disk, write the harmonic metric as h_0(id + t g + t̄ g^{⋆h0}); the obstruction to holomorphicity is the class [(η̄(θ^{⋆h0}),0)] in H^1(X_0,(End E^∨, ad(θ^{⋆h0}))). When this class vanishes, the gauge transformation G = id + t(λ−1)/2 g + t̄(λ̄−1)/2 g^{⋆h0} conjugates the rescaled flat connection D_t^{(λ)} = D_{h_0} + λθ + λ̄ θ^{⋆h0} to a constant connection, showing isomonodromy up to first order; conversely, gauge triviality of D_t^{(λ)} forces the obst
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For every quasi-projective family defined over a number field, the algebraic points of the base at which the associated variation of Hodge structures is Hodge-generic form a set that is analytically dense in the complex points of the base. Under a large-monodromy hypothesis the same density holds for the points that satisfy no extra period relations of degree at most any fixed δ and for the points that satisfy the Mumford–Tate conjecture.
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Theorem 1.2 states: if (V, F•, ∇, Q) is a polarized complex variation of Hodge structures on the central fiber and U ⊂ S is a closed complex analytic subvariety such that the real analytic isomonodromic family of Higgs bundles restricts to a holomorphic family on U, then U is contained in the non-abelian Noether–Lefschetz locus. Equivalently, holomorphicity of the isomonodromic Higgs family forces the deformed Higgs bundles to be graded, so they underlie polarized complex variations of Hodge structures. The proof is carried out through a truncated statement: a holomorphic order-n isomonodromic deformation of a graded stable Higgs bundle over an n-th order deformation of the central fiber is
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We produce refined index obstructions, generalizing recently constructed index obstructions due to de Jong and Perry, for topologically trivial Brauer classes on smooth and projective complex varieties. We show that our refined obstructions are more stringent than previous obstructions and, as a consequence, we produce more counterexamples to the integral Hodge conjecture. Throughout this work, we focus on algorithmic aspects of these obstructions and we illustrate many of these aspects through the concrete examples of complex abelian varieties.
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The paper's central claim is that the classical uniformizing lattice of a hyperbolic log-orbi curve is not an extra datum but a reconstruction. It constructs, for every such curve C, a canonical maximal principal PSL2-Higgs object (U_C, ϑ_C), characterized etale-locally as the pushout along SL2→PSL2 of the standard rank-two model with Θ^{⊗2}≃ω_C. Because the μ2 ambiguity in choosing Θ is killed by passage to PSL2, this object exists even when no square root of the orbifold canonical bundle exists globally. A Tannakian parahoric non-abelian Hodge–Riemann–Hilbert correspondence transports the object through de Rham to a Betti representation ρ_C: π_1^orb(C)→PSL2(C), and maximality forces ρ_C to
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The inductive structure of the boundary stratification verifies these conjectures in a surprisingly wide range of cases by relating Hodge structures and l-adic Galois representations on the cohomology to algebraic cycles in a manner compatible with the stratification.
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The central claim is that the Bockstein Δ_n of the cup-product class Θ_n = π_1^* α_1 ∪ ⋯ ∪ π_n^* β_n lies in H^{2n}(X_n, Z(n)) and maps nontrivially into the distinguished Enriques-Brauer component of the MV obstruction channel; once the Brauer-separation hypothesis is assumed, this implies that Δ_n is a non-algebraic 2-torsion integral Hodge class on the product of n Enriques surfaces.
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For every g,n with 2g-2+n>0, every degrees d,d' and every pair of Pagani–Tommasi stability conditions σ,σ', the compactified Jacobians J̄^d_{g,n}(σ) and J̄^{d'}_{g,n}(σ') have isomorphic rational cohomology as mixed Hodge structures; when both stability conditions are invariant under a product of symmetric groups S_λ the isomorphism may be chosen S_λ-equivariant.
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The classical semiregularity theorem of Buchweitz and Flenner continues to hold for equivariant noncommutative varieties. Specializing to twisted derived categories answers Markman’s question and streamlines a step in the proof of the Hodge conjecture for abelian fourfolds; in addition, for many finite group actions the invariant category is of geometric origin.
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The central claim is a canonical isomorphism of pure Hodge structures: the graded piece GrW_{k+w} of the weight filtration on the weighted cohomology W0H^k(X,E) of the reductive Borel–Serre compactification is isomorphic to the intersection cohomology IH^k(X^BB,E) of the Baily–Borel compactification. The map is constructed Hodge-theoretically as the natural morphism from the weight-zero weighted complex to the intersection complex; injectivity follows from weight filtrations. Surjectivity is proved analytically by showing that the weight-zero complex is quasi-isomorphic for large p to the sheaf of L^p-forms, and that every L^2-harmonic form (which represents intersection cohomology) is such
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The Hodge bundle on the moduli space of genus-g curves (g ≥ 2) is simple: it contains no non-trivial vector sub-bundles.
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Every torsion-free, graded-polarizable, admissible variation of mixed Hodge structures of type (0,0), (-1,0), (0,-1), (-1,-1) on the complex analytic space of a connected smooth finite-type C-scheme S arises, up to isogeny, from a 1-motive over S. Under suitable hypotheses on S and on the lattices and tori, the enriched Hodge realization is an equivalence between the category of 1-motives over S and the category of such variations; in general the equivalence holds only after isogeny. The global Mumford–Tate group of a 1-motive over S has neutral connected component equal to the Mumford–Tate group of its generic fibre.
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The central claim is that the deformation of (p,p)-classes along a family of compact Kähler manifolds is governed by explicit sections of Hodge bundles of the form H(e^{i_φ(t)}(I+T i_φ(t))^{-1}eη), where φ(t) is the Beltrami differential realizing the nearby complex structures, T = ∂*G∂ is the Green-operator contraction, and H is harmonic projection on the central fiber. These sections coincide with the period-matrix blocks and give, via an implicit-function argument, a real-analytic Hodge map H(σ,t) that sends any Kähler class σ on the central fiber to a (1,1)-class on X_t with a positive definite representative. From this the authors derive upper semicontinuity of Kähler cones, equality of
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The central claim, Theorem 1.8, is that the vanishing of H^p(Y,R^qπ_*(K_X⊗L⊗I(h))) depends only on the numerical dimension nd(T) of the positive current T on the base, not on the fiber geometry or the singularities of h, as long as the curvature of h dominates π^*T. The proof shows that the natural map φ^{p,q} from H^p(Y,R^qπ_*(K_X⊗L⊗I(h))) into H^{p+q}(X,K_X⊗L⊗I(h)) is the zero map. To do this, the paper constructs a sequence of quasi-plurisubharmonic approximants φ_k of the potential of T, with analytic singularities and controlled lower eigenvalue bounds, and proves (Lemma 4.1) that after pulling back to X, the multiplier ideal sheaf I(ψ' + π^*φ_k) equals I(ψ' + π^*φ) for all large k. Thi
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Theorem 1.3 is the central claim: under Assumption 1.1 — smooth total space near the boundary, trivial relative canonical bundle, and a singular fiber with only isolated singularities — if that fiber has a concentrated mixed Hodge spectrum, then the family is rigid. The proof runs by contradiction. Assuming non-rigidity produces a variation of Hodge structures over a product of curves that decomposes as a tensor product of two nontrivial factors. At the boundary point, this decomposition passes to the limiting mixed Hodge structure and clashes with the vanishing-cycle image: the isolated singularity's spectrum is concentrated (one Hodge filtration per weight), but the tensor product would ge
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The paper proves that the Fano eightfold F—defined as a connected component of the fixed locus of an antisymplectic involution on a variety deformation equivalent to Hilb^8(K3)—admits a semistable degeneration F→D whose central fibre is the transverse union Σ ∪ (Q4/Y), with intersection Q3/Y. Here Y is a smooth cubic fourfold, Σ is a flop of the blow-up of P^8 in a genus-eight K3 surface, and Q4/Y (resp. Q3/Y) is a fibration in smooth quadric fourfolds (resp. threefolds) over Y. From this degeneration the author derives the Hodge diamond of F: the non-zero entries are concentrated in even bidegrees, the largest being h^{4,4}=253, and h^{1,1}=1 (Picard rank one); odd cohomology vanishes.
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The paper's central claim is Conjecture 4.1: the canonical isomorphism between combinatorial and geometric intersection cohomology maps the span of combinatorial cycle classes [V(tau)]_comb onto the geometric Hodge classes. Since projective toric intersection cohomology is Hodge-Tate, this is equivalent to asserting that the classes [V(tau)]_comb span IH^{2k}_comb(Sigma, Q) for every k. The paper proves this for all projective toric varieties of dimension n <= 3, assuming the stated BBFK-BL compatibility, and unconditionally for simplicial fans. In the simplicial case the proof uses the presentation of rational cohomology as a quotient of the ring generated by ray variables, so the combinato
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Theorem: Let A0 be an abelian variety over a subfield k of C finitely generated over Q. If the integral Hodge conjecture fails for A = (A0)_C, then the integral Tate conjecture fails for A0 over some finite extension of k. If the paper is correct, any integral Hodge class on the complexification that is not algebraic produces, after descent via Deligne's absolute Hodge theory, an integral Galois-invariant l-adic class on a finite extension of k that is not in the image of the cycle class map.
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For a very general cubic fourfold X containing two planes P1, P2 that intersect (in a point or along a line), the K3 surfaces SP1 and SP2 obtained by resolving the double covers of the plane branched along the sextic discriminant curves are not isomorphic (Theorem 3.4); since each has no nontrivial Fourier–Mukai partner, they are not derived equivalent either (Corollary 3.5). When P1 and P2 meet along a line L, the paper constructs a smooth minimal surface FL of Kodaira dimension one with two double covers f_i: FL -> SPi, an elliptic fibration over L, and identical branch loci (Theorem 3.7). This correspondence yields a Hodge isometry T(SP1, αP1) ≃ T(X) ≃ (f_1^*T(SP1) ∩ f_2^*T(SP2), 1/2(.))
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Let X be a smooth Fano threefold, Z the parameter space of a family of degree d curves C_z, and Y ∈ |-K_X| a smooth anti-canonical divisor meeting each curve transversely. Intersection induces a morphism f: Z → Y^{(d)}, and Y^{(d)} is hyperkähler with symplectic form ψ. Theorem A asserts that f(Z) is a maximal-dimensional Lagrangian submanifold of Y^{(d)}, and that for every tangent vector ξ of Z and every first-order deformation class η ∈ H^1(Ω^2_X), the identity ψ(α(ξ), β(η)) = ⟨dAJ_X(ξ), η⟩ holds, where α = f_* and β is a naturally defined map from H^1(Ω^2_X) to the normal space of f(Z) in Y^{(d)}. In words: the infinitesimal Abel-Jacobi map is the symplectic pairing between how the inter
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The central claim is Theorem 3.4: for a special modular family Y→C of K3 surfaces with rational total space, the geometric Gross–Zagier conjecture holds in weight four and level Γ̂—meaning the higher Green's function is realized, up to a rational multiple, by the real regulator of a higher cycle. Concretely, at a CM point c (a fiber of Picard rank 20), the anti-invariant Green's function Ĝ_c equals ⟨r_W, η0⟩, the real-regulator pairing of a cycle W∈CH^2(Y\Y_Σ,1) with a canonical (1,1)-form η0. Such a cycle restricts to a CM fiber as divisors paired with algebraic functions, so the value at CM points lands in Q log Q. Corollary 3.6 applies this to the 23 mirror families in Table 1, five of wh
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The central claim is Theorem 4.2: let f,g:X→A^1 have f^{-1}(0)_red=g^{-1}(0)_red. Then in the Grothendieck group K_0(X_0), [ψ^u_f(M)]=[ψ^u_g(M)] for every M in D(X). The proof restricts to pure objects M supported on the complement X^* of the central fibre, and expresses the class of pΨ^u_f(M)=ψ^u_f(j_*M)[-1] as a sum over graded pieces of the weight filtration of the intermediate extension i_*j_{!*}M, with Tate twists. Since the filtration induced on i_*j_{!*}M is independent of the defining function, the class is equation-independent. The paper also proves Theorem 5.2: for proper f and pure M, the sequence H^k(X_0;i^*M) → H^k(X_0;ψ^u_f(M)) --N--> H^k(X_0;ψ^u_f(M))(-1) is exact for each k,
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The central claim is that the rational Hodge conjecture for smooth projective threefolds reduces to a statement about moving Hodge classes into families of curves: if a codimension-$2$ Hodge class is a specialization of complete-intersection curves, then the class is algebraic, i.e., a rational linear combination of algebraic cycles. The argument captures the Hodge class with a family of curves and shows it is algebraic. This reduction is then made effective: the paper gives sufficient deformation-theoretic criteria (normal-bundle cohomology vanishing and surjectivity of the restriction map) that are shown to hold for the class of a line on a general quintic threefold, and generically for Ca
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The central claim, as stated in the abstract, is that the general Gross-Zagier log-algebraicity conjecture is not an isolated analytic statement: it is a consequence of a standard conjecture in the theory of motives. The proof idea is to attach to the relevant modular data a motive from a moduli stack of elliptic curves with marked points, whose single-valued periods include a newly defined class of matrix-valued higher Green's functions for odd and even weights. The motive carries a biextension structure built from symmetric powers of the motives of elliptic curves. In the special case of level 1 and weight 4, the claim is that the motive of M_{1,3} is mixed Tate, which yields a completely
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The paper establishes that there exists a smooth projective variety X over the complex numbers with representable CH_0-group but no universal 0-cycle. Representability means that the Albanese morphism induces an isomorphism CH_0(X)^0 ≅ Alb(X), so all degree-zero zero-cycles are accounted for by the Albanese variety. A universal 0-cycle would be a degree-one zero-cycle defined in a way that survives arbitrary base change; its absence is detected by an integral cohomology obstruction imported from a known counterexample to the integral Hodge conjecture. The construction transfers that counterexample into the zero-cycle setting while preserving representability of CH_0, thereby answering the question in the negative.
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The paper claims: 'Starting with an arbitrary rational (1,1)-class α, we algorithmically build a one-parameter family of quartic K3's acquiring at most ten A1-nodes. On the central fibre X~0, the class α specializes to a Q-linear combination of the hyperplane class and the exceptional (-2)-curves... This yields an explicit, finite-step procedure that realizes any rational (1,1)-class by an algebraic cycle.' This is a constructive proof of the Hodge conjecture for K3 surfaces without Torelli-type results.
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The central discovery is that the Fourier–Sato transform, for sheaves whose cohomology is locally constant along $\mathbb{C}^*$-orbits, is canonically a vanishing-cycles construction, and that this construction can be lifted verbatim to mixed Hodge modules. Concretely, for $M\in\mathrm{DM}^b_{\mathrm{mon}}(V)$ the paper sets $M^\wedge=\varphi_t\,\gamma_*\,p^*M$ and proves Theorem 6.4, the canonical isomorphism $\mathrm{rat}(M^\wedge)\simeq\mathrm{rat}(M)^\wedge$, where $\mathrm{rat}$ is the conservative, perverse-exact rational realization functor. Theorem 6.5 adds Verdier duality compatibility: $\mathrm{D}(M^\wedge)\simeq(\mathrm{D}M)^\wedge[2r](r)$ for a bundle of rank $r$. The paper is explicit that the key sheaf-level ingredients are not new and are drawn from the cited literature; the contribution is the observation that this particular formula can serve as the definition of the mixed-Hodge-module transform, bypassing explicit filtrations.
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The paper's central claim is that the $p$-adic period conjectures for 1-motives over number fields with good reduction at $p$ hold at depth 2 for the period spaces built from the integration pairing $\varpi_M$ and from Fontaine's map $\varphi_M$, and at depth 1 for the period space built from rational-point logarithms $h_p(M)$. Concretely, Theorem 4 and Theorem 5 assert that every linear relation among $H^\varpi_p$-periods or $H^\varphi_p$-periods is a formal consequence of bilinearity and functoriality applied through a short exact sequence of 1-motives, and the same holds for $h_p$-periods already at depth 1. These results are established relative to the canonical $\mathbb{Q}$-structures $H^\varphi_p(M)$, $H^\varpi_p(M)$, and $h_p(M)$ constructed in Section 4.2, which the authors obtain by pulling back rational points of the formal $p$-divisible group along the integration and Fontaine maps. The proof routes through a $p$-adic analytic subgroup theorem for 1-motives that controls the annihilator of any period class by the de Rham realization of a subquotient motive, mirroring Wüstholz's classical theorem.
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The paper's central claim is the Constructive Hodge Degeneration Principle. For a rational class $\alpha \in H^{p,p}(X_t,\mathbb{Q})$ on a smooth fibre, if $N(\alpha) \neq 0$ and the class $\delta = N(\alpha)$ is algebraic in the limiting mixed Hodge structure, then $\alpha$ is a limit of algebraic classes as $t \to 0$. The paper further claims that $\delta$ is of geometric origin: it lies in the image of the specialization map from the exceptional cohomology of a semistable model of the central fibre, so the algebraicity of $\delta$ is inherited from an actual algebraic component. This is packaged as Conjecture 3.1, that every rational Hodge class is degenerationally realizable through families with suitable vanishing cycles or exceptional divisors. The supporting computation is a quartic K3 family where one ordinary double point produces a vanishing cycle, the monodromy logarithm has rank one, and the limiting Picard number rises from 1 to 2, with the new class represented by an exceptional curve.
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The central claim, on the paper's own terms, is that every rational cohomology class of type $(k,k)$ on a smooth projective complex variety is algebraic, and the proof vehicle is a spanning statement: the projected Gauss-Manin derivatives of holomorphic forms generate the whole orthogonal complement of $H^{k,k}(X)$ in $H^{2k}(X,\mathbb{C})$. The Hermitian spectral fingerprint vanishes exactly on the $(k,k)$ part, so rational $(k,k)$ classes are marked by zero fingerprint. The paper asserts this dichotomy is stable across realizations, yielding absolute Hodge behavior, and then invokes deep results in arithmetic geometry to conclude algebraicity. The conclusion is stated plainly: the Hodge conjecture is resolved within this framework.
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The central claim is that the Hodge Conjecture holds unconditionally for smooth projective complete intersections, proved through derived categories, and that the general case can be reduced to the statement that limits of algebraic cycles in a flat degeneration remain algebraic. The author presents this as a deformation-theoretic bridge: any smooth projective variety sits in a flat family with complete intersection fibers, so the conjecture for the generic fiber transfers to the special fiber provided the limiting cycle is algebraic.
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The central discovery is Theorem A: for $X=\Sigma_{g,r}$ with $g\ge0$, $r\ge1$, the signed Hilbert series $f_X(x,y,u,t)=\sum_{p,q,i,n}(-1)^i h^{p,q;i}(\operatorname{Conf}_n(X))\,x^p y^q u^i t^n$ equals $$\frac{1}{(1+xyut)^{r-1}}\cdot\frac{\Phi_g\{(1-$xyz^{2}$)(1-xz)^g(1-yz)^g\}}{(1-t)(1-$x^{2}$$yu^{2}$$t^{2}$)^g(1-$xy^{2}$$u^{2}$$t^{2}$)^g},$$ where $\Phi_g$ is the $\mathbb{Z}[x,y]$-linear map sending $z^j$ to $u^j t^j$ for $0\le j\le g$ and to $u^{j-1}t^j$ for $g+2\le j\le 2g+2$; the $z^{g+1}$ coefficient automatically vanishes and need not be specified. Equality means the coefficient of $x^p y^q u^i t^n$ is exactly $(-1)^i h^{p,q;i}(\operatorname{Conf}_n(X))$, so the formula determines every mixed Hodge number. The genuinely new content is the full range $g\ge2$.
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Theorem 1.3: gr^W_13 H^*_c(M_{g,n}) vanishes outside degrees k1=3g+n-2 and k2=3g+n-3, with Sn-equivariant isomorphisms gr^W_13 H^{k1} = Z_{g,n} ⊗ LS12 and gr^W_13 H^{k2} = W_{g,n} ⊗ LS12 for (g,n) = (2,11), (4,8), (6,5), (8,2), where Z and W are the listed representations.
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The central claim is that the Miyaoka–Yau inequality $(2(n+1)\widehat{c}_2(X)-n\widehat{c}_1(X)^2)\cdot\langle c_1(K_X)^{n-2}\rangle\ge0$ remains true for $n$-dimensional projective klt varieties with big $K_X$, and analogously with $-K_X$ for $K$-semistable klt varieties. The proof's engine is a Bogomolov–Gieseker inequality: any rank $r$ reflexive Higgs sheaf that is semistable with respect to the movable class $\langle\alpha^{n-1}\rangle$ of a big class $\alpha$ must satisfy $(2r\widehat{c}_2(E)-(r-1)\widehat{c}_1(E)^2)\cdot\langle\alpha^{n-2}\rangle\ge0$. Applying this to the Higgs sheaf $E=\Omega^{[1]}_X\oplus\mathcal{O}_X$ with Higgs field $(a,b)\mapsto(0,a)$, whose stability follows from the semistability of $\Omega^{[1]}_X$, yields the inequality. The formulation via non-pluripolar products is essential because big classes can have negative top self-intersections; the product $\langle\alpha^{n-2}\rangle$ discards the divisorial negative part and lands in homology, so the paper defines the needed intersection numbers through an orbifold modification.
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The central claim is a structural translation: the Hodge conjecture would be reduced to showing that algebraic Hodge cycles have algebraic spectral fingerprints. The paper presents this as a hypothesis rather than a proof, and its evidence is a simplified torus model together with a conceptual outline for K3 surfaces. In the proposed picture, the characteristic form $\eta_Z$ of a Hodge cycle $Z$ is expanded as $\eta_Z = \sum_k a_k \phi_k$ in the eigenbasis of the Laplace-Beltrami operator on $Z$, and the arithmetic nature of the coefficients $a_k$ becomes the new invariant proposed to detect algebraicity. For families of varieties, $\eta_Z$ is replaced by a dynamic characteristic form varying through moduli space, so the $a_k$ become functions on the moduli space that are expected to be algebraic when the family of cycles is algebraic.
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The paper establishes a structure theorem: for a klt pair (X, Δ) with X a compact Kähler manifold and -(K_X + Δ) nef, there exists a locally constant fibration f: X → Y such that the general fiber F is a rationally connected manifold and Y is a compact Kähler manifold with c1(Y) = 0. The proof proceeds by constructing, along a resolution of the MRC fibration, a direct image sheaf that is pseudo-effective with vanishing first Chern class, then applying the new flatness criterion to conclude it is locally free and numerically flat. The flatness of this sheaf yields a flat connection that descends to a splitting of the tangent bundle of X into the fibration direction and a numerically trivial part; Ehresmann's theorem then upgrades the fibration to a locally constant one. The same circle of ideas yields the Beauville–Bogomolov–Yau decomposition for klt Kähler pairs, the Hacon–McKernan inequalities in the Kähler setting, and the characterization of equality c2 = 0 in terms of tori and P1-bundles over tori.
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Theorem 6.1: 'Let X be any 5-dimensional Calabi-Yau manifold with Hodge number h^{1,1}(X)=56. On this manifold, there exists a constraint-coupled Spencer theoretical framework with E7 as the structure group that rigorously satisfies the three preconditions of the Spencer-Hodge verification criterion. Therefore, for any manifold X belonging to this category, its (1,1)-type Hodge conjecture holds.' If correct, every rational (1,1) Hodge class on such X is algebraic, proven without constructing cycles.
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The paper's central discovery is Theorem 3.4(4): if $X$ is an irreducible projective variety over $\mathbb{C}$ with at worst isolated singularities, then the extended cycle class map $\mathrm{cl}^1: A^1_{\mathrm{ext}}(X) \to H^2_{\mathrm{Hdg}}(X) = \mathrm{Gr}^W_2 H^2(X,\mathbb{Q}) \cap H^{1,1}$ is surjective. Here $A^1_{\mathrm{ext}}(X)$ is the inverse limit over all resolutions $\phi: \widetilde X_\phi \to X$ of the subgroup $A^1_{\mathrm{ext}}(\phi)$ of $A^1(\widetilde X_\phi)$ consisting of classes whose cycle class restricts to zero on the exceptional divisor $E_\phi$. The same theorem also shows that $A^p(X)$ embeds into $A^p_{\mathrm{ext}}(X)$, that the BGS map extends to the extended group, and that for $p=1$ the extended group is functorial under morphisms preserving smooth loci. Along the way, the paper proves that the usual BGS map is already surjective for normal projective surfaces with rational singularities, and gives an explicit contracted-cubic example where the usual map is not surjective but the extended map is.
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The central claim is Theorem 5.4: for a projective algebraic manifold $X$, if a Lie group $G$ and associated structures satisfy Premise 5.1, Premise 5.2, and Premise 5.3, then $H^{p,p}(X)\cap H^{2p}(X,\mathbb{Q})=H^{2p}_{\mathrm{alg}}(X,\mathbb{Q})$. The core technical result behind it is Theorem 6.4, which asserts that the space $H^{2p}_{\mathrm{constraint}}(X)$ generated by constraint-coupled Spencer kernels coincides exactly with $H^{2p}_{\mathrm{alg}}(X,\mathbb{Q})$ under the same premises. A class is Spencer-Hodge when it admits a nonzero certification tensor $s$ in $\mathrm{Sym}^{2p}(\mathfrak{h})$ with $s\in K^{2p}_\lambda$ and $\delta_\mathfrak{g}^\lambda(\omega\otimes s)=0$; from these conditions the paper derives flatness under the Spencer-Gauss-Manin connection, and Premise 5.3 converts that flatness into algebraicity. The K3 chapter then computes the $\mathfrak{su}(2)$ kernels explicitly and claims that for rank-one Picard K3 surfaces the constraint direction of the kernel maps precisely onto the one-dimensional algebraic class space.
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On the paper's own terms, the discovery explained is that polytopes in projective space and planar polypols are positive geometries in both senses considered, with explicit formulas for the canonical form. For a simple polytope, the canonical form is the sum over vertices of determinants divided by products of facet factors; for any polytope it is the dual volume function $f(y)=\operatorname{vol}((P-y)^\circ)$ continued meromorphically. For a nodal quasi-regular rational polypol, the canonical form is $\omega(P)=\alpha\cdot \frac{\mathrm{adj}_P}{f_1\cdots f_r}(x\,dy\wedge dz-y\,dx\wedge dz+z\,dx\wedge dy)$, up to the constant $\alpha$ fixed by the iterated residues at the vertices. The same form is shown to be the image of the relative class $[P]$ under the Hodge-theoretic map of [9], so the two definitions agree on these families.
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The central claim is the Main Theorem: given representations (γ_i, ω_i) of 1-periods α_i, Algorithm 2.0.1 computes a Q-basis for the space rel_Q(α_1,...,α_k) of Q-linear relations. In particular, it decides whether a given 1-period is transcendental and whether two 1-periods are equal. The proof makes the dimension estimate for saturated 1-motives effective: it constructs the mixed Hodge structure H1(C\D,E) of punctured marked curves, builds the associated Jacobian motive, supersaturates it into a Baker motive plus a saturated motive, and reads off period relations from endomorphisms and trivial relations. This gives the first effective resolution of the period-equality problem for 1-periods.
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Theorem A (Theorem 3.6) states: for a smooth projective variety $X$ of dimension $n$ and a supercritical collection $\mathbf{L} = (L_1,\ldots,L_{n-2})$ of free divisor classes, $\ker \mathbf{L} = \operatorname{span}_{\mathbb{R}}\{[D] : D \in \operatorname{Prime}(X),\ \mathbf{L}\cdot[D]=0\}$. In particular, $\mathbf{L}$ is a hard Lefschetz class if and only if $\mathbf{L}\cdot[D]\neq 0$ for every prime divisor $D$. The same conclusion holds when the classes are only semiample (Remark 1.2). Corollary A derives the equality case of the Alexandrov-Fenchel inequality for a supercritical collection of rational convex polytopes: equality holds exactly when the two polytopes share their supporting hyperplanes in all active normal directions. Corollary B characterizes equality in the Khovanskii-Teissier inequality for two free divisor classes by writing their difference as a combination of prime divisors annihilated by the remaining intersection numbers. The proof is inductive, with a threefold base case and a restriction-to-a-general-hypersurface step.
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The central claim, stated as Theorem 1.9, is that uniqueness of tangent currents follows from an integrability condition on the generalized Lelong numbers. For a positive closed (p,p)-current T on a complex manifold X, a Kähler submanifold V of dimension l, and a relatively compact piecewise $C^{2}$ domain B⊂V, assume T=T^+-T^- near B with T^± in the class CL_{p;1,1}(B), meaning each is weakly approximable by smooth closed forms with uniformly bounded $C^{1}$ norms near the boundary of B. If for every j between max(0,l-p) and min(l,k-p) either the square-root integral of the difference between ν_j at radii r and r/2 or the plain integral of the difference between ν_j(r) and the limit ν_j is finite, then T admits a unique tangent current along B. The paper also establishes a finite-cover version with conditions on local mass indicators and proves that for the current of integration over a pure-codimension-p analytic set S the average means converge with rate O(r^ρ), so uniqueness holds there and the convergence speed is quantified.
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The core thesis is that bi-algebraic geometry supplies the right definition of special in both Hodge theory and Teichmüller theory. For $\mathcal{M}_g$ and the Hodge bundle $\Omega\mathcal{M}_g(\kappa)$, a subvariety is bi-algebraic if its lift to the universal cover is locally defined by algebraic equations in period coordinates; the claim is that CM Jacobians, sub-Shimura varieties generically contained in the Torelli locus, Teichmüller curves, and $\mathrm{GL}_2(\mathbb{R})^+$-orbit closures are all bi-algebraic, and that the genuinely special ones are atypical intersections. The survey proves or cites that these objects satisfy the same dimensional-excess inequality, and are therefore governed by the Zilber-Pink conjecture; it predicts what ZP implies for real-multiplication loci, and it states a new functional-transcendence result for meromorphic differentials (Theorem 9.4) that would extend the framework to strata with poles.
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The central claim is Theorem 1.2.12: for an almost orthogonal oriented $(-1)$-shifted symplectic stack $X$ satisfying assumptions (i)-(iv), there is an isomorphism of monodromic mixed Hodge complexes on the good moduli space $\underline{X}$, $$\bigoplus_{(F,\$\alpha$)\in \mathrm{Faces}^{\mathrm{sp}}(X)} \left(g_{\$\alpha$,*} \mathrm{BPS}_{X_\$\alpha$} \otimes H^*(B\mathbb{G}$_m^{{\dim F}}$)^{\mathrm{vir}}\right)^{\mathrm{Aut}(\$\alpha$)} \simeq p_* \phi_X,$$ where $\phi_X$ is the Donaldson-Thomas vanishing-cycle sheaf, $\mathrm{BPS}_{X_\alpha}$ is the zeroth perverse cohomology of the localized DT sheaf (the BPS sheaf), and the sum runs over special faces of the component lattice. The smooth-stack analogue replaces BPS sheaves by intersection complexes, expressing $p_* \mathrm{IC}_U$ as the same kind of direct sum. The paper argues that this decomposition is induced by the cohomological Hall induction, and that the almost-orthogonal condition makes the relevant pushforwards small, so the perverse-degenerate CoHI is symmetric and an isomorphism.
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The central claim is Theorem 1.5.1: the Hodge-Weil classes of polarized abelian sixfolds of Weil type with complex multiplication by $K$ and discriminant $-1$ are algebraic, for every imaginary quadratic field $K$. The proof shows that, starting from a $Q(\sqrt{-d})$-secant plane $P$ spanned by the Chern characters of two ideal sheaves on a genus-3 Jacobian $X$, Orlov's derived equivalence $\Phi:D^b(X\times X)\to D^b(X\times\widehat{X})$ produces an object whose dual $E$ is a simple reflexive sheaf of rank $8d$. The characteristic class $\kappa(E)$ is $\mathrm{Spin}(V)_P$-invariant and therefore remains of Hodge type under every deformation of $X\times\widehat{X}$ as a polarized abelian sixfold of Weil type. The paper proves a semiregularity theorem for $\mu_r$-twisted sheaves on abelian varieties, applies it to a twisted reflexive sheaf descended from $E$, and concludes that $E$ deforms locally over the 9-dimensional moduli space. The deformed sheaves carry the algebraic cycles whose classes span the Hodge-Weil subspace, and Corollary 1.6.1 then states that the Hodge conjecture holds for all abelian fourfolds.
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The central claim around which the notes are organized is Theorem 2.6, the geometric Zilber-Pink theorem. Let $(S,\mathbb{V})$ be a polarized integral variation of Hodge structures on a smooth quasi-projective base, with period map $\Phi$ into a Hodge variety, and let $Z$ be an irreducible component of the Zariski closure of the union of atypical special subvarieties of positive period dimension. The theorem asserts that either $Z$ is itself a maximal atypical special subvariety, or the adjoint Mumford-Tate group of $Z$ decomposes as a nontrivial product $H^{\mathrm{ad}}_Z \times L_Z$ and $Z$ contains a Zariski-dense set of fibers of the $L_Z$-factor of $\Phi$, each of which is an atypical weakly special subvariety; furthermore $Z$ is Hodge generic in a monodromically typical (hence typical) special subvariety. In words, the atypical Hodge locus is finitely generated up to families of atypical fibers forced by product decompositions of the ambient group. This is the geometric form of the Zilber-Pink finiteness expectation: atypical intersections are not arbitrary, they are organized by the structure of the Mumford-Tate group.
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The central claim is the theorem that for every genus zero pair (X,Y) — a compact complex variety X of dimension n together with a closed subvariety Y such that X\Y is smooth and H_n(X,Y) has vanishing Hodge numbers $h^{{-p,0}}$ for all p>0 — the iterated residue map R: $\Omega$^n_log(X\Y) -> gr^W_0 H_n(X,Y)_C is an isomorphism. The paper defines the canonical map can: H_n(X,Y) -> $\Omega$^n_log(X\Y) by composing the quotient to gr^W_0 with the inverse of R, and calls can($\sigma$)=omega_sigma the canonical form of $\sigma$. The map is shown to be linear, compatible with residues of boundaries via Res(omega_sigma)=omega_{d $\sigma$}, invariant under modifications, functorial for pushforwards and pullbacks, and multiplicative under products of pairs. The genus zero condition is equivalent to uniqueness of canonical forms, while existence is equivalent to the non-vanishing of the combinatorial rank $h^{{0,0}}$. In particular, every logarithmic n-form on X\Y is the canonical form of some relative cycle with complex coefficients.
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The paper's central claim is Theorem 1.0.1. Let $X$ be a smooth projective $n$-fold, $(Y,\Upsilon)$ a polarized smooth projective variety, and $d\in N^1(X)$ acyclic and separating $r(d)$ points. Then there is a first-quadrant spectral sequence of Galois representations and mixed Hodge structures $E_1^{p,q}\Rightarrow H_c^{p+q}(\operatorname{Mor}_d(X,Y);\mathbb{Q})$, and for $p\le r(d)+1$ the page is $$$E_1^{{p,*}}$ = \big((H^*(\mathring{X}^p;\mathbb{Q})\otimes \operatorname{sgn}_{S_p})^{S_p}\big)\otimes H^*(\operatorname{Pic}^d(X);\mathbb{Q})\otimes H_c^*(Y(D_{p-1});\mathbb{Q}),$$ where $\mathring{X}^p$ is $X^p$ with all diagonals removed and $Y(D_{p-1})$ are auxiliary schemes defined in §2.5.4. When $Y=\mathbb{P}^N$, the sequence collapses at $E_2$ for $0\le p\le r(d)+1$ and $\dim\operatorname{Mor}_d(X,Y)-r(d)\le q\le \dim\operatorname{Mor}_d(X,Y)$, yielding homological stability in that range. If $\delta=d-c_1(K_X)$ is ample, the stable bound satisfies $$r(d)=\left\lfloor\min_{[W]\in CH^k(X),\,1\le k\le n}\left(2(\delta^k.[W])^{1/k}-\frac{n}{2}+\frac{n-1}{2}\right)\right\rfloor-1.$$ Thus the cohomology of algebraic map spaces is controlled by the configuration spaces of the domain, the Picard variety, and auxiliary moduli of $Y$, with arithmetic and Hodge structure intact.
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The authors define the Lichtenbaum-Quillen dimension of a smooth complex variety as the threshold degree above which the natural comparison map from algebraic K-theory to topological K-theory with finite coefficients becomes an isomorphism. They establish that this dimension is compatible with semi-orthogonal decompositions, which allows them to compute it for Kuznetsov components of Fano varieties and to obtain new cases of the integral Hodge conjecture by applying homological projective duality.
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We prove the generalised Hodge conjecture, the (motivated) Mumford-Tate conjecture, and the generalised Tate conjecture for all GM varieties. We compute all integral Chow groups of GM varieties, except for the only two infinite-dimensional cases (1-cycles on GM fourfolds and 2-cycles on GM sixfolds). We prove that if two GM varieties are generalised partners or generalised duals, their rational Chow motives in middle degree are isomorphic.