Algebraic ansatz

The primary quantity we try to learn is the Hermitian matrix \(H\) in the following algebraic ansatz for the Kähler potential:

\[ K(z, \bar{z}) = \frac{1}{\pi k} \log \left( \sum_{\alpha\bar{\beta}} s_\alpha(z) H^{\alpha \bar{\beta}} s_{\bar{\beta}}(\bar{z}) \right) \,. \]

One has to choose what set of polynomials \(s_\alpha\) to use. For Donaldson’s algorithm, these have to form a basis on the variety. For machine learning algorithms, it is also possible to use the full set of monomials of some chosen degree in ambient projective space. A basic algorithm to obtain a basis is implemented.

To add a numerically more advantageous choices, LBSections can be subclassed.

donaldson.LBSections(degree)	General structure representing set of line bundle sections.
donaldson.MonomialBasisFull(dim_proj, degree)
donaldson.MonomialBasisReduced(variety, ...)

The components required to compute the metric, the Ricci curvature, etc. are the variety (containing the defining polynomial) and the choice of sections. These are stored together in the AlgebraicMetric object, which has methods to compute these quantities.

donaldson.AlgebraicMetric(variety, sections)
donaldson.AlgebraicMetric.eta(h, zs, params)	Compute the eta ratio.
donaldson.AlgebraicMetric.sigma_accuracy(...)	The \(\sigma\) accuracy measure.
donaldson.AlgebraicMetric.ricci(h, zs, params)	Compute the Ricci curvature tensor in local coordinates.
donaldson.AlgebraicMetric.ricci_scalar(h, ...)	Compute the Ricci scalar.

Internal: geometric objects

Internally, the computation of geometric objects is contained in a computational graph. This avoids duplicate code and allows easier testing of intermediate values.

donaldson.GeometricObjects(h, zs, params, ...)

Below is a summary of quantities which can be computed, for reference.