lefschetz-family 0.1.6

This package implements algorithms relying on Lefschetz fibration theory to compute periods of algebraic varieties.

lefschetz-family

Description

This Sage package provides a means of efficiently computing periods of complex projective hypersurfaces and elliptic surfaces over $\mathbb P^1$ with certified rigorous precision bounds. It implements the methods described in https://arxiv.org/abs/2306.05263. Here is a runtime benchmark for various examples:

Variety (generic)	Time (on 4 M1 cores)
Elliptic curve	8 seconds
Quartic curve	4 minutes
Cubic surface	4 minutes 20
Quartic surface	1 hour (for holomorphic periods)
Cubic threefold	1 hour (for holomorphic periods)

How to install

Follow the installation procedure of the numperiods package, then run

sage -pip install lefschetz-family

or

sage -pip install --user lefschetz-family

Requirements

Sage 9.0 and above is recommended. Furthermore, this package has the following dependencies:

• Ore Algebra.
• numperiods.
• The delaunay-triangulation package from PyPI.

Usage

Hypersurface

The first step is to define the polynomial $P$ defining the projective hypersurface $X=V(P)$. For instance, the following gives the Fermat elliptic curve:

R.<X,Y,Z> = PolynomialRing(QQ)
P = X**3+Y**3+Z**3

Then the following creates an object representing the hypersurface:

from period import LefschetzFamily
X = Hypersurface(P)

The period matrix of $X$ is the simply given by:

X.period_matrix

See the computation of the periods of the Fermat quartic surface for a usage example.

See the computation of the periods of the Fermat quartic surface for a usage example.

Options

The object Hypersurface can be called with several options:

• method ("voronoi" by default/"delaunay"/"delaunay_dual"): the method used for computing a basis of homotopy. voronoi uses integration along paths in the voronoi graph of the critical points; delaunay uses integration along paths along the delaunay triangulation of the critical points; delaunay_dual paths are along the segments connecting the barycenter of a triangle of the Delaunay triangulation to the middle of one of its edges. In practice, delaunay is more efficient for low dimension and low order varieties (such as degree 3 curves and surfaces, and degree 4 curves). This gain in performance is however hindered in higher dimensions because of the algebraic complexity of the critical points (which are defined as roots of high order polynomials, with very large integer coefficients). "delaunay" method is not working for now
• nbits (positive integer, 400 by default): the number of bits of precision used as input for the computations. If a computation fails to recover the integral monodromy matrices, you should try to increase this precision. The output precision seems to be roughly linear with respect to the input precision.
• debug (boolean, False by default): whether coherence checks should be done earlier rather than late. We recommend setting to true only if the computation failed in normal mode.

Properties

The object Hypersurface has several properties. Fibration related properties, in positive dimension:

• fibration: a list of independant hyperplanes defining the iterative pencils. The first two element of the list generate the pencil used for the fibration.
• critical_values: the list critical values of that map.
• basepoint: the basepoint of the fibration (i.e. a non critical value).
• fiber: the fiber above the basepoint as a Hypersurface object.
• fundamental_group: the class computing representants of the fundamental group of $\mathbb P^1$ punctured at the critical values.
• paths: the list of simple loops around each point of critical_values. When this is called, the ordering of critical_values changes so that the composition of these loops is the loop around infinity.
• family: the one parameter family corresponding to the fibration.

Homology related properties:

• monodromy_matrices: the matrices of the monodromy action of paths on $H_{n-1}(X_b)$.
• vanishing_cycles: the vanshing cycles at each point of critical_values along paths.
• thimbles: the thimbles of $H_n(Y,Y_b)$. They are represented by a starting cycle in $H_n(Y_b)$ and a loop in $\mathbb C$ avoiding critical_values and pointed at basepoint.
• kernel_boundary: linear combinations of thimbles with empty boundary.
• extensions: integer linear combinations of thimbles with vanishing boundary.
• infinity_loops: extensions around the loop at infinity.
• homology_modification: a basis of $H_n(Y)$.
• intersection_product_modification: the intersection product of $H_n(Y)$.
• fibre_class: the class of the fibre in $H_n(Y)$.
• section: the class of a section in $H_n(Y)$.
• thimble_extensions: couples (t, T) such that T is the homology class in $H_n(Y)$ representing the extension of a thimble $\Delta \in H_{n-1}(X_b, X_{bb'})$ over all of $\mathbb P^1$, with $\delta\Delta =$t. Futhermore, the ts define a basis of the boundary map $\delta$.
• invariant: the intersection of section with the fibre above the basepoint, as a cycle in $H_{n-2}({X_b}_{b'})$.
• exceptional_divisors: the exceptional cycles coming from the modification $Y\to X$, given in the basis homology_modification.
• homology: a basis of $H_n(X)$, given as its embedding in $H_2(Y)$.
• intersection_product: the intersection product of $H_n(X)$.
• lift: a map taking a linear combination of thimbles with zero boundary (i.e. an element of $\ker\left(\delta:H_n(Y, Y_b)\to H_{n-1}(Y_b)\right)$) and returning the homology class of its lift in $H_2(Y)$, in the basis homology_modification.
• lift_modification: a map taking an element of $H_n(Y)$ given by its coordinates in homology_modification, and returning its homology class in $H_n(X)$ in the basis homology.

Cohomology related properties:

• cohomology: a basis of $PH^n(X)$, represented by the numerators of the rational fractions.
• holomorphic_forms: the indices of the forms in cohomology that form a basis of holomorphic forms.
• picard_fuchs_equation(i): the picard fuchs equation of the parametrization of i-th element of cohomology by the fibration

Period related properties

• period_matrix: the period matrix of $X$ in the aforementioned bases homology and cohomology
• period__matrix_modification: the period matrix of the modification $Y$ in the aforementioned bases homology_modification and cohomology
• holomorphic_periods: the periods of holomorphic_forms in the basis homology.
• holomorphic_periods: the periods of the pushforwards of holomorphic_forms in the basis homology_modification.

Miscellaneous properties:

• P: the defining equation of $X$.
• dim: the dimension of $X$.
• degree: the degree of $X$.
• ctx: the options of $X$, see related section above.

The computation of the exceptional divisors can be costly, and is not always necessary. For example, the Picard rank of a quartic surface can be recovered with holomorphic_periods_modification.

EllipticSurface

The defining equation for the elliptic surface should be given as a univariate polynomial over a trivariate polynomial ring. The coefficients should be homogeneous of degree $3$.

R.<X,Y,Z> = PolynomialRing(QQ)
S.<t> = PolynomialRing(R)
P = X^2*Y+Y^2*Z+Z^2*X+t*X*Y*Z 

Then the following creates an object representing the hypersurface:

from period import LefschetzFamily
X = EllipticSurface(P)

Options

The options are the same as those for Hypersurface (see above).

Properties

The object ElliptcSurface has several properties. Fibration related properties, in positive dimension:

• fibration: the two linear maps defining the map $X\dashrightarrow \mathbb P^1$.
• critical_values: the list critical values of that map.
• basepoint: the basepoint of the fibration (i.e. a non critical value).
• fiber: the fiber above the basepoint as a LefschetzFamily object.
• paths: the list of simple loops around each point of critical_points. When this is called, the ordering of critical_points changes so that the composition of these loops is the loop around infinity.
• family: the one parameter family corresponding to the fibration.

Homology related properties:

• extensions: the extensions of the fibration.
• extensions_morsification: the extensions of the morsification of the fibration.
• homology: the homology of $X$.
• singular_components: a list of lists of combinations of thimbles of the morsification, such that the elements of singular_components[i] form a basis of the singular components of the fibre above critical_values[i]. To get their coordinates in the basis homology, use X.lift(X.singular_components[i][j]).
• fibre_class: the class of the fibre in homology.
• section: the class of the zero section in homology.
• intersection_product: the intersection matrix of the surface in the basis homology.
• morsify: a map taking a combination of extensions and returning its coordinates on the basis of thimbles of the morsification.
• lift: a map taking a combination of thimbles of the morsification with empty boundary and returning its class in homology.

Cohomology related properties:

• holomorphic_forms: a basis of rational functions $f(t)$ such that $f(t) \operatorname{Res}\frac{\Omega_2}{P_t}\wedge\mathrm dt$ is a holomorphic form of $S$.
• picard_fuchs_equations: the list of the Picard-Fuchs equations of the holomorphic forms mentionned previously.

Period related properties:

• period_matrix: the holomorphic periods of $X$ in the bases self.homology and self.holomorphic_forms.
• effective_periods: the holomorphic periods $X$ in the bases self.effective_lattice and self.holomorphic_forms

Lattices. Unless stated otherwise, lattices are given the coordinates of a basis in the basis homology.

• effective_lattice: The lattice of effective cycles of $X$, consisting of the concatenation of extensions, singular_components, fibre_class and section.
• neron_severi: the Néron-Severi group of $X$.
• trivial: the trivial lattice.
• essential_lattice: the essential lattice.
• mordell_weil: the Mordell-Weil group of $X$, described as the quotient module neron_severi/trivial.
• mordell_weil_lattice: the intersection matrix of the Mordell-Weil lattice of $X$.

Miscellaneous properties:

• dim: the dimension of $X$.
• ctx: the options of $X$, see related section above.

Contact

For any question, bug or remark, please contact [e-mail].

Roadmap

Near future milestones:

• Encapsulate integration step in its own class
• Certified computation of the exceptional divisors
• Making Delaunay triangulation functional again
• Saving time on differential operator by precomputing cache before parallelization

Middle term goals include:

• Having own implementation of 2D voronoi graphs/Delaunay triangulation

Long term goals include:

• Computing periods of elliptic fibrations.
• Tackling higher dimensional varieties (most notably cubics in $\mathbb P^5$).
• Computing periods of singular varieties.
• Computing periods of complete intersections.
• Computing periods of weighted projective hypersurfaces, notably double covers of $\mathbb P^2$ ramified along a cubic.

Other directions include:

• Computation of homology through braid groups instead of monodromy of differential operators.

Project status

This project is actively being developped.