60,573 papers in this slice of arXiv.
Arvid Siqveland
We give the minimum of category theory necessary for understanding the definition and applications of Grothendieck topos. We state the basic properties of sites and sheaves, and give applications to the theory of moduli. We prove that for categories C with explicit stated properties, we can construct a category of schemes of objects in C which is a proper category rather than a 2
Moritz Hartlieb, Saket Shah
Following ideas of Kapustka-Kapustka, we use Lagrangian fibrations to construct twisted hyperholomorphic sheaves on products of hyperkähler manifolds of K3[n]- and OG10-type. As applications, we prove the Lefschetz standard conjecture and the D-equivalence conjecture for hyperkähler manifolds of OG10-type.
Sanjay Amrutiya, Umesh Dubey
In this note, we construct moduli spaces of super-representations of quivers by extending King's Geometric Invariant Theory (GIT) framework to the super setting. Using the even part of super general linear groups and a parity-shifting operator, we construct moduli spaces that topologically parameterize super-representations of a fixed super-dimension. We also outline the construction of super quiver varieties for super-representations of framed quivers (without doubling); and realise super-Grassmannians and super-flag varieties as geometric quotients within this setting.
Stefano Vigni
Let f be a non-CM newform of weight k≥4, level N and trivial Nebentypus. Let p∤N be an odd prime number that is ordinary for f and denote by f(p) the p-adic Hida family passing through f. Assuming very specific instances of two general conjectures in arithmetic algebraic geometry (injectivity of p-adic Abel-Jacobi maps, positive definiteness of archimedean height pairings à la Gillet-Soulé), we prove that, for all but finitely many p as above, if the analytic rank of f is 1, then all but finitely many specializations of f(p) of even weight and trivial Nebentypus have analytic rank 1. This result provides evidence for Greenberg's "minimality conjecture" on analytic ranks in families of modular forms.
Alessio Cela, Sae Koyama
We tropically enumerate planar well-spaced elliptic curves with a fixed j-invariant. Combined with the recent genus 1 correspondence theorem of Cela--Koyama, this yields an alternative proof of Pandharipande's algebraic enumeration.
Boming Jia
Let g be a complex simple Lie algebra of type G2, F4, or E8. We prove that the closure Omin(g) of its minimal nilpotent orbit is not isomorphic to (T∗X)aff for any smooth quasi-affine variety X. We do not assume that the isomorphism is equivariant or Poisson. We also prove the same statement in types A1 and A2.
Robert Laterveer
Let Y be a very general prime Fano threefold of genus 7. We exhibit an explicit 2-cycle on Y×Y that is Abel-Jacobi trivial but non-torsion in the Chow group A4(Y×Y). As a consequence, Y does not admit a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. We also show that any Fano threefold has a multiplicative Chow-Künneth decomposition modulo algebraic equivalence.
Mikhail Kapranov
In 1972, M. Sato proposed an approach to proving modularity of forms like Thetanullwerte by characterizing them via certain differential operators of infinite order (DOI) in the modular variable(s) alone. A DOI is an infinite series in derivatives decreasing so fast that it acts on holomorphic functions by a sheaf morphism. This approach was developed by several authors including Kashiwara, Kawai, Takei and Yoshida. We give an interpretation of this approach using supersymmetry which provides a natural source of DOIs: the naive exponential of any odd supersymmetry generator is a DOI. The case of the Riemann theta function of genus n is governed by the supergroup OSp(1|2n) (and its metaplectic cover) acting on a natural super-thickening of the Siegel plane. For n=2 this is the 3-dimensional N=1 superconformal group and the structure at hand is precisely the free massless scalar supermultiplet (combining the Laplace and Dirac equations). For n>2 we get a super-extension of the generalized conformal structure existing on the Lagrangian Grassmannian as on any Hermitian symmetric space. An additional interesting feature here is that the odd supersymmetry generators acting ``on-shell'' (i.e., in the space of solutions of the equations of motion) satisfy even-style Heisenberg commutation relations. These equations of motion upgrade to a complex of differential operators corresponding to a natural BGG-type resolution of the super-Weil representation of osp(1|2n).
Ping Li
We study whether the topology underlying a Calabi-Yau manifold can support natural geometric positivity or negativity structures. In even complex dimension n≥4 (assuming b2=1 when n≥6), we strengthen a theorem of Oguiso-Peternell by proving that a Calabi-Yau manifold is not homeomorphic to a weak Fano n-fold. The same obstruction applies to Kähler manifolds with quasi-positive holomorphic sectional curvature. A transformation-group analogue excludes, in particular, symplectic manifolds admitting Hamiltonian circle actions with isolated fixed points. On the negative side, we show that the fundamental group of a Calabi-Yau manifold is not isomorphic to that of a Kähler hyperbolic manifold. Taken together, these results exhibit a common topological rigidity separating Calabi-Yau manifolds from several fundamental classes governed by geometric positivity or negativity.
Viatcheslav Kharlamov, Rareş Răsdeaconu
We establish lower bounds for the Smith--Thom deficiency of the Hilbert square of a maximal nonsingular real projective variety. These bounds recover the known surface case and provide new obstructions to Smith-Thom maximality in every dimension at least two. As applications, we prove that the Hilbert square of any real abelian variety of dimension at least two is not Smith-Thom maximal. We further show that the deficiencies of the Hilbert squares of maximal real abelian varieties grow exponentially with the dimension. We also obtain nonmaximality results for the Hilbert squares of certain Cartesian products.
Piotr Achinger, Katharina Hübner, Marcin Lara +1
We introduce the tame étale fundamental group π1t(X/K) of a rigid space X over a non-archimedean field K. We show that if X is qcqs and K has topologically finitely generated tame Galois group (e.g. algebraically closed or a local field), then π1t(X/K) is topologically finitely generated. If X is moreover the rigid generic fibre of a strictly semistable formal scheme such that the smooth locus of its special fibre admits a projective snc compactification, then π1t(X/K) is topologically finitely presented. The proofs rely on techniques of logarithmic geometry (extended beyond its usual scope of finitely generated monoids), in particular on an analogous finiteness statement for the tame log étale fundamental group, and on the 'vertical compactification' of a map of adic spaces.
Mohamed Ali Belabbas
Let A be a finitely generated commutative algebra over a field K of characteristic zero. We prove that every finitely generated Lie subalgebra L⊆DerK(A) whose elements are locally finite on A is finite-dimensional. Consequently, for a Lie subalgebra generated by finitely many locally finite derivations, the following are equivalent: it is finite-dimensional, it acts locally finitely on A, and all its elements are locally finite. A key ingredient is a second theorem of independent interest: every Lie subalgebra of DerK(A) whose elements are locally nilpotent is solvable; when A is reduced, its derived length is at most dimA, and this bound is sharp. For an affine variety X over an algebraically closed field, we deduce that a subgroup of Aut(X) generated by finitely many connected algebraic subgroups is algebraic if and only if every element of the Lie algebra generated by their tangent algebras is locally finite. For two unipotent one-parameter subgroups, this provides an answer to a problem posed by Popov in 2005. Our results also characterize polynomial control systems admitting an exact finite-dimensional bilinear realization by polynomial observables containing the state coordinates. The proofs rest on the introduction of a cofinite ideal meeting the closure of every associated point of SpecA: on the subalgebra of derivations vanishing to second order along the corresponding finite subscheme, local finiteness forces local nilpotence. The intersection of that subalgebra with L is therefore solvable by the second theorem, and has finite codimension in L; together with local finiteness of the adjoint action, this yields finite-dimensionality.
Thomas Prellberg
We formulate an elementary collision-generated compression principle for homogeneous Keller maps. In the language of polarization algebras, the construction takes the subalgebra generated by a collision; the additional point is that this subalgebra carries a noninjective Keller restriction and controls the dimension of the standard symmetric lift. For Thompson's 24-variable cubic-homogeneous map, exact polarization gives growth 2,4,11,20,20, and the resulting subalgebra is exactly MacFarlane's 20-dimensional invariant subspace. Thus the known 24-to-20 compression is recovered canonically from the collision itself. Although the resulting 40-variable lift had already been noted, we give an explicit 350-monomial homogeneous quartic for which Zhao's Vanishing Conjecture fails, together with an exact gradient collision over (i). Over , (i), and , no proper invariant linear subspace containing this collision is available before the unchanged symmetric lift. No global minimality is claimed. The principal explicit calculations are checked by an accompanying exact calculation.
Hadi Hajieghrary, Benedikt Walter, Chaitanya Shinde +1
Automated-driving rulebooks rank rule violations lexicographically, and model predictive control enforces that ranking either exactly, through L+1 sequential programs per tick, or approximately, through a weighted sum tuned by the separation heuristic w1≫w2≫⋯≫wL
You-Cheng Chou, Hsian-Hua Tseng
We study descendant integrals on moduli stacks of simplicially stable curves. For each finite collision complex, we factor the reduction morphism into elementary wall crossings and obtain a reconstruction formula in terms of ordinary Witten-Kontsevich correlators. We use the formula to derive corrected Virasoro recursions. We then package all finite-support theories in a completed disjoint-support Fock space, where the extended potential is a cumulant translate of the Witten-Kontsevich potential.
Roger Casals
The main result of this article shows that the quiver mutation class of a malleable real Morsification uniquely determines the integral monodromy module of a plane curve singularity. In particular, we show that the quiver mutation class of a malleable divide determines the complex topological type of an irreducible plane curve singularity. This establishes the algebraic-to-topological implication of a conjecture of S.~Fomin, P.~Pylyavskyy, E.~Shustin and D.~Thurston in the malleable irreducible case. The result is proven by developing representation-theoretic techniques based on an equivariant Euler pairing in the derived category of continuous finite-dimensional dg modules over a differential bigraded Ginzburg algebra. A key step uses these techniques to show that the quiver mutation class of a plabic fence uniquely recovers the torsion part of the Alexander module of the associated smooth link.
Lev Borisov, Carlos Rito
We construct quintic surfaces in the three-dimensional projective space P3 with 18 ordinary cusps. Our starting point is the Barth--Rams description of quintics containing a 3-divisible set of 12 cusps. A specialization in which the two contact cubics are singular along two skew lines produces a family with 16 cusps, and examples with 18 cusps can be found over small finite fields. Our main construction is based on quintics admitting two Barth--Rams decompositions. The corresponding sets of 12 cusps meet in 7 points, and we prove that the locus of quintics admitting two such decompositions contains a 6-dimensional component in the moduli space whose general member has 17 cusps. This makes it possible to find members with 18 cusps efficiently over finite fields. We lift one of these surfaces to characteristic zero using Newton--Hensel lifting and LLL reconstruction, obtaining a quintic over a number field of degree 22. We verify that this surface has 18 ordinary cusps and no other singularities.
Lev Borisov, Carlos Rito
We compute explicit defining equations for eight fake quadrics arising as Z/2×Z/4-covers of two singular Z/2-Godeaux surfaces obtained in earlier work of the second author. Starting from explicit equations for the universal covers of the Godeaux surfaces, we reconstruct the relevant character eigenspaces and determine the homogeneous ideals of the bicanonical models of the eight fake quadrics in P8. All eight models are defined over Q. We prove that the surfaces are pairwise non-isomorphic and rigid. Combined with the non-product result established in the earlier work, this gives the first explicit projective models of fake quadrics which are not isogenous to a product of curves.
Tomasz Kowalczyk
Let (p,q) be a Jacobian pair. We show that the real Jacobian conjecture holds if the degree of p is 7 and the highest degree homogenous part is of the form αx7+βx6y for α2+β2=0. We then show that there are no atypical Jacobian pairs such that degp=7 and degq is even and coprime with 7.
Steven Dale Cutkosky
Let R be a d-dimensional Noetherian local ring with maximal ideal mR and I be an ideal of R. The epsilon multiplicity ε(I) of I is shown in a recent paper of Stephen Landsittel to be the limit of Amao multiplicities ε(I)=m→∞limmda(Im,(Im)sat) if the dimension of the nilradical of R^ is less than d. It is shown in this paper that for an arbitrary local ring R which is of finite type over a field, the limit m→∞limmda(Im,(Im)sat) always exists.