Quantum codes linked to abelian varieties, providing mathematical rigor.
problem Quantum error correction through complex abelian varieties.
method Mathematical formulation of Gottesman-Kitaev-Preskill codes using abelian varieties.
result Asymptotic isometry of encoding, precise gate realizations, and failure probability optimization.
Complex projective manifolds without rational curves are quotients of Abelian varieties.
problem Characterizing complex projective manifolds without rational curves.
method Using conjectures about rational and entire curves on Calabi-Yau varieties.
result Non-hyperbolic complex projective manifolds contain the image of an Abelian variety.
Introduces semi-abelian generalized complex structures.
problem Deformation theory of abelian complex structures.
method Definition and examples of semi-abelian generalized complex structures.
result Illustration of new concept with examples.
Galois action on manifold structures of complex varieties is abelian.
problem Understanding the Galois action on topological manifold structures of complex varieties.
method Definition of profinite normal structure set and Galois action analysis.
result Galois action on manifold structures of simply-connected varieties is abelian.
Study non-existence of complex ball quotients in Torelli locus.
problem Non-existence of totally geodesic complex ball quotients in Torelli locus.
method Analytic techniques.
result Analytic techniques used to study non-existence.
Compactifies moduli of abelian varieties and curves by attaching flat tori.
problem Compactifying moduli spaces of abelian varieties and curves.
method Explicitly attaching moduli of flat tori to abelian varieties and curves.
result Explicit determination of Gromov-Hausdorff limits of abelian varieties and curves.
The study classifies holomorphic projective connections on complex threefolds.
problem Characterizing holomorphic projective connections on complex threefolds.
method Analyzing properties of holomorphic projective connections on complex projective threefolds.
result Holomorphic projective connections on complex threefolds are either flat or translation invariant on abelian threefolds.
The paper shows how certain complex projective varieties can be broken down into simpler types.
problem Understanding the structure of complex projective varieties with pseudo-effective tangent sheaves.
method Developed a theory of pseudo-effective sheaves and applied the minimal model program.
result Projective klt varieties with pseudo-effective tangent sheaves can be decomposed into Fano varieties and Q-abelian varieties.
Non-Archimedean balanced metrics approximate cscK metrics for totally degenerate abelian varieties
problem Non-Archimedean balanced metrics for polarized abelian varieties
method Non-Archimedean analogue of the cscK metric
result Uniform estimate for Calabi-Yau metrics on fibers
Study on moduli spaces of Higgs bundles over Abelian varieties, focusing on their topology and polynomials.
problem Determine the topology and polynomials of moduli spaces of Higgs bundles over Abelian varieties.
method Analyzing the Poincaré polynomials and mixed Hodge polynomials of moduli spaces MAH(G) for various groups G and dimensions d. result Explicit formulas for Poincaré polynomials and mixed Hodge polynomials in specific cases, including rank 2 and 3 Higgs bundles.
We prove that one can obtain natural bundles of Lie algebras on rank two s-Kähler manifolds, whose fibres are isomorphic to so(s+1,s+1), su(s+1,s+1) and sl(2s + 2,\R). In the most rigid case (which includes complex tori and abelian varieties) these bundles have natural flat connections, whose flat global sections act n…
The paper studies spectral analysis on complex spaces and finds explicit formulas for eigensections.
problem Understanding eigensections on complex projective spaces and Grassmannians.
method Using creation and annihilation operators, converting higher energy eigensections to lower energy holomorphic sections.
result Explicit formulas for the dimension of higher-level eigensections on Pn. Study shows how to separate jets on complex varieties using coverings.
problem Separating jets on complex varieties of maximal Albanese dimension.
method Using finite abelian étale covers and moving Seshadri constants.
result Existence of abelian covers that separate jets.
We give a new proof of a slightly weaker form of a theorem of P. Colmez. This theorem gives a formula for the Faltings height of abelian varieties with complex multiplication by a C.M. field whose Galois group over Q is abelian; it reduces to the formula of Chowla and Selberg in the case of elliptic curves. We sh…
We show that a general n-dimensional polarized abelian variety (A,L) of a given polarization type and satisfying h0(A,L)≥28n⋅n!nn is projectively normal. In the process, we also obtain a sharp lower bound for the volume of a purely one-dimensional complex analytic subvariety i…
Complete Calabi-Yau metrics on abelian fibrations over complex space.
problem Constructing complete Calabi-Yau metrics on noncompact abelian fibrations.
method Using abelian fibrations over C, we construct complete Calabi-Yau metrics and provide compactification. result We provide a compactification for the abelian fibration X such that the compactified variety has a negative canonical bundle. The paper proves structures for complex projective varieties with certain tangent bundle properties.
problem Characterizing complex projective varieties based on properties of their tangent bundles.
method Analyzing the positivity of exterior powers of tangent bundles and using étale covers.
result Complex projective varieties with nef exterior powers of tangent bundles are Fano fiber spaces over Abelian varieties.
The geometric torsion conjecture asserts that the torsion part of the Mordell--Weil group of a family of abelian varieties over a complex quasiprojective curve is uniformly bounded in terms of the genus of the curve. We prove the conjecture for abelian varieties with real multiplication, uniformly in the field of multi…
This paper formalizes manifolds in positive characteristic varieties.
problem Establishing l-adic formal manifold structures on positive characteristic varieties.
method Develops and proves the existence of l-adic formal manifold structures and abelianized Galois symmetries.
result Proves l-adic homotopic equivalence and l-local lifting for simply-connected varieties.
This thesis explores algebraic cycles and moduli spaces over real numbers.
problem Understanding the cycle class map and its image in real algebraic geometry.
method Constructing integral Fourier transforms on Chow rings of abelian varieties over any field.
result Proof of integral Hodge conjecture for real abelian threefolds and moduli space properties.
The paper classifies complex Abelian varieties with pseudo-real multiplication and describes their moduli spaces.
problem Classifying three-dimensional complex Abelian varieties with pseudo-real multiplication.
method Explicit construction and analysis of moduli spaces of these varieties.
result The boundary of the eigenform locus for pseudo-real multiplication coincides with cross-ratio equations for certain stable curves.
Abstract: Study of metrics on line bundles over complex varieties.
problem Chern-Weil and Hilbert-Samuel formulae for singular metrics.
method Theory of b-divisors and multiplier ideal volume function.
result Generalization of a result for elliptic curves to higher degrees.
Quantum trace maps abelian character varieties to SL2 character varieties.
problem Classifying irreducible representations of Chekhov-Fock algebras.
method Non-commutative deformation of algebraic morphisms.
result Induces birational morphism between torus and SL2 character variety.
Fibers of abelian varieties over curves can be approximated algebraically.
problem Approximating fibers of abelian varieties over curves algebraically.
method Proving the existence of algebraic approximations for fibrations.
result Fibrations with abelian varieties over curves have algebraic approximations.
Let G be a complex reductive algebraic group (not necessarily connected), let K be a maximal compact subgroup, and let A be a finitely generated Abelian group. We prove that the conjugation orbit space Hom(A,K)/K is a strong deformation retract of the GIT quotient space Hom(A,G)//G. As a corollary, we determine necessa…
Study of hyperkähler reduction on abelian varieties and toric manifolds.
problem Understanding hyperkähler reduction on specific manifolds.
method Lifts canonical Kähler reduction to hyperkähler, studies on abelian varieties and toric manifolds.
result Obtains decoupling result, variational characterisation, relation to K-stability, and proves existence and uniqueness under suitable assumptions. Using gauge theory for Spin(7)-manifolds of dimension 8, we develop a procedure, called Spin-rotation, which transforms a (stable) holomorphic structure on a vector bundle over a complex torus of dimension 4 into a new holomorphic structure over a different complex torus. We show non-trivial examples of this procedure …
Study abelian varieties' Weil-Petersson metric asymptotics.
problem Asymptotic behavior of Weil-Petersson metric on abelian varieties.
method Linking asymptotic with multi-scale collapsing limits of parametrized flat tori.
result Refined description of Weil-Petersson metric on abelian varieties.
Introduces non-abelian Hodge correspondence linking algebraic structures to geometry.
problem None explicitly stated, focuses on introduction.
method Pedagogical introduction of concepts linking algebraic structures to geometry.
result Explains the non-abelian Hodge correspondence and its connections.
Unique map connects curve pairs to abelian varieties.
problem Uniqueness of the Prym map between curve pairs and abelian varieties.
method Holomorphic map construction, classification of homomorphisms, geometric group theory.
result Prym map is the unique nonconstant holomorphic map for g≥4 and h≤g−1. Construct special Lagrangian fibrations on abelian varieties using retraction techniques.
problem Constructing special Lagrangian fibrations on abelian varieties.
method Explicit construction using special techniques in non-Archimedean geometry.
result Solved a conjecture of Kontsevich-Soibelman for finite quotients of abelian varieties.
Explicit formula for Bergman kernel of abelian varieties proved.
problem Explicit formula for Bergman kernel of polarized abelian varieties.
method Explicit formula for Bergman kernel of polarized abelian varieties.
result Explicit formula for Bergman kernel of polarized abelian varieties.
Resolves conjectures on non-abelian Hodge loci for quasi-projective varieties.
problem Understanding non-abelian Hodge loci for quasi-projective varieties.
method Analyzes Z-local systems and polarized variations of Hodge structures. result Proves algebraicity of non-abelian Hodge loci for Q-anisotropic monodromy. Study balancing embedding of degenerating Abelian varieties.
problem Balancing embedding of degenerating Abelian varieties.
method Maximal unipotent family over punctured disc, base change, theta functions, Gromov-Hausdorff limit.
result Balanced embedding of degeneration into projective space.
The study bounds growth of Hodge numbers and computes L2-Betti numbers for irregular varieties.
problem Bounding growth of normalized Hodge numbers and computing L2-Betti numbers for irregular varieties. method Analysis of abelian covers, weak generic Nakano vanishing theorem, and convergence of plurigenera.
result Optimal bounds on the growth of normalized Hodge numbers and computation of L2-Betti numbers. We look into a construction of principal abelian varieties attached to certain spin manifolds, due to Witten and Moore-Witten around 2000 and try to place it in a broader framework. This is related to Weil intermediate Jacobians but it also suggests to associate abelian varieties to polarized even weight Hodge structur…
We discuss properties of complex algebraic orbifold groups, their characteristic varieties, and their abelian covers. In particular, we deal with the question of (quasi)-projectivity of orbifold groups. We also prove a structure theorem for the variety of characters of normal-crossing quasi-projective orbifold groups. …
Study theta functions and adiabatic curvature on Abelian varieties.
problem Explicitly compute curvature of direct image bundles on Pic^0(M).
method Use theta functions associated with line bundles and adiabatic limit.
result Explicit curvature computation of direct image bundle on Pic^0(M).
The paper studies gradings on nilpotent Lie algebras linked to smooth algebraic varieties.
problem Understanding gradings on nilpotent Lie algebras associated with algebraic varieties.
method Analyzing lattice structures in nilpotent Lie groups and their fundamental groups.
result Conditions for a lattice to be the fundamental group of a smooth complex algebraic variety.
We present a method for deciding when a regular abelian cover of a finite CW-complex has finite Betti numbers. To start with, we describe a natural parameter space for all regular covers of a finite CW-complex X, with group of deck transformations a fixed abelian group A, which in the case of free abelian covers of ran…
We consider the self-dual vortex equations on a positive line bundle L --> M over a compact Kaehler manifold of arbitrary dimension. When M is simply connected, the moduli space of vortex solutions is a projective space. When M is an abelian variety, the moduli space is the projectivization of the Fourier-Mukai transfo…
In this thesis we study asymptotic behavior of projective embeddings of abelian varieties and their amoebas. The projective embeddings are given by theta functions. It is known that a Lagrangian fibration of the abelian variety determines a basis of theta functions. After reviewing the relation from the viewpoint of ge…
Global rigidity theorem for curve to abelian variety maps.
problem Characterizing maps between moduli spaces of curves and abelian varieties.
method Analyzing nonconstant holomorphic maps between moduli spaces.
result Holomorphic maps between specific moduli spaces are rigid, with only one possible form.
The study shows boundedness of certain fibered varieties in algebraic geometry.
problem Bounding fibered varieties in algebraic geometry.
method Analyzing Calabi-Yau varieties and their fibrations by abelian or symplectic varieties.
result There are only finitely many deformation classes of certain fibered varieties.
The paper proves properties of strata of differentials, showing they are affine and extremal.
problem Properties of strata of differentials, particularly their geometry and tautological rings.
method Analyzing the tautological rings and using Teichmüller dynamics to prove properties.
result Strata of differentials are affine and their stratification is extremal.
The paper extends structure theorem to projective klt varieties with specific tangent sheaf properties.
problem Understanding the structure of projective klt varieties with nef tangent sheaves.
method Developing theory of positivity of coherent sheaves and proving structure theorem.
result Projective klt varieties with specific tangent sheaf properties admit rationally connected fibrations onto abelian varieties.
The paper constructs a free abelian group from Anosov representations on bundles.
problem Constructing a free abelian group from Anosov representations on bundles.
method Extending the Γ-action via ρ to the space of connections on pullbacks of tangent bundles. result A free abelian group Fab is constructed and acts properly discontinuously on it. Describes spectral data for singular fibres of a specific Hitchin system.
problem Characterizing singular fibres of the SL(2,C)-Hitchin system. method Using Hecke transformations and analysis of parameter spaces, the paper stratifies and compactifies the singular spaces.
result Large classes of singular fibres are shown to be fibre bundles over Prym varieties.