cyjax.donaldson.AlgebraicMetric#

class cyjax.donaldson.AlgebraicMetric(variety, sections)#

Bases: object

__init__(variety, sections)#

Algebraic metric object.

This wraps the variety and linen bundle sections objects and provides functions to compute geometric objects based on the algebraic ansatz.

All functions can be jit-compiled and automatically handle any combination of batch dimensions for the inputs.

Methods

__init__(variety, sections)

Algebraic metric object.

donaldson_step(key, h, params, vol_cy, ...)

Single step of Donaldson's algorithm.

eta(h, zs, params[, patch, dependent, grad_def])

Compute the eta ratio.

kahler_potential(h, zs, params[, patch, ...])

Compute the Kahler potential.

metric(h, zs, params[, patch, dependent, ...])

Compute the metric in terms of local coordinates.

ricci(h, zs, params[, patch, dependent, ...])

Compute the Ricci curvature tensor in local coordinates.

ricci_scalar(h, zs, params[, patch, ...])

Compute the Ricci scalar.

sigma_accuracy(key, params, h, count)

The \(\sigma\) accuracy measure.

Attributes

degree

Degree of algebraic variety.

variety

Complex projective variety.

sections

Choice of line bundle sections.
property degree: int#

Degree of algebraic variety.

donaldson_step(key, h, params, vol_cy, batches, batch_size)#

Single step of Donaldson’s algorithm.

Parameters:

  • key – Random key for point-sampling used in MC integration.

  • h – Hermitian matrix.

  • params – Complex moduli parameters of variety.

  • vol_cy – Volume of the variety using the canonical volume form.

  • batches – Number of batches used in the Monte Carlo integration.

  • batch_size – Number of points sampled in each batch of the Monte Carlo integration.

Returns:

New value of the H matrix.

eta(h, zs, params, patch=None, dependent=None, grad_def=None)#

Compute the eta ratio.

Parameters:

  • h – Hermitian matrix.

  • zs – affine or homogeneous coordinates.

  • params – Complex moduli parameters of variety.

  • patch – Affine patch index for affine coordinates.

  • dependent – Optional, dependent coordinate index.

  • grad_def – Optional, gradient of defining polynomial.

Returns:

The eta ratio evaluated at given point(s).

kahler_potential(h, zs, params, patch=None, dependent=None, grad_def=None)#

Compute the Kahler potential.

Parameters:

  • h – Hermitian matrix.

  • zs – affine or homogeneous coordinates.

  • params – Complex moduli parameters of variety.

  • patch – Affine patch index for affine coordinates.

  • dependent – Optional, dependent coordinate index.

  • grad_def – Optional, gradient of defining polynomial.

Returns:

Kahler potential evaluated at given point(s).

metric(h, zs, params, patch=None, dependent=None, grad_def=None)#

Compute the metric in terms of local coordinates.

Parameters:

  • h – Hermitian matrix.

  • zs – affine or homogeneous coordinates.

  • params – Complex moduli parameters of variety.

  • patch – Affine patch index for affine coordinates.

  • dependent – Optional, dependent coordinate index.

  • grad_def – Optional, gradient of defining polynomial.

Returns:

Metric, affine patch index, dependent coordinate index.

ricci(h, zs, params, patch=None, dependent=None, grad_def=None)#

Compute the Ricci curvature tensor in local coordinates.

Parameters:

  • h – Hermitian matrix.

  • zs – affine or homogeneous coordinates.

  • params – Complex moduli parameters of variety.

  • patch – Affine patch index for affine coordinates.

  • dependent – Optional, dependent coordinate index.

  • grad_def – Optional, gradient of defining polynomial.

Returns:

Ricci curvature tensor, affine patch index, dependent coordinate index.

ricci_scalar(h, zs, params, patch=None, dependent=None, grad_def=None)#

Compute the Ricci scalar.

Parameters:

  • h – Hermitian matrix.

  • zs – affine or homogeneous coordinates.

  • params – Complex moduli parameters of variety.

  • patch – Affine patch index for affine coordinates.

  • dependent – Optional, dependent coordinate index.

  • grad_def – Optional, gradient of defining polynomial.

Returns:

Ricci scalar evaluated at given point(s).

sections: LBSections#

Choice of line bundle sections.

sigma_accuracy(key, params, h, count)#

The \(\sigma\) accuracy measure.

The \(\sigma\) measure is the integral of \(|1-\eta|\) over the manifold with respect to the holomorphic volume form.

Parameters:

  • key – Random key for point-sampling used in MC integration.

  • h – Hermitian matrix.

  • params – Complex moduli parameters of variety.

  • count – Number of points used in Monte Carlo approximation of the integral.

Returns:

The \(sigma\) accuracy measure.

variety: VarietySingle#

Complex projective variety.