Complex Algebraic Geometry Research Articles

Some interesting birational morphisms of smooth affine quadric 3-folds *

We study a family of birational maps of smooth affine quadric 3-folds, over the complex numbers, of the form x1x4−x2x3= constant, which seems to have some (among many others) interesting/unexpected characters: (a) they are cohomologically hyperbolic, (b) their second dynamical degree is an algebraic number but not an algebraic integer, and (c) the logarithmic growth of their periodic points is strictly smaller than their algebraic entropy. These maps are restrictions of a polynomial map on C4 preserving each of the quadrics. The study in this paper is a mixture of rigorous and experimental ones, where for the experimental study we rely on Bertini which is a reliable and fast software for expensive numerical calculations in complex algebraic geometry.

Hurwitz moduli varieties parameterizing Galois covers of an algebraic curve

Given a smooth, projective curve \(Y\), a finite group \(G\) and a positive integer $n$ we study smooth, proper families \(X\to Y\times S\to S\) of Galois covers of \(Y\) with Galois group isomorphic to $G$ branched in \(n\) points, parameterized by algebraic varieties \(S\). When \(G\) is with trivial center we prove that the Hurwitz space \(H^G_n(Y)\) is a fine moduli variety for this moduli problem and construct explicitly the universal family. For arbitrary \(G\) we prove that \(H^G_n(Y)\) is a coarse moduli variety. For families of pointed Galois covers of \((Y,y_0)\) we prove that the Hurwitz space \(H^G_n(Y,y_0)\) is a fine moduli variety, and construct explicitly the universal family, for arbitrary group \(G\). We use classical tools of algebraic topology and of complex algebraic geometry.

Asymptotics of multivariate sequences in the presence of a lacuna

We explain a discontinuous drop in the exponential growth rate for certain multivariate generating functions at a critical parameter value in even dimensions d \geq 4 . This result depends on computations in the homology of the algebraic variety where the generating function has a pole. These computations are similar to, and inspired by, a thread of research in applications of complex algebraic geometry to hyperbolic PDEs, going back to Leray, Petrowski, Atiyah, Bott and Gårding. As a consequence, we give a topological explanation for certain asymptotic phenomena appearing in the combinatorics and number theory literature. Furthermore, we show how to combine topological methods with symbolic algebraic computation to determine explicitly the dominant asymptotics for such multivariate generating functions, giving a significant new tool to attack the so-called connection problem for asymptotics of P-recursive sequences. This in turn enables the rigorous determination of integer coefficients in the Morse–Smale complex, which are difficult to determine using direct geometric methods.

Formality and finiteness in rational homotopy theory

We explore various formality and finiteness properties in the differential graded algebra models for the Sullivan algebra of piecewise polynomial rational forms on a space. The 1 -formality property of the space may be reinterpreted in terms of the filtered and graded formality properties of the Malcev Lie algebra of its fundamental group, while some of the finiteness properties of the space are mirrored in the finiteness properties of algebraic models associated with it. In turn, the formality and finiteness properties of algebraic models have strong implications on the geometry of the cohomology jump loci of the space. We illustrate the theory with examples drawn from complex algebraic geometry, actions of compact Lie groups, and 3 -dimensional manifolds.

New Directions in Real Algebraic Geometry

This workshop explored the forefront of connections of real algebraic geometry with convex analysis, combinatorics, and computational complexity. Important aspects have been promising interactions with the fields of quantum information theory, discrete geometry, complex and random algebraic geometry.

A path forward: Tropicalization in extremal combinatorics

Many important problems in extremal combinatorics can be stated as proving a pure binomial inequality in graph homomorphism numbers, i.e., proving that hom(H1,G)a1⋯hom(Hk,G)ak≥hom(Hk+1,G)ak+1⋯hom(Hm,G)am holds for some fixed graphs H1,…,Hm and all graphs G. One prominent example is Sidorenko's conjecture. For a fixed collection of graphs U={H1,…,Hm}, the exponent vectors of valid pure binomial inequalities in graphs of U form a convex cone. We compute this cone for several families of graphs including complete graphs, even cycles, stars and paths; the latter is the most interesting and intricate case that we compute. In all of these cases, we observe a tantalizing polyhedrality phenomenon: the cone of valid pure binomial inequalities is actually rational polyhedral, and therefore all valid pure binomial inequalities can be generated from the finite collection of exponent vectors of the extreme rays. Using the work of Kopparty and Rossman ([17]), we show that the cone of valid inequalities is indeed rational polyhedral when all graphs Hi are series-parallel and chordal, and we conjecture that polyhedrality holds for any finite collection U. We demonstrate that the polyhedrality phenomenon also occurs in matroids and simplicial complexes. Our description of the inequalities for paths involves a generalization of the Erdős-Simonovits conjecture recently proved in its original form in [29] and a new family of inequalities not observed previously. We also solve an open problem of Kopparty and Rossman on the homomorphism domination exponent of paths. One of our main tools is tropicalization, a well-known technique in complex algebraic geometry, first applied in extremal combinatorics in [3]. We prove several results about tropicalizations which may be of independent interest.

Riemann-Roch theorem and Kodaira-Serre duality

Abstract The Riemann-Roch theorem is of utmost importance and a vital tool to the fields of complex analysis and algebraic geometry, specifically in the algebraic geometric theory of compact Riemann surfaces. It tells us how many linearly independent meromorphic functions there are having certain restrictions on their poles. The aim of this paper is to give two proofs of this important theorem and explore some of its numerous consequences. As an application, we compute the genus of some interesting algebraic curves or Riemann surfaces.

Discovering Geometric Inequalities: The Concourse of GeoGebra Discovery, Dynamic Coloring and Maple Tools

Recently developed GeoGebra tools for the automated deduction and discovery of geometric statements combine in a unique way computational (real and complex) algebraic geometry algorithms and graphic features for the introduction and visualization of geometric statements. In our paper we will explore the capabilities and limitations of these new tools, through the case study of a classic geometric inequality, showing how to overcome, by means of a double approach, the difficulties that might arise attempting to ‘discover’ it automatically. On the one hand, through the introduction of the dynamic color scanning method, which allows to visualize on GeoGebra the set of real solutions of a given equation and to shed light on its geometry. On the other hand, via a symbolic computation approach which currently requires the (tricky) use of a variety of real geometry concepts (determining the real roots of a bivariate polynomial p(x,y) by reducing it to a univariate case through discriminants and Sturm sequences, etc.), which leads to a complete resolution of the initial problem. As the algorithmic basis for both instruments (scanning, real solving) are already internally available in GeoGebra (e.g., via the Tarski package), we conclude proposing the development and merging of such features in the future progress of GeoGebra automated reasoning tools.

Komplexe Analysis - Algebraicity and Transcendence

This is the report of the Oberwolfach workshop Komplexe Analysis 2020. It was mainly devoted to the transcendental methods of complex algebraic geometry and featured eighteen talks about recent important developments in Hodge theory, moduli spaces, hyperbolicity, Fano varieties, algebraic foliations, algebraicity theorems for subvarieties and their applications to transcendance proofs for numbers. Two talks were more algebraic in nature and devoted to non-commutative deformations and syzygies of secant varieties.

Characterization of quantum entanglement via a hypercube of Segre embeddings

A particularly simple description of separability of quantum states arises naturally in the setting of complex algebraic geometry, via the Segre embedding. This is a map describing how to take products of projective Hilbert spaces. In this paper, we show that for pure states of n particles, the corresponding Segre embedding may be described by means of a directed hypercube of dimension $$(n-1)$$ , where all edges are bipartite-type Segre maps. Moreover, we describe the image of the original Segre map via the intersections of images of the $$(n-1)$$ edges whose target is the last vertex of the hypercube. This purely algebraic result is then transferred to physics. For each of the last edges of the Segre hypercube, we introduce an observable which measures geometric separability and is related to the trace of the squared reduced density matrix. As a consequence, the hypercube approach allows to measure entanglement, naturally relating bipartitions with q-partitions for any $$q\ge 1$$ . We test our observables against well-known states, showing that these provide well-behaved and fine measures of entanglement.

Sums of Squares: A Real Projective Story

The relationship between nonnegative polynomials and sums of squares is one of the central questions in real algebraic geometry. A modern approach is to look at nonnegative polynomials and sums of squares on a real variety. We survey the rich history of this area and focus on recent developments highlighting unexpected links to complex algebraic geometry and commutative algebra.

An arithmetic count of the lines on a smooth cubic surface

We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field$k$, generalizing the counts that over${\mathbf {C}}$there are$27$lines, and over${\mathbf {R}}$the number of hyperbolic lines minus the number of elliptic lines is$3$. In general, the lines are defined over a field extension$L$and have an associated arithmetic type$\alpha$in$L^*/(L^*)^2$. There is an equality in the Grothendieck–Witt group$\operatorname {GW}(k)$of$k$,\[ \sum_{\text{lines}} \operatorname{Tr}_{L/k} \langle \alpha \rangle = 15 \cdot \langle 1 \rangle + 12 \cdot \langle -1 \rangle, \]where$\operatorname {Tr}_{L/k}$denotes the trace$\operatorname {GW}(L) \to \operatorname {GW}(k)$. Taking the rank and signature recovers the results over${\mathbf {C}}$and${\mathbf {R}}$. To do this, we develop an elementary theory of the Euler number in$\mathbf {A}^1$-homotopy theory for algebraic vector bundles. We expect that further arithmetic counts generalizing enumerative results in complex and real algebraic geometry can be obtained with similar methods.

Topics on the Geometry of Rational Homogeneous Spaces

This is a survey paper about a selection of results in complex algebraic geometry that appeared in the recent and less recent litterature, and in which rational homogeneous spaces play a prominent role. This selection is largely arbitrary and mainly reflects the interests of the author.

Singularities of Hermitian–Yang–Mills connections and Harder–Narasimhan–Seshadri filtrations

This is the first of a series of papers where we relate tangent cones of Hermitian-Yang-Mills connections at an isolated singularity to the complex algebraic geometry of the underlying reflexive sheaf, when the sheaf is locally modelled on the pull-back of a holomorphic vector bundle from the projective space. In this paper we shall impose an extra assumption that the graded sheaf determined by the Harder-Narasimhan-Seshadri filtrations of the vector bundle is reflexive. In general we conjecture that the tangent cone is uniquely determined by the double dual of the associated graded object of a Harder-Narasimhan-Seshadri filtration of an algebraic tangent cone, which is a certain torsion-free sheaf on the projective space. In this paper we also prove this conjecture when there is an algebraic tangent cone which is locally free and stable.

Level curves of rational functions and unimodular points on rational curves

We obtain an improvement and broad generalisation of a result of N. Ailon and Z. Rudnick (2004) on common zeros of shifted powers of polynomials. Our approach is based on reducing this question to a more general question of counting intersections of level curves of complex functions. We treat this question via classical tools of complex analysis and algebraic geometry.

Automorphisms of surfaces: Kummer rigidity and measure of maximal entropy

We classify complex projective surfaces X with an automorphism f of positive entropy for which the unique measure of maximal entropy is absolutely continuous with respect to the Lebesgue measure. As a byproduct, if X is a K3 surface and is not a Kummer surface, the periodic points of f are equidistributed with respect to a probability measure which is singular with respect to the canonical volume of X . The proof is based on complex algebraic geometry and Hodge theory, Pesin’s theory and renormalization techniques. A crucial argument relies on a new compactness property of entire curves parametrizing the invariant manifolds of the automorphism.

Aspherical 4-manifolds of odd Euler characteristic

An explicit construction of closed, orientable, smooth, aspherical 4-manifolds with any odd Euler characteristic greater than 12 is presented. The constructed manifolds are all Haken manifolds in the sense of B. Foozwell and H. Rubinstein and can be systematically reduced to balls by suitably cutting them open along essential codimension-one submanifolds. Euler characteristics divisible by 3 are known to arise from complex algebraic geometry considerations. Examples with Euler characteristic 1, 5, 7, or 11 appear to be unknown.

Generic Finiteness for a Class of Symmetric Planar Central Configurations of the Six-Body Problem and the Six-Vortex Problem

A symmetric planar central configuration of the Newtonian six-body problem $x$ is called cross central configuration if there are precisely four bodies on a symmetry line of $x$. We use complex algebraic geometry and Groebner basis theory to prove that for a generic choice of positive real masses $m_1,m_2,m_3,m_4,m_5=m_6$ there is a finite number of cross central configurations. We also show one explicit example of a configuration in this class. A part of our approach is based on relaxing the output of the Groebner basis computations. This procedure allows us to obtain upper bounds for the dimension of an algebraic variety. We get the same results considering cross central configurations of the six-vortex problem.

Three Topological Results on the Twistor Discriminant Locus in the 4-Sphere

We exploit techniques from classical (real and complex) algebraic geometry for the study of the standard twistor fibration $\pi:\mathbb{CP}^{3}\to S^{4}$. We prove three results about the topology of the twistor discriminant locus of an algebraic surface in $\mathbb{CP}^{3}$. First of all we prove that, with the exception of two exceptional cases, the real dimension of the twistor discriminant locus of an algebraic surface is always equal to 2. Secondly we describe the possible intersections of a general surface with the family of twistor lines: we find that only 4 configurations are possible and for each of them we compute the dimension. Lastly we give a decomposition of the twistor discriminant locus of a given cone in terms of its singular locus and its dual variety.

The N-point gravitational lens as cover and his the profile cover

The study of mathematical models of gravitational lenses are not direct observations. A special place in such studies is the visualization of the lens model. The image of the source and its images in the N-point gravitational lens, in the picture plane, visualizes the mathematical model - the algebraic equation of the lens. Recently, the number of studies of the equation of the N-point gravitational lens by algebraic methods has increased [6–8]. Such studies make it possible to consider the gravitational lens not only as an algebraic, but also as a topological object. In the work, the equation of the N-point gravitational lens in the complex form is studied. A bundle above the source plane is assigned to it. We investigated one subfamily of lens equations. A critical set of equations of this subfamily is a closed Jordan curve. To the equations of this subfamily we put in correspondence not only a vector bundle, but also a covering. A method for describing coverings is developed for equations whose caustic in the finite plane is a closed Jordan curve (Jordan caustic). A special case of such coverings is coverings for the equation of an N-point gravitational lens, the critical set of which is a closed Jordan curve. These equations, also, have Jordan caustics. The method is similar to the method for describing Riemann surfaces of algebraic functions, graphs ‒ profiles. The algorithm for constructing coverings and the developed method for describing these coverings illustrates an example of a cover given by a rational non-analytic function of a complex variable The covering surface has not only a Jordan caustic, but also a second-order branch point at an infinitely distant point. The methods of the theory of functions of a complex variable, algebraic geometry, algebraic topology and graph theory are used.