Algebraic and complex geometry

We prove the rational Hodge conjecture for every finite product of projective complex K3 surfaces: every rational Hodge class is algebraic. The factors may be distinct or repeated, with no restrictions on their Picard numbers, periods, or endomorphism fields.

We prove that the Kuga–Satake correspondence is algebraic for every smooth projective complex K3 surface. More precisely, the prescribed embedding of its transcendental cohomology into the cohomology of its Kuga–Satake abelian variety is induced by a rational algebraic cycle, with the fixed normalization and full even-Clifford target. The result also holds for isogenous Kuga–Satake models with the transported embedding.

We prove the rational Hodge conjecture in every codimension on every self-power of a complex abelian sixfold with an imaginary-quadratic action and a compatible polarization whose rational homological Hermitian form is hyperbolic of signature \((3,3)\). We also prove it in every codimension on every self-power of a complex abelian variety of dimension at most five admitting an imaginary-quadratic action. Both results include nonsimple varieties and special periods with additional endomorphisms. The proof uses the companion theorem on the rational Hodge conjecture for CM abelian varieties.

We prove the rational Hodge conjecture for complex abelian varieties with complex multiplication: every rational Hodge class on such a variety is a rational linear combination of algebraic cycle classes. As consequences, we obtain the generalized Hodge conjecture for CM abelian varieties, the Tate conjecture for abelian varieties over finite fields, and the Hodge standard conjecture for abelian varieties in arbitrary characteristic.

We prove the rational Hodge and generalized Hodge conjectures in every cohomological degree of every self-power of a projective complex K3 surface whose transcendental quadratic space, with its cup-product form, admits a rational isometric embedding in \(\mathbb U_{\mathbb Q}^{\oplus2}\perp\langle-1\rangle^4\). This includes every ample P-polarized K3 surface, including all Picard jumps, for \(P=\mathbb U\oplus D_8(-1)\oplus D_4(-1)\). On this locus we construct algebraic correspondences inducing every prescribed standard even-Clifford Kuga–Satake tensor. More generally, for any projective complex K3 surface, algebraicity of one exact standard even-Clifford Kuga–Satake tensor implies both conjectures for every self-power.

We prove the rational Hodge conjecture on every self-power of the Jacobian at each tensor Hodge-generic point of the full marked variation of a connected abelian cover of curves. This holds in every base genus and for every compatible branching pattern. For any CM abelian variety, the conclusion also holds for every self-power of its product with the Jacobian, on the same Hodge-generic locus. We also prove the conjecture on every self-power of a very general member of the full smooth labelled family of diagonal complete intersections cut out by at most two equations of a common degree, in every dimension and every degree at least two.

We prove that every rational Weil class on a split abelian eightfold of Weil type is algebraic. The result holds for every imaginary quadratic field, every compatible polarization type, and every member of the split family, including those with additional endomorphisms. Thus the full two-dimensional rational Weil space in codimension four is generated by algebraic cycle classes.

For a polarized K3 surface whose primitive cohomology has full orthogonal Hodge group, one algebraic correspondence inducing a nonzero map from that cohomology to the second cohomology of an abelian variety suffices to recover the prescribed full Kuga–Satake correspondence. We give an equivalent condition using holomorphic one-forms on a generically finite surface cover. If this input holds very generally in a polarized component, the prescribed correspondence is algebraic throughout that component, for every choice of standard data on the transcendental part. The existence of the initial correspondence remains a hypothesis.

We prove the ordinary Iitaka subadditivity conjecture for surjective projective morphisms with connected fibers between smooth connected projective varieties over algebraically closed fields of characteristic zero. If F is the geometric generic fiber of \(f:X\to Z\), then \(\kappa(X)\geq\kappa(F)+\kappa(Z)\).

We prove the reverse logarithmic Kodaira inequality for a surjective connected-fiber morphism \(f:(X,E)\to(Y,D)\) of smooth projective reduced simple-normal-crossing pairs, with \(\mathop{\mathrm{Supp}}\nolimits (f^*D)\subseteq\mathop{\mathrm{Supp}}\nolimits E\), such that X and every boundary stratum are smooth over \(Y\setminus\mathop{\mathrm{Supp}}\nolimits D\). Together with logarithmic subadditivity, the inequality gives additivity, including both negative-infinity cases. This resolves Popa's logarithmic additivity conjecture positively in the projective reduced-SNC, stratum-smooth setting.

We prove Campana's orbifold Iitaka subadditivity conjecture for rational simple normal crossing boundaries on compact manifolds in Fujiki class \(\mathcal C\), including coefficient one. Ordinary and logarithmic subadditivity follow.

We prove the logarithmic Iitaka–Viehweg inequality for projective surjective morphisms with connected fibers between smooth complex quasi-projective varieties whose base has nonnegative logarithmic Kodaira dimension. The variation measures the birational field of definition of the whole geometric generic fiber. This resolves Popa's logarithmic variation conjecture positively.

We prove the compact log-smooth Kähler case of b-semiampleness. Let \(f:Y\to X\) be a surjective holomorphic map with connected fibers between smooth compact connected Kähler manifolds, and let Δ be an effective rational divisor with simple normal crossing support and coefficients in \([0,1]\), with \(K_Y+\Delta\sim_{\mathbb Q}f^*L\) for \(L\in\mathop{\mathrm{Pic}}\nolimits (X)_{\mathbb Q}\). There is a smooth compact Kähler modification \(S\to X\) for which the threshold-moduli line satisfies \(M_{S_1}=\nu^*M_S\) in \(\mathop{\mathrm{Pic}}\nolimits (S_1)_{\mathbb Q}\) for every smooth compact Kähler modification \(\nu:S_1\to S\), and some positive multiple of M_S is represented by a holomorphic line bundle generated by global sections. Horizontal components of coefficient one are allowed; neither projectivity nor a Campana orbifold Iitaka hypothesis is assumed.

We prove a uniform index theorem for connected projective semi-log-canonical log Calabi–Yau pairs in every fixed dimension at least four over an algebraically closed field of characteristic zero. For boundary coefficients in a fixed finite rational set, a single multiple of the log canonical divisor is Cartier and linearly trivial. The multiple depends only on the dimension and coefficient set, not on the number of irreducible components. Together with the established theorem in dimensions at most three, this resolves the finite-rational-coefficient semi-log-canonical index conjecture.

Assuming orbifold Iitaka subadditivity, the specified pseudo-effective fourfold minimal model program, and abundance for nef fourfold adjoints of nonnegative Kodaira dimension, we prove the existence of good minimal models for globally strongly ℚ-factorial compact Kähler klt fourfold pairs with effective rational boundary and analytically pseudo-effective actual ℚ-Cartier adjoint. The additional step is nonvanishing. We prove it by fibration arguments and, in algebraic dimension zero, by singular metrics, holomorphic foliations, and extension from a reduced boundary. The projective abundance argument used in the proof is included in full.

For normal projective log canonical pairs over algebraically closed fields of characteristic zero, of fixed dimension d ≥ 5 and with effective boundary coefficients in a fixed finite rational set, we prove that one complete rounded pluricanonical system generates the full Iitaka field whenever the ℚ-Cartier log canonical divisor has nonnegative Kodaira dimension. Its degree depends only on the dimension and coefficient set. For the paper's normalized canonical bundle formulae over ℂ, we also bound the Cartier denominators of the moduli divisors on smooth projective determining models in terms of the dimension and coefficient set.

Assume logarithmic Iitaka subadditivity for surjective morphisms with connected fibers between smooth projective complex varieties with compatible reduced simple normal crossing boundaries. We prove log abundance for normal irreducible compact Kähler spaces in every dimension: for a log canonical pair \((X,\Delta)\) with effective rational boundary and \(K_X+\Delta\) ℚ-Cartier, analytic nefness of \(K_X+\Delta\) implies semiampleness.

We prove the effective Iitaka fibration conjecture in characteristic zero. For each dimension, one pluricanonical degree defines the Iitaka fibration of every smooth integral projective variety of that dimension and nonnegative Kodaira dimension over an algebraically closed field. The associated sections generate the full Iitaka function field.

We prove uniform denominator and effective-system bounds for log Calabi-Yau fibrations from projective log canonical complex pairs of dimension at most four onto positive-dimensional bases, with boundary coefficients in a fixed finite rational set. The bounds give a uniform trivializing degree and a uniform b-Cartier multiple of the moduli b-divisor. When the base divisor is big, a uniform complete rounded adjoint system has section ratios generating the full base function field.

Let \((H,\Theta)\) be a projective complex klt pair with effective rational boundary and nef ℚ-Cartier adjoint. On any projective log resolution, minimal semipositive metrics on the pulled-back adjoint exist and have zero Lelong numbers everywhere. On smooth projective complex varieties, we also prove an H¹-injectivity theorem for rational interior boundaries with simple-normal-crossing support when both endpoint bundles carry zero-Lelong semipositive metrics.

For a projective ℚ-factorial dlt pair with effective rational boundary over an algebraically closed field of characteristic zero, we prove that the adjoint has positive Iitaka dimension whenever a nonzero effective Cartier multiple is supported on the coefficient-one boundary and restricts to a semiample line bundle on its whole reduced support. This gives log abundance after nonvanishing in dimension at most four over ℂ.

We prove canonical nonvanishing for smooth connected projective complex fourfolds: if K_X is pseudo-effective, then \(H^0(X,mK_X)\ne0\) for some positive integer m.

We prove semiampleness of the actual ℚ-Cartier adjoint \(K_X+\Delta\) of a normal connected compact Kähler klt fourfold whenever it is analytically nef and some positive Cartier multiple has a nonzero section. The boundary is effective and rational; the fourfold need not be projective or ℚ-factorial. In Iitaka dimension zero, a positive Cartier multiple is the trivial holomorphic line bundle.

We prove the finite-rational-coefficient case of the effective log Iitaka conjecture in dimension four. For normal projective log canonical complex fourfolds with boundary coefficients in a fixed finite rational set and pseudo-effective rational Cartier adjoint, one uniform degree makes the complete rounded reflexive system nonempty and its section ratios generate the full Iitaka field. We also prove a uniform canonical index bound for projective klt complex fourfolds with rationally trivial canonical divisor.

Fix d ≥ 1 and t > 0. Let \((X,B)\) be an ordinary projective log canonical ℚ-pair of dimension d over a characteristic-zero field k, with X normal and integral, \(H^0(X,\mathcal O_X)=k\), B effective, and \(K_X+B\sim_{\mathbb Q}0\). We prove that every prime component S of B with coefficient at least t satisfies \([k_S:k]\leq N(d,t)\), where k_S is the relative algebraic closure of k in \(k(S)\). This also bounds the Stein degree of S over k, proving the contraction-to-a-point formulation of Birkar's Stein-degree conjecture for ordinary ℚ-pairs.

We prove the Campana–Peternell inequality \(\kappa(X)\geq\kappa(D)\) for a smooth connected projective complex variety X and an effective Cartier divisor D whenever \(m_0K_X-D\) is pseudo-effective for some positive integer m₀. For an algebraic fiber space \(f\colon X\to Y\) between smooth connected projective complex varieties, the hypothesis that \(m_0K_X-f^*H\) is pseudo-effective with H ample Cartier gives \(\kappa(X)=\kappa(F)+\dim Y\) for a very general smooth fiber F, as well as nonzero sections of \(mK_X-f^*H\) for all sufficiently large divisible m.

We prove the rational-boundary log abundance conjecture in every dimension over algebraically closed fields of characteristic zero: every nef ℚ-Cartier log canonical divisor on a projective log canonical pair with effective rational boundary is semiample. Over ℂ, the proof also establishes canonical nonvanishing for smooth projective varieties in every dimension.

We prove the numerical-dimension-one case of log abundance for threefolds over algebraically closed fields of characteristic p > 3. If \((X,B)\) is a projective log canonical threefold pair with effective rational boundary, and \(K_X+B\) is ℚ-Cartier, nef, and of numerical dimension one, then \(K_X+B\) is semiample. Neither terminality nor ℚ-factoriality of X is required.

Let X be a projective ℚ-factorial terminal threefold over an algebraically closed field of characteristic p > 3. We prove the numerical-dimension-one case of abundance: if K_X is nef with \(\nu(K_X)=1\), then K_X is semiample and \(\kappa(X,K_X)=1\).

Let X be a smooth connected compact Kähler manifold, let B be an effective rational simple normal crossing divisor with coefficients less than one, and let M be a nef rational holomorphic line bundle on X. If \(K_X+B\) is pseudo-effective and \(K_X+B+M\) is nef, we prove that its first Chern class in real Bott–Chern cohomology is represented by a semiample rational line bundle. This numerical statement allows a flat change of line bundle; it does not assert semiampleness of the original adjoint.

We prove the Generalised Abundance Conjecture: if \((X,B)\) is a projective klt ℚ-pair over an algebraically closed field of characteristic zero, \(K_X+B\) is pseudo-effective, M is a nef ℚ-Cartier divisor on X, and \(K_X+B+M\) is nef, then \(K_X+B+M\) is numerically equivalent to a semiample ℚ-Cartier divisor on X.

We resolve the numerical-dimension-one case of the minimal-model conjecture for smooth connected complex projective varieties of dimension at least three. If K_X is pseudo-effective and \(\kappa_\sigma(X,K_X)=1\), with κ_σ defined by section growth with a fixed ample twist, then X admits a projective ℚ-factorial terminal minimal model.

We resolve the existence form of the minimal-model conjecture for projective generalized log canonical ℚ-pairs over algebraically closed fields of characteristic zero. Such a pair admits a minimal model when its adjoint divisor is pseudo-effective, and a Mori fibre space otherwise, while keeping the nef ℚ-Cartier b-divisor fixed. Taking the nef part to be zero gives the corresponding result for ordinary log canonical ℚ-pairs.

We prove the ordinary-double-point gap conjecture for boundary-zero complex algebraic klt germs in every dimension: a singular n-dimensional germ has normalized volume at most \(2(n-1)^n\), with equality precisely at an analytic ordinary double point.

We prove that every singular complex algebraic klt fourfold germ with zero boundary has normalized volume at most 162, with equality precisely for an analytic ordinary double point. This resolves the ordinary-double-point volume-gap conjecture in dimension four.

We prove Fujita's freeness conjecture. If X is a smooth complex projective variety of dimension n and L is an ample line bundle, then \(K_X+mL\) is globally generated for every integer \(m\geq n+1\).

Let L be an ample line bundle on a smooth integral projective variety X over an algebraically closed field of positive characteristic, with \(\dim X=n\ge3\). We prove that there is a threshold \(r_0=r_0(X,L)\) such that for every integer \(r\ge r_0\), the ordinary multipoint Seshadri constant at the geometric generic tuple of r points equals the volume bound \((L^n/r)^{1/n}\). The same conclusion holds in dimension two. This establishes the positive-characteristic form of the qualitative Nagata–Biran–Szemberg assertion at geometric generic tuples.

Let L be an ample line bundle on a smooth integral complex projective variety X of dimension n ≥ 3. We prove that there is a threshold \(r_0=r_0(X,L)\) such that for every integer \(r\ge r_0\), the ordinary multipoint Seshadri constant at r very general points equals the volume bound \((L^n/r)^{1/n}\). This establishes the qualitative Nagata–Biran–Szemberg assertion in these dimensions.

We prove that a nonzero effective plane curve of degree d at r ≥ 10 very general complex points has total multiplicity strictly less than \(d\sqrt r\). The inequality holds simultaneously for all curves, including reducible and nonreduced curves, and establishes Nagata's conjecture in its strict, nonhomogeneous form.

We prove that, for every smooth integral complex projective surface S and every ample line bundle L, the multipoint Seshadri constant at r very general points equals \(\sqrt{L^2/r}\) for every sufficiently large integer r. This resolves positively the qualitative Nagata–Biran conjecture for surfaces.

We prove Bloch's conjecture for smooth connected projective complex surfaces with \(p_g=0\): the Albanese homomorphism on integral degree-zero zero-cycles is an isomorphism.

We prove the strong hyperkähler SYZ conjecture: every holomorphic line bundle with nonzero nef isotropic first Chern class on a compact irreducible holomorphic symplectic Kähler manifold is semiample.

We prove that the normal projective base of a projective Lagrangian fibration from a compact irreducible holomorphic symplectic Kähler manifold is projective space. This resolves the projective-space base conjecture for such fibrations in every dimension and deformation type.

We prove that every complex K3 surface X is Oka, resolving the K3 Oka conjecture. Equivalently, for every m ≥ 1, every holomorphic map to X from a neighborhood of a compact convex set in ℂ^m can be approximated uniformly on that set by entire maps \(\mathbb C^m\to X\).

We prove the P = W conjecture on the full rational cohomology of fixed-determinant, trace-free Higgs moduli spaces in composite rank and coprime degree, for smooth projective complex curves of genus at least two. Together with the established prime-rank cases, this gives the equality in every coprime rank.

We prove the equivariant cohomological Hikita conjecture for arbitrary finite quivers, including loops and multiple arrows. For any dimension and framing vectors, any commuting flavor torus, and a stability character whose semistable locus is stable and has free gauge action, the equivariant cohomology of the Nakajima variety is canonically isomorphic to the coordinate ring of the scheme-theoretic fixed locus of the stability cocharacter on the flavor-deformed Coulomb branch. This is an isomorphism of graded algebras over the common coefficient ring, retaining nilpotents and including the empty case.

We construct a smooth projective complex fourfold with large fundamental group whose universal cover is not Stein. The universal cover contains no positive-dimensional compact complex-analytic subvariety, so this disproves Shafarevich's holomorphic-convexity conjecture even under the large-fundamental-group hypothesis.

We construct a smooth connected projective complex surface whose universal cover is not holomorphically convex. This disproves the Shafarevich conjecture on holomorphic convexity in complex dimension two.

We construct an explicit integral complex affine fourfold X with \(X\times\mathbb A^1\cong\mathbb A^5\) but \(X\not\cong\mathbb A^4\). This gives a negative answer to Zariski's affine-space cancellation problem over ℂ in dimension four. The same construction disproves the Stable Coordinate Conjecture in ambient dimension five and yields smooth 𝔸³-fibrations over 𝔸¹ and 𝔸² that are not Zariski-locally trivial. These fibrations disprove the Dolgachev–Weisfeiler affine-fibration conjecture over these bases.

We construct a singular normal affine complex surface whose tangent sheaf is free of rank two. This disproves the Lipman–Zariski conjecture in characteristic zero.

We construct an explicit degree-five polynomial over \(\mathbf C\) that is not a coordinate in four variables but becomes one after adjoining a single variable. This gives a counterexample to the stable coordinate conjecture in four variables. Every fiber is isomorphic to affine three-space, yet its embedding in affine four-space is not rectifiable. Thus the example also disproves the Abhyankar–Sathaye embedding conjecture in ambient dimension four.

We construct an explicit counterexample to the Abhyankar–Sathaye conjecture: a noncoordinate polynomial in four complex variables whose zero fibre is affine three-space. Adjoining variables gives counterexamples in every ambient dimension at least four.

We give counterexamples to Griffiths' conjecture in rank two on the quadric surface \(\mathbb P^1\times\mathbb P^1\). We construct an explicit bundle G whose coordinatewise power pullbacks, tensored with \(\mathcal O(1,1)\), are ample for every positive power, but admit no smooth strictly Griffiths-positive Hermitian metric for all sufficiently large powers.

We prove that every compact connected Kähler manifold of positive complex dimension containing no nonconstant entire curve has ample canonical bundle and is projective. This resolves positively Kobayashi's canonical-ampleness conjecture in the smooth compact Kähler category.

We prove the two-summand form of Beauville's compatible splitting conjecture. If the tangent bundle of a compact connected Kähler manifold decomposes into two integrable holomorphic subbundles of positive rank, then its ordinary universal cover admits a product decomposition whose factor tangent bundles are the lifted specified summands.

We prove that both summands of every specified holomorphic splitting of the tangent bundle of a smooth rationally connected projective complex manifold into two positive-rank subbundles are integrable. This proves Höring's conjecture on rationally connected projective manifolds. Höring's product theorem then gives a product decomposition compatible with the specified splitting.

We disprove the Chow and rational-cohomological forms of Pixton's completeness conjecture for his original relation system. We construct a formal tautological class outside the original Pixton relation span whose image is zero in the Chow ring and in rational cohomology. The example has genus \(10^{60}\) and \(3\binom{10^{60}}3\) markings.

For every sufficiently large admissible Hassett discriminant, we prove that a very general cubic fourfold of that discriminant is irrational, although its Kuznetsov component is equivalent to the ordinary derived category of a projective K3 surface. The discriminant threshold is ineffective. This disproves Kuznetsov's rationality conjecture. The same cubics have associated untwisted polarized K3 surfaces in the Hodge-theoretic sense, so they also disprove the sufficiency direction of the associated-K3 rationality prediction.

Let X be a smooth projective complex threefold with trivial canonical bundle. We construct numerical Bridgeland stability conditions with the exact ordinary and square-root-Todd central charges at every sufficiently large volume. One volume threshold works on an open set of real twists and ample directions. The resulting stability conditions have the support property on the full numerical Grothendieck group and stable point sheaves. Separately, on every smooth projective complex threefold we prove a strong tilt inequality above a volume threshold uniform in the object and twist along a fixed polarization.

We prove Toda's normalized quintic Gepner conjecture. On every smooth complex quintic threefold, we construct a numerical Bridgeland stability condition for which tensoring by the hyperplane bundle, followed by the spherical twist at the structure sheaf, increases phase by 2/5.

Every sequence of generalized-canonical flips on compact Kähler fourfolds with rational boundary and fixed rational analytically nef b-divisor terminates when the small morphisms are projective. We prove this for normal globally Weil ℚ-factorial models with boundary coefficients less than one, allowing exceptional log discrepancies equal to one. No scaling rule or pseudo-effectivity assumption is required. The theorem applies to existing projective small diagrams with the specified opposite ample signs.

Every sequence of generalized log canonical flips on a normal irreducible globally Weil ℚ-factorial compact Kähler fourfold terminates, provided the flips are projective small diagrams with the stated opposite ample signs. The boundary is rational, and the nef b-divisor is fixed and represented by an analytically nef ℚ-Cartier divisor on a projective modification. No scaling rule or pseudo-effectivity assumption is required. The theorem concerns existing flip sequences; it does not assert the existence of all contractions or flips.

We prove that every maximal ordinary negative-ray program starting from a compact Kähler klt fourfold pair with effective rational boundary in the global Weil-divisor ℚ-factorial category terminates. It ends at a nef model when the adjoint is pseudo-effective and at a projective Mori fibre space otherwise.

We prove that every permitted minimal model program for a projective log canonical fourfold with rational boundary over an algebraically closed field of characteristic zero terminates. The result allows arbitrary negative extremal rays and mixed birational steps on the given models, without pseudo-effectivity or initial ℚ-factoriality.

We prove that a globally Weil-ℚ-factorial compact Kähler klt fourfold pair with effective rational boundary and canonical rational line bundle admits a finite ordinary minimal model program starting on the given pair. It ends at a nef model when the adjoint class is pseudo-effective and at a Mori fibre space otherwise.

Using the abelianity theorem for special compact Kähler manifolds in every dimension, we prove virtual abelianity of the entire orbifold fundamental group of the special order-two root orbifold associated with a pair \((X,\tfrac12D)\), where X is a smooth projective fourfold and D is a nonempty smooth connected divisor. We construct a special smooth projective eightfold whose fundamental group surjects onto the required group.

We give an independent proof that every complex linear representation of the ordinary fundamental group of a connected smooth special complex quasi-projective variety has virtually nilpotent image of class at most two. This conclusion was previously announced by Cao–Deng–Hacon–Păun. We also construct special open surfaces whose general quasi-Albanese fibres are not special.

We prove that the fundamental group of every special compact Kähler manifold is virtually abelian, resolving Campana's abelianity conjecture in all dimensions. In particular, every smooth compact connected Kähler manifold of Kodaira dimension zero has virtually abelian ordinary fundamental group.

Every nonempty connected semialgebraic bounded open subset of a complex affine variety admitting a properly discontinuous cocompact group of biholomorphisms is smooth and biholomorphic to a bounded symmetric domain. This answers the bounded-domain question of Kollár and Pardon affirmatively.

We prove the Kollár–Pardon conjecture: the universal cover of a connected normal projective complex variety is biholomorphic to a semialgebraic open subset of a projective variety if and only if it is a product of a bounded symmetric domain, a complex affine space, and a simply connected normal projective variety.

We give a negative answer to Zariski's multiplicity question in four complex variables. We construct two reduced convergent holomorphic function germs with isolated critical points and multiplicities four and five whose zero-set germs are ambiently homeomorphic.

We give a negative answer to the embedded Zariski multiplicity conjecture. We construct two reduced hypersurface germs that are ambiently homeomorphic but have multiplicities two and three. Both have isolated singularities and lie in a common complex affine space of dimension divisible by eight. Their defining function germs are also topologically right equivalent, and the same examples answer Arnold's corank problem negatively for ambient topological equivalence.

We prove the Global Spherical Shell conjecture: every connected minimal compact complex surface of class VII with positive second Betti number contains a global spherical shell. This is a holomorphically embedded neighborhood of the standard three-sphere in \(\mathbb C^2\setminus\{0\}\) whose complement is connected.

We resolve the contact-Fano homogeneity conjecture and the Riemannian LeBrun–Salamon conjecture positively. Every smooth connected complex projective contact Fano manifold of complex dimension at least three, with its given contact distribution, is contact-isomorphic to the adjoint variety of a simple complex Lie algebra. Consequently, every closed connected smooth positive quaternionic-Kähler manifold of real dimension \(4m\geq8\) is homothetic to a compact symmetric Wolf space.

We prove the generalized Mukai conjecture: every positive-dimensional smooth complex Fano manifold of dimension n, Picard number ρ, and pseudoindex ι satisfies \(\rho(\iota-1)\le n\). Equality holds precisely for the product of ρ copies of \(\mathbb P^{\iota-1}\).

We prove that every holomorphic one-parameter family of isolated hypersurface singularities in ℂ³ with constant Milnor number is topologically right-trivial. This gives a positive answer to the surface case of the μ-constant problem. The trivialization fixes the parameter and the origin section, and preserves the defining functions.

We prove that full ordinary descendant Virasoro constraints pass from a smooth projective complex base to the projectivization of any algebraic vector bundle of rank at least two. The bundle need not split and satisfies no positivity requirement. Assuming the full constraints on the base, the conclusion includes every genus, each individual integral curve class, and all cohomology insertions, including primitive and odd classes. The result also applies successively to towers of projective bundles.

We prove the Virasoro conjecture for the ordinary descendant Gromov–Witten theory of smooth complete intersections in projective space, in every genus and curve class and with arbitrary cohomology insertions. This includes primitive and odd cohomology classes, with no semisimplicity assumption.

We prove the Cartier-divisor conjecture of Birkar and Shokurov for rational boundaries in characteristic zero and for real boundaries over ℂ. For an ϵ-lc Fano type contraction with positive-dimensional base and nef negative log canonical class, a Cartier divisor through any prescribed base point can be chosen with pullback log canonical threshold bounded below in terms of the dimension and ϵ alone.

For fixed dimension d and positive rational ϵ, we prove that every ϵ-log-canonical Fano contraction over an algebraically closed field of characteristic zero admits, near each closed base point, a klt complement of index bounded only by d and ϵ. Over ℂ, we also obtain monotone klt complements for Fano type pairs with nef anti-log-canonical divisor and coefficients in a fixed finite rational set. This proves the finite-rational-coefficient form of Shokurov's bounded-klt-complement conjecture.

We prove that every smooth connected complex projective Fano sixfold with nef tangent bundle is rational homogeneous, resolving the Campana–Peternell conjecture in complex dimension six. As a consequence, a connected compact Kähler manifold X with nef holomorphic tangent bundle and \(\dim_{\mathbb C}X-\widetilde q(X)\leq6\) has ordinary universal cover \(F\times\mathbb C^{\widetilde q(X)}\), where F is a rational homogeneous manifold and \(\widetilde q(X)\) is the maximal irregularity of a connected finite étale cover.

From a smooth semipositive anticanonical metric on a smooth projective complex variety, we construct an unweighted integrable semipositive metric on a corrected anticanonical line of a smooth base, using a normal equidimensional toroidal model and full generic adjoint rank one at exponent zero. The explicit rational boundary correction pulls back to an exceptional divisor. On varieties without positive-degree holomorphic forms, two-metric transfer gives sections with prescribed boundary poles from a finite-volume log-anticanonical metric and a target line with bounded semipositive weights. In particular, a smooth projective complex variety with a smoothly semipositive anticanonical bundle and no such forms has a nonzero invariant section of a positive anticanonical multiple for every torus action with its natural linearization. On this class of varieties, generic section and adjoint-rank hypotheses give contractions of invariant fibrations that preserve both conditions unless an invariant global section already exists.

For a holomorphic action of a compact torus on a compact complex manifold, we prove that the invariant Euler characteristic of naturally linearized anticanonical powers is polynomial on a divisible progression, with its actual value at exponent zero. Independently, on a smooth projective variety with smoothly semipositive anticanonical bundle, we remove a fixed pseudoeffective error from an unbounded sequence of effective twists. These results convert invariant cohomology into twisted differential forms and prove anticanonical nonvanishing on smooth projective varieties with smoothly semipositive anticanonical bundle, by descending the forms before conversion.

Let Y be smooth projective and let C be an effective integral divisor. If \(-K_Y+C\) admits a metric with a global strict curvature lower bound for which the canonical section of C is locally square integrable, every pseudoeffective rational divisor becomes rationally effective after an effective correction supported on C. For a smooth projective X with smoothly semipositive \(L=-K_X\), sections of \(m_jL-P\) at unbounded positive exponents, with any fixed pseudoeffective Cartier error P, yield a section of a positive multiple of L.

On a smooth connected projective complex variety with smoothly semipositive anticanonical bundle, we prove that asymptotically zero valuation on the curvature-null face is attained by an effective rational anticanonical divisor. The fixed auxiliary line bundle is arbitrary.

Let X be a smooth projective complex variety with smoothly semipositive anticanonical bundle. For any torus linearization of that bundle, we show that invariant sections in unbounded degrees with one fixed negative pseudoeffective error yield an invariant section in a positive untwisted degree. Combining the natural invariant index with compact-monodromy structure, we deduce that every such X has a nonzero section of some positive anticanonical power.

Let \((W,D)\) be a projective ℚ-factorial klt pair with effective rational boundary. We prove that a positive Cartier multiple of \(-(K_W+D)\) has a nonzero section if this divisor is nef, its pullback to a resolution admits a semipositive metric with locally bounded weights, and the resolution has nonzero structure-sheaf Euler characteristic. Consequently a smooth rationally connected projective variety with smoothly semipositive anticanonical bundle has a nonzero naturally invariant anticanonical plurisection for every algebraic torus action.

Every smooth connected projective complex variety with smoothly semipositive anticanonical bundle has a nonzero section of some positive anticanonical power.

We prove the unramified quantum geometric Langlands equivalence for every connected simple complex algebraic group G and every level \(c\in\mathbb C\setminus\mathbb Q\), including non-real levels. For every smooth projective connected complex curve X, it identifies the full twisted D-module categories on \(\mathop{\mathrm{Bun}}\nolimits _G(X)\) and \(\mathop{\mathrm{Bun}}\nolimits _{G^\vee}(X)\) at the dual shifted levels c and \(-1/(rc)\), where r is the lacing number. The equivalence uses the given global forms and includes all connected components.