Abstract Let π : X → Δ m \pi\colon X\to\Delta^{m} be a proper smooth Kähler morphism from a complex manifold 𝑋 to the unit polydisc Δ m \Delta^{m} . Suppose the fibers over the complement of a proper analytic subset are biholomorphic to a fixed projective manifold 𝑆. If the canonical line bundle of 𝑆 is semiample, then we show that all fibers over Δ m \Delta^{m} are biholomorphic to 𝑆. As an application, we obtain that, for smooth families where the canonical line bundle of the generic fiber is semiample, birational isotriviality is equivalent to isotriviality. Moreover, we establish a new Parshin–Arakelov type isotriviality criterion.
Abstract For n ≤ 6 n\leq 6 , we compute the integral Chow ring of every modular compactification of M 1 , n \mathcal{M}_{1,n} parametrising only Gorenstein curves with smooth, distinct markings. These include the Deligne–Mumford, Schubert, and Smyth compactifications, and many more. They can all be excised from the stack of log-canonically polarised Gorenstein curves. The Chow ring of the latter admits a simple, combinatorial description, which we compute by patching along a natural stratification by core level . We deduce that all these modular compactifications satisfy the Chow–Künneth generation property, that the cycle class map is an isomorphism, and for n = 4 n=4 , we study whether Getzler’s relation holds integrally and for other compactifications.
Abstract In this article, we establish an L 2 L^{2} extension theorem for Nakano semi-positive singular Hermitian metrics on holomorphic vector bundles, and the strong openness and stability properties for the multiplier submodule sheaves associated to such singular metrics. We solve affirmatively a question of Lempert on the preservation of Nakano semi-positivity under limit of an increasing sequence of metrics based on Deng–Ning–Wang–Zhou’s characterization of Nakano positivity.
We state a correction of an error in the n = 4 n=4 case of Theorem 3 of the paper [L. Esser, L. Ji and J. Moraga, Symmetries of Fano varieties, J. Reine Angew. Math. 819 2025, 89-133].
Abstract We conjecture that a natural twisted derived category of any hyper-Kähler variety of K 3 [ n ] K3^{[n]} -type is controlled by its Markman–Mukai lattice. We prove the conjecture under numerical constraints, and our proof relies on Markman’s projectively hyperholomorphic bundle and a recently proven twisted version of the D-equivalence conjecture. In particular, we prove a conjecture of Huybrechts, stating that any two fine moduli spaces of stable sheaves on a K 3 K3 surface are derived equivalent if they are of the same dimension.
Abstract We compute the S n S_{n} -equivariant topological Euler characteristic of the Kontsevich moduli space M ̄ 1 , n ( P r , d ) \overline{\mathcal{M}}_{1,n}(\mathbb{P}^{r},d) . Letting M ̄ 1 , n nrt ( P r , d ) ⊂ M ̄ 1 , n ( P r , d ) \overline{\mathcal{M}}_{1,n}^{\mathrm{nrt}}(\mathbb{P}^{r},d)\subset\overline{\mathcal{M}}_{1,n}(\mathbb{P}^{r},d) denote the subspace of maps from curves without rational tails, we solve for the motive of M ̄ 1 , n ( P r , d ) \overline{\mathcal{M}}_{1,n}(\mathbb{P}^{r},d) in terms of M ̄ 1 , n nrt ( P r , d ) \overline{\mathcal{M}}_{1,n}^{\mathrm{nrt}}(\mathbb{P}^{r},d) and plethysm with a genus-zero contribution determined by Getzler and Pandharipande. Fixing a generic C ⋆ \mathbb{C}^{\star} -action on P r \mathbb{P}^{r} , we derive a closed formula for the Euler characteristic of M ̄ 1 , n nrt ( P r , d ) C ⋆ \overline{\mathcal{M}}_{1,n}^{\mathrm{nrt}}(\mathbb{P}^{r},d)^{\mathbb{C}^{\star}} as an S n S_{n} -equivariant virtual mixed Hodge structure, which leads to our main formula for the Euler characteristic of M ̄ 1 , n ( P r , d ) \overline{\mathcal{M}}_{1,n}(\mathbb{P}^{r},d) . Our approach connects the geometry of torus actions on Kontsevich moduli spaces with symmetric functions in Coxeter types 𝐴 and 𝐵, as well as the enumeration of graph colourings with prescribed symmetry.
We prove that a sub-Riemannian manifold with a full-support Radon measure is never CD (K, N) for any K is an element of R and N is an element of (1, infinity) unless it is Riemannian. This generalizes previous non-CD results for sub-Riemannian manifolds, where a measure with smooth and positive density is considered. Our proof is based on the analysis of the tangent cones and the geodesics within. Secondly, we construct new RCD structures on R-n, named cone-Grushin spaces, that fail to be sub-Riemannian due to the lack of a scalar product along a curve, yet exhibit characteristic features of sub-Riemannian geometry, such as horizontal directions, large Hausdorff dimension, and inhomogeneous metric dilations.
For n <= 6 n\leq 6 , we compute the integral Chow ring of every modular compactification of M 1 , n \mathcal{M}_{1,n} parametrising only Gorenstein curves with smooth, distinct markings. These include the Deligne-Mumford, Schubert, and Smyth compactifications, and many more. They can all be excised from the stack of log-canonically polarised Gorenstein curves. The Chow ring of the latter admits a simple, combinatorial description, which we compute by patching along a natural stratification by core level. We deduce that all these modular compactifications satisfy the Chow-K & uuml;nneth generation property, that the cycle class map is an isomorphism, and for n = 4 n=4 , we study whether Getzler's relation holds integrally and for other compactifications.
We compute the Sn-equivariant topological Euler characteristic of the Kontsevich moduli space & Mscr; (1;n)(P (R), d). Letting & Mscr;(1,n)(& Popf;(r), d) C M--(1;n)(P (R), d) denote the subspace x of maps from curves without rational tails, we solve for the motive of of & Mscr;(1,n)(& Popf;(r), d) and plethysm with a genus-zero contribution determined by Getzler and Pandharipande. Fixing a generic C*-action on P (R), we derive a closed formula for the Euler characteristic of M--(nrt) (1;n)(P- r, d)as an Sn-equivariant virtual mixed Hodge structure, which leads to our main formula for the Euler characteristic of M--(nrt) (1;n)(P- r, d)(C*) in terms & Mscr; (1;n)(P (R), d). Our approach connects the geometry of torus actions on Kontsevich moduli spaces with symmetric functions in Coxeter types A and B, as well as the enumeration of graph colourings with prescribed symmetry.
We investigate the asymptotic behavior of the SU ( 2 ) \operatorname{SU}(2) -Yang-Mills-Higgs energy E ( Phi , A ) = integral M | d A Phi | 2 + | F A | 2 E(\Phi,A)=\int_{M}\lvert d_{A}\Phi vert{2}+\lvert F_{A} vert{2} in the large mass limit, proving convergence to the codimension-three area functional in the sense of De Giorgi's Gamma-convergence. More precisely, for a compact manifold with boundary M and any family of pairs Phi m is an element of Omega 0 ( M ; s u ( 2 ) ) \Phi_{m}\in\Omega{0}(M;\mathfrak{su}(2)) and A m is an element of Omega 1 ( M ; s u ( 2 ) ) A_{m}\in\Omega{1}(M;\mathfrak{su}(2)) indexed by a mass parameter m -> infinity m o\infty , satisfyingE ( Phi m , A m ) <= C m and lim m -> infinity 1 m integral M ( m - | Phi m | ) 2 = 0 , see text E(\Phi_{m},A_{m})\leq Cm\quad ext{and}\quad\lim_{m o\infty}\frac{1}{m}\int_{M}(m-\lvert\Phi_{m} vert){2}=0,we prove that the ( n - 3 ) (n-3) -currents dual to 1 2 pi m tr ( d A m Phi m boolean AND F A m ) \frac{1}{2\pi m}\operatorname{tr}(d_{A_{m}}\Phi_{m}\wedge F_{A_{m}}) converge subsequentially to a relative integral ( n - 3 ) (n-3) -cycle T of massM ( T ) <= lim inf m -> infinity 1 4 pi m E ( Phi m , A m ) , see text \mathbb{M}(T)\leq\liminf_{m o\infty}\frac{1}{4\pi m}E(\Phi_{m},A_{m}),and show conversely that any integral ( n - 3 ) (n-3) -current T with [ T ] = 0 is an element of H n - 3 ( M , partial derivative M ; Z ) [T]=0\in H_{n-3}(M,\partial M;\mathbb{Z}) admits such an approximation, with equality above. In the special case of pairs ( Phi m , A m ) (\Phi_{m},A_{m}) satisfying the generalized monopole equation & lowast; d A m Phi m = F A m boolean AND Theta *d_{A_{m}}\Phi_{m}=F_{A_{m}}\wedge\Theta for a calibration form Theta is an element of Omega n - 3 ( M ) \Theta\in\Omega{n-3}(M) , we deduce that the limit nu = lim m -> infinity 1 2 pi m | d A m Phi m | 2 u=\lim_{m o\infty}\frac{1}{2\pi m}\lvert d_{A_{m}}\Phi_{m} vert{2} of the Dirichlet energy measures satisfies nu <= | T | u\leq\lvert T vert , with equality if and only if T is calibrated by Theta, giving evidence for predictions of Donaldson-Segal in the settings of G 2 G_{2} -manifolds and Calabi-Yau 3-folds.
A degree d genus g cover of the complex projective line by a smooth curve C yields a vector bundle on the projective line by pushforward of the structure sheaf. Which bundles are possible? Equivalently, which Pd-2-bundles over P(1)contain such covers? (In the language of many previous papers: what are the scrollar invariants of the cover?) We give a complete answer in degree4, which exhibits the expected pathologies. We describe a polytope(one per degree) which we propose gives the complete answer for primitive covers, i.e., covers that do not factor through a subcover. We show that all such bundles (for primitive covers) liein this polytope, and that a "positive proportion" of the polytope arises from smooth covers.Moreover, we show the necessity of the primitivity assumption. Finally, we show that the image of the map from the Hurwitz space of smooth covers to the space of bundles is not preserved by generization (for d > 4 and g >> d(1)).
Let ( X , omega ) (X,\omega) be a compact K & auml;hler manifold, ( L , h L ) (L,h{L}) a positive line bundle, and ( E , h E ) (E,h{E}) a Hermitian holomorphic vector bundle of rank r on X. We prove that the pullback by the Kodaira embedding associated to L p circle times E L{p}\otimes E of the k-th Chern form of the dual of the universal bundle over the Grassmannian converges as p -> infinity p o\infty to the k-th power of the Chern form c 1 ( L , h L ) c_{1}(L,h{L}) for 0 <= k <= r 0\leq k\leq r . If c 1 ( L , h L ) = omega c_{1}(L,h{L})=\omega , we also determine the second term in the semi-classical expansion, which involves c 1 ( E , h E ) c_{1}(E,h{E}) . As a consequence, we show that the limit distribution of zeros of random sequences of holomorphic sections of high powers L p circle times E L{p}\otimes E is c 1 ( L , h L ) r c_{1}(L,h{L}){r} . Furthermore, we compute the expectation of the currents of integration along degeneracy sets of random holomorphic sections.
We establish that all rings of S-integers are universally definable in function fields in one variable over certain ground fields including global and non-archimedean local fields. That is, we show that the complement of such a ring of S-integers is always a diophantine set. As a technical tool, we use a reciprocity exact sequence for quadratic Witt groups in function fields over almost arbitrary base fields (of any characteristic), which is new and of potentially independent interest.
We give a complete description of the local geometry of the p-adic eigencurve at p-irregular classical weight one cusp forms under the assumption that certain Gross-Stark regulators do not vanish, as predicted by classical conjectures in p-adic transcendental number theory. We also provide several applications to the Hecke structure of the ordinary p-adic & eacute;tale cohomology of towers of modular curves, as well as to Beilinson-Flach elements.
In this paper, we show that, for any finite subgroup Gamma < O(4) acting freely on S & sup3;, there exists a 4-dimensional complete Riemannian manifold (M, g) with Ric >= 0 such that the asymptotic cone of (M, g) is C(S & sup3;/Gamma) for some delta = delta(Gamma) > 0. This answers a question of Brue-Pigati-Semola [E. Brue, A. Pigati and D. Semola, Topological regularity and stability of noncollapsed spaces with Ricci curvature bounded below, preprint 2024, https://arxiv.org/abs/2405.03839] about the topological obstructions of 4-dimensional non-collapsed tangent cones. Combining this result with the work of Brue-Pigati-Semola, one can classify the 4-dimensional non-collapsed tangent cone in the topological sense.
In this article, we establish an L 2 L<^>{2} extension theorem for Nakano semi-positive singular Hermitian metrics on holomorphic vector bundles, and the strong openness and stability properties for the multiplier submodule sheaves associated to such singular metrics. We solve affirmatively a question of Lempert on the preservation of Nakano semi-positivity under limit of an increasing sequence of metrics based on Deng-Ning-Wang-Zhou's characterization of Nakano positivity.
Based on the formalism of rigid-analytic motives of [J. Ayoub, M. Gallauer and A. Vezzani, The six-functor formalism for rigid analytic motives, Forum Math. Sigma 10 (2022), Paper no. e61], we extend our previous work [L. Fargues and P. Scholze, Geometrization of the local Langlands correspondence, preprint (2021), https://arxiv.org/abs/2102.134592021; to appear in Ast & eacute;risque] from & ell;-adic sheaves to motivic sheaves. In particular, we prove independence of & ell; of the L-parameters constructed there.
The goal of this work is to study some aspects of the geometry of the first cover Sigma(1) in the Drinfeld tower over H(K )(d)the Drinfeld symmetric space over K a finite extension of Q(p). It is a cyclic & eacute;tale cover of order prime to p and even of Kummer type from the vanishing of the Picard group of H-K(d) shown in a previous work of the author. It is then completely described by a certain class of invertible functions on H-K(d) via the Kummer exact sequence and the main result of this article gives an explicit description of this class thus providing "equations" for Sigma(1). This statement extends and uses crucially the local description over a vertex obtained by Wang (and originally by Teitelbaum in dimension 1). One of the main consequences of our global equation is the description of invertible functions of Sigma(1) in terms of the invertible functions of H-K(d).
We prove Vojta's abc conjecture for the projective space P n ( C ) \mathbb{P}{n}(\mathbb{C}) , assuming that the entire curves in P n ( C ) \mathbb{P}{n}(\mathbb{C}) are highly ramified over the coordinate hyperplanes. This extends the results of Guo Ji and the second-named author for the case n = 2 n=2 (see [J. Guo and J. T.-Y. Wang, A complex case of Vojta's general ABC conjecture and cases of Campana's orbifold conjecture, Trans. Amer. Math. Soc. 377 (2024), 7, 4961-4991]). We also explore the corresponding results for projective toric varieties. Consequently, we establish a version of Campana's orbifold conjecture for finite coverings of projective toric varieties.
In [B. Andrews, Classification of limiting shapes for isotropic curve flows, J. Amer. Math. Soc. 16 (2003), 2, 443-459], Ben Andrews obtained the complete classification of the solutions of the planar isotropic L p L_{p} Minkowski problem. In this paper, by generalizing his result, we obtain the complete classification of the solutions of the planar isotropic L p L_{p} dual Minkowski problem for any p , q is an element of R p,q\in\mathbb{R} . That is, we classify all embedded solutions of the equationu 1 - p ( u theta 2 + u 2 ) q - 2 2 ( u theta theta + u ) = 1 on S 1 . u{1-p}(u_{\theta}{2}+u{2}){\frac{q-2}{2}}(u_{\theta\theta}+u)=1\quad\text{on}\ \mathbb{S}{1}.To establish the classification, we convert the ODE for the solution into an integral and study its asymptotic behavior, duality and monotonicity.
