KODAIRA DIMENSION OF UNIVERSAL HOLOMORPHIC SYMPLECTIC VARIETIES

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748021000013 ◽

2021 ◽

pp. 1-18

Author(s):

Shouhei Ma

Keyword(s):

Crucial Role ◽

Transition Point ◽

Kodaira Dimension ◽

Period Map ◽

Universal Family ◽

Finite Period ◽

Symplectic Varieties

Abstract We prove that the Kodaira dimension of the n-fold universal family of lattice-polarised holomorphic symplectic varieties with dominant and generically finite period map stabilises to the moduli number when n is sufficiently large. Then we study the transition of Kodaira dimension explicitly, from negative to nonnegative, for known explicit families of polarised symplectic varieties. In particular, we determine the exact transition point in the Beauville–Donagi and Debarre–Voisin cases, where the Borcherds $\Phi _{12}$ form plays a crucial role.

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The Completed Finite Period Map and Galois Theory of Supercongruences

International Mathematics Research Notices ◽

10.1093/imrn/rny004 ◽

2018 ◽

Vol 2019 (23) ◽

pp. 7379-7405

Author(s):

Julian Rosen

Keyword(s):

Algebraic Number ◽

Complex Number ◽

Galois Theory ◽

Rational Numbers ◽

Multiple Zeta Values ◽

Period Map ◽

Transcendental Numbers ◽

Multiple Zeta ◽

Zeta Values ◽

Finite Period

Abstract A period is a complex number arising as the integral of a rational function with algebraic number coefficients over a region cut out by finitely many inequalities between polynomials with rational coefficients. Although periods are typically transcendental numbers, there is a conjectural Galois theory of periods coming from the theory of motives. This paper formalizes an analogy between a class of periods called multiple zeta values and congruences for rational numbers modulo prime powers (called supercongruences). We construct an analog of the motivic period map in the setting of supercongruences and use it to define a Galois theory of supercongruences. We describe an algorithm using our period map to find and prove supercongruences, and we provide software implementing the algorithm.

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Unobstructedness of hyperkähler twistor spaces

Mathematische Zeitschrift ◽

10.1007/s00209-021-02841-4 ◽

2021 ◽

Author(s):

Ana-Maria Brecan ◽

Tim Kirschner ◽

Martin Schwald

Keyword(s):

Open Neighborhood ◽

Projective Line ◽

Twistor Spaces ◽

Period Map ◽

Universal Family ◽

Complex Projective Line ◽

Complex Projective

AbstractA family of irreducible holomorphic symplectic (ihs) manifolds over the complex projective line has unobstructed deformations if its period map is an embedding. This applies in particular to twistor spaces of ihs manifolds. Moreover, a family of ihs manifolds over a subspace of the period domain extends to a universal family over an open neighborhood in the period domain.

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Polarized microbeam FT-IR analysis of single fibers

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100163496 ◽

1996 ◽

Vol 54 ◽

pp. 206-207

Author(s):

Liling Cho ◽

David L. Wetzel

Keyword(s):

Molecular Orientation ◽

Transition Point ◽

Polymer Fibers ◽

Single Fiber ◽

Wire Grid ◽

Polyester Fibers ◽

Ft Ir ◽

Ir Analysis ◽

The Relationship ◽

Wire Grid Polarizer

Polarized infrared microscopy has been used for forensic purposes to differentiate among polymer fibers. Dichroism can be used to compare and discriminate between different polyester fibers, including those composed of polyethylene terephthalate that are frequently encountered during criminal casework. In the fiber manufacturering process, fibers are drawn to develop molecular orientation and crystallinity. Macromolecular chains are oriented with respect to the long axis of the fiber. It is desirable to determine the relationship between the molecular orientation and stretching properties. This is particularly useful on a single fiber basis. Polarized spectroscopic differences observed from a single fiber are proposed to reveal the extent of molecular orientation within that single fiber. In the work presented, we compared the dichroic ratio between unstretched and stretched polyester fibers, and the transition point between the two forms of the same fiber. These techniques were applied to different polyester fibers. A fiber stretching device was fabricated for use on the instrument (IRμs, Spectra-Tech) stage. Tension was applied with a micrometer screw until a “neck” was produced in the stretched fiber. Spectra were obtained from an area of 24×48 μm. A wire-grid polarizer was used between the source and the sample.

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