Theta Functions on Varieties with Effective Anti-Canonical Class.

Format:

Book

Author/Creator:

Gross, Mark.

Contributor:

Hacking, Paul.

Siebert, Bernd.

Series:

Memoirs of the American Mathematical Society

Memoirs of the American Mathematical Society ; v.278

Language:

English

Subjects (All):

Functions, Theta.

Surfaces, Algebraic.

Mirror symmetry.

Calabi-Yau manifolds.

Physical Description:

1 online resource (122 pages)

Edition:

1st ed.

Place of Publication:

Providence : American Mathematical Society, 2022.

Summary:

"We show that a large class of maximally degenerating families of n-dimensional polarized varieties comes with a canonical basis of sections of powers of the ample line bundle. The families considered are obtained by smoothing a reducible union of toric varieties governed by a wall structure on a real n-(pseudo-)manifold. Wall structures have previously been constructed inductively for cases with locally rigid singularities [Gross and Siebert, From real affine geometry to complex geometry (2011)] and by Gromov-Witten theory for mirrors of log Calabi-Yau surfaces and K3 surfaces [Gross, Pandharipande and Siebert, The tropical vertex ; Gross, Hacking and Keel, Mirror symmetry for log Calabi-Yau surfaces (2015); Gross, Hacking, Keel, and Siebert, Theta functions and K3 surfaces (In preparation)]. For trivial wall structures on the n-torus we retrieve the classical theta functions. We anticipate that wall structures can be constructed quite generally from maximal degenerations. The construction given here then provides the homogeneous coordinate ring of the mirror degeneration along with a canonical basis. The appearance of a canonical basis of sections for certain degenerations points towards a good compactification of moduli of certain polarized varieties via stable pairs, generalizing the picture for K3 surfaces [Gross, Hacking, Keel, and Siebert, Theta functions and K3 surfaces (In preparation)]. Another possible application apart from mirror symmetry may be to geometric quantization of varieties with effective anti-canonical class"-- Provided by publisher.

Contents:

Cover

Title page

Introduction

Chapter 1. The affine geometry of the construction

1.1. Polyhedral affine pseudomanifolds

1.2. Convex, piecewise affine functions

Chapter 2. Wall structures

2.1. Construction of ₀

2.2. Monomials, rings and gluing morphisms

2.3. Walls and consistency

2.4. Construction of \foX^{∘}

Chapter 3. Broken lines and canonical global functions

3.1. Broken lines

3.2. Consistency and rings in codimension two

3.3. The canonical global functions _{ }

3.4. The conical case

3.5. The multiplicative structure

Chapter 4. The projective case -theta functions

4.1. Conical affine structures

4.2. The cone over a polyhedral pseudomanifold

4.3. Theta functions and the Main Theorem

4.4. The action of the relative torus

4.5. Jagged paths

Chapter 5. Additional parameters

5.1. Twisting the construction

5.2. Twisting by gluing data

Chapter 6. Abelian varieties and other examples

Appendix A. The GS case

A.1. One-parameter families

A.2. The universal formulation

A.3. Equivariance

A.4. The non-simple case in two dimensions

Bibliography

Back Cover.

Notes:

Description based on publisher supplied metadata and other sources.

Other Format:

Print version: Gross, Mark Theta Functions on Varieties with Effective Anti-Canonical Class

ISBN:

9781470471675

1470471671

OCLC:

1336954643