scholarly journals Serre–Tate theory for Calabi–Yau varieties

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Piotr Achinger ◽  
Maciej Zdanowicz
Keyword(s):  
Weak Sense ◽  
Higher Dimensions ◽  
Explicit Construction ◽  
K3 Surfaces ◽  
Canonical Coordinates ◽  
Canonical Class ◽  
Witt Vectors ◽  
Hodge Filtration ◽  
Relative Version ◽  
Simply Connected

Abstract Classical Serre–Tate theory describes deformations of ordinary abelian varieties. It implies that every such variety has a canonical lift to characteristic zero and equips the base of its universal deformation with a Frobenius lifting and canonical multiplicative coordinates. A variant of this theory has been obtained for ordinary K3 surfaces by Nygaard and Ogus. In this paper, we construct canonical liftings modulo p 2 {p^{2}} of varieties with trivial canonical class which are ordinary in the weak sense that the Frobenius acts bijectively on the top cohomology of the structure sheaf. Consequently, we obtain a Frobenius lifting on the moduli space of such varieties. The quite explicit construction uses Frobenius splittings and a relative version of Witt vectors of length two. If the variety has unobstructed deformations and bijective first higher Hasse–Witt operation, the Frobenius lifting gives rise to canonical coordinates. One of the key features of our liftings is that the crystalline Frobenius preserves the Hodge filtration. We also extend Nygaard’s approach from K3 surfaces to higher dimensions, and show that no non-trivial families of such varieties exist over simply connected bases with no global one-forms.

scholarly journals Type II degenerations of K3 surfaces

1985 ◽  
Vol 99 ◽  
pp. 11-30 ◽  
Author(s):  
Shigeyuki Kondo
Keyword(s):  
Number Field ◽  
Complex Manifold ◽  
Complex Number ◽  
Three Dimensional ◽  
K3 Surfaces ◽  
Type Ii ◽  
Holomorphic Map ◽  
Canonical Class ◽  
Proper Holomorphic Map ◽  
Complex Number Field

A degeneration of K3 surfaces (over the complex number field) is a proper holomorphic map π: X→Δ from a three dimensional complex manifold to a disc, such that, for t ≠ 0, the fibres Xt = π-1(t) are smooth K3 surfaces (i.e. surfaces Xt with trivial canonical class KXt = 0 and dim H1(Xt, Oxt) = 0).

scholarly journals Admissibility for a Class of Quasiregular Representations

2007 ◽  
Vol 59 (5) ◽  
pp. 917-942 ◽  
Author(s):  
Bradley N. Currey
Keyword(s):  
Wavelet Transform ◽  
Semidirect Product ◽  
Explicit Construction ◽  
Vector Group ◽  
Almost Everywhere ◽  
Orthogonality Relations ◽  
Completely Reducible ◽  
Multiplicity Free ◽  
Simply Connected

AbstractGiven a semidirect product G = N ⋊ H where N is nilpotent, connected, simply connected and normal in G and where H is a vector group for which ad() is completely reducible and R-split, let τ denote the quasiregular representation of G in L2(N). An element ψ ∈ L2(N) is said to be admissible if the wavelet transform f ⟼ 〈 f, τ(·)ψ 〉 defines an isometry from L2(N) into L2(G). In this paper we give an explicit construction of admissible vectors in the case where G is not unimodular and the stabilizers in H of its action on are almost everywhere trivial. In this situation we prove orthogonality relations and we construct an explicit decomposition of L2(G) into G-invariant, multiplicity-free subspaces each of which is the image of a wavelet transform . We also show that, with the assumption of (almost-everywhere) trivial stabilizers, non-unimodularity is necessary for the existence of admissible vectors.

Central Extensions of Loop Groups and Obstruction to Pre-Quantization

2013 ◽  
Vol 56 (1) ◽  
pp. 116-126 ◽  
Author(s):  
Derek Krepski
Keyword(s):  
Riemann Surface ◽  
Lie Group ◽  
Moduli Space ◽  
Explicit Construction ◽  
Loop Groups ◽  
Central Extensions ◽  
Simply Connected

AbstractAn explicit construction of a pre-quantumline bundle for themoduli space of flat G-bundles over a Riemann surface is given, where G is any non-simply connected compact simple Lie group. This work helps to explain a curious coincidence previously observed between Toledano Laredo's work classifying central extensions of loop groups LG and the author's previous work on the obstruction to pre-quantization of the moduli space of flat G-bundles.

scholarly journals Explicit Spectral Decimation for a Class of Self-Similar Fractals

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Sergio A. Hernández ◽  
Federico Menéndez-Conde
Keyword(s):  
Laplace Operator ◽  
Sierpinski Gasket ◽  
Higher Dimensions ◽  
Explicit Construction ◽  
Sierpiński Gasket ◽  
Infinite Collection ◽  
Self Similar ◽  
The Laplace Operator

The method of spectral decimation is applied to an infinite collection of self-similar fractals. The sets considered are a generalization of the Sierpinski Gasket to higher dimensions; they belong to the class of nested fractals and are thus very symmetric. An explicit construction is given to obtain formulas for the eigenvalues of the Laplace operator acting on these fractals.

scholarly journals Computable Constants for Korn’s Inequalities on Riemannian Manifolds

2021 ◽  
Author(s):  
R. J. Knops
Keyword(s):  
Boundary Conditions ◽  
Minimal Surface ◽  
Riemannian Manifolds ◽  
Dirichlet Boundary ◽  
Explicit Construction ◽  
Dirichlet Boundary Conditions ◽  
Two Dimensional ◽  
Spherical Cap ◽  
Connected Manifold ◽  
Simply Connected

AbstractA method is presented for the explicit construction of the non-dimensional constant occurring in Korn’s inequalities for a bounded two-dimensional Riemannian differentiable simply connected manifold subject to Dirichlet boundary conditions. The method is illustrated by application to the spherical cap and minimal surface.

scholarly journals RELATIVE CYCLES WITH MODULI AND REGULATOR MAPS

2017 ◽  
Vol 18 (06) ◽  
pp. 1233-1293 ◽  
Author(s):  
Federico Binda ◽  
Shuji Saito
Keyword(s):  
Cartier Divisor ◽  
Explicit Construction ◽  
Chow Groups ◽  
Chow Group ◽  
Deligne Cohomology ◽  
Fundamental Class ◽  
Normal Crossing ◽  
Rham Complex ◽  
Relative Version ◽  
Higher Chow Groups

Let $\overline{X}$ be a separated scheme of finite type over a field $k$ and $D$ a non-reduced effective Cartier divisor on it. We attach to the pair $(\overline{X},D)$ a cycle complex with modulus, those homotopy groups – called higher Chow groups with modulus – generalize additive higher Chow groups of Bloch–Esnault, Rülling, Park and Krishna–Levine, and that sheafified on $\overline{X}_{\text{Zar}}$ gives a candidate definition for a relative motivic complex of the pair, that we compute in weight $1$ . When $\overline{X}$ is smooth over $k$ and $D$ is such that $D_{\text{red}}$ is a normal crossing divisor, we construct a fundamental class in the cohomology of relative differentials for a cycle satisfying the modulus condition, refining El Zein’s explicit construction of the fundamental class of a cycle. This is used to define a natural regulator map from the relative motivic complex of $(\overline{X},D)$ to the relative de Rham complex. When $\overline{X}$ is defined over $\mathbb{C}$ , the same method leads to the construction of a regulator map to a relative version of Deligne cohomology, generalizing Bloch’s regulator from higher Chow groups. Finally, when $\overline{X}$ is moreover connected and proper over $\mathbb{C}$ , we use relative Deligne cohomology to define relative intermediate Jacobians with modulus $J_{\overline{X}|D}^{r}$ of the pair $(\overline{X},D)$ . For $r=\dim \overline{X}$ , we show that $J_{\overline{X}|D}^{r}$ is the universal regular quotient of the Chow group of $0$ -cycles with modulus.

DOUBLING HOMOTOPY K3 SURFACES

2003 ◽  
Vol 12 (03) ◽  
pp. 347-354
Author(s):  
B. DOUG PARK
Keyword(s):  
K3 Surfaces ◽  
New Family ◽  
Simply Connected

We perform certain doubling operation on the homotopy K3 surfaces of R. Fintushel and R. J. Stern to obtain a new family of smooth closed simply-connected irreducible spin 4-manifolds indexed by knots in S3.

Kernels in the Category of Formal Group Laws

2016 ◽  
Vol 68 (2) ◽  
pp. 334-360 ◽  
Author(s):  
Oleg Demchenko ◽  
Alexander Gurevich
Keyword(s):  
Finite Field ◽  
Formal Group ◽  
Explicit Construction ◽  
Formal Groups ◽  
Witt Vectors ◽  
Formal Group Law ◽  
Formal Group Laws ◽  
Strict Isomorphism ◽  
Dieudonné Module ◽  
Group Law

AbstractFontaine described the category of formal groups over the ring of Witt vectors over a finite field of characteristic p with the aid of triples consisting of the module of logarithms, the Dieudonné module, and the morphism from the former to the latter. We propose an explicit construction for the kernels in this category in term of Fontaine's triples. The construction is applied to the formal norm homomorphism in the case of an unramified extension of ℚp and of a totally ramiûed extension of degree less or equal than p. A similar consideration applied to a global extension allows us to establish the existence of a strict isomorphism between the formal norm torus and a formal group law coming from L-series.

scholarly journals Every algebraic Kummer surface is the K3-cover of an Enriques surface

1990 ◽  
Vol 118 ◽  
pp. 99-110 ◽  
Author(s):  
Jong Hae Keum
Keyword(s):  
Crucial Role ◽  
K3 Surfaces ◽  
Abelian Surface ◽  
Enriques Surface ◽  
Kummer Surface ◽  
3 Dimensional ◽  
Complex Torus ◽  
Kummer Surfaces ◽  
Dimension 2 ◽  
Simply Connected

A Kummer surface is the minimal desingularization of the surface T/i, where T is a complex torus of dimension 2 and i the involution automorphism on T. T is an abelian surface if and only if its associated Kummer surface is algebraic. Kummer surfaces are among classical examples of K3-surfaces (which are simply-connected smooth surfaces with a nowhere-vanishing holomorphic 2-form), and play a crucial role in the theory of K3-surfaces. In a sense, all Kummer surfaces (resp. algebraic Kummer surfaces) form a 4 (resp. 3)-dimensional subset in the 20 (resp. 19)-dimensional family of K3-surfaces (resp. algebraic K3 surfaces).

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