arXiv subjects

Grigory Mikhalkin

Publications and source records attributed to Grigory Mikhalkin.

At least 19 recordsLinked to original sources

arXiv2024

Wave fronts and caustics in the tropical plane

The paper studies intrinsic geometry in the tropical plane. Tropical structure in the real affine $n$-space is determined by the integer tangent vectors. Tropical isomorphisms are affine transformations preserving the integer lattice of the tangent space, they may be identified with the group $\operatorname{GL_n}(\mathbb{Z})$ extended by arbitrary real translations. This geometric structure allows one to define wave front propagation for boundaries of convex domains. Interestingly enough, an arbitrary compact convex domain in the tropical plane evolves to a finite polygon after an arbitrarily small time. The caustic of a wave front evolution is a tropical analytic curve. The paper studies geometry of the tropical wave fronts and caustics. In particular, we relate the caustic of a tropical angle to the continued fraction expression of its slope, and treat it as a tropical trigonometry notion.

math.AG↗

arXiv2023

Ellipsoidal superpotentials and stationary descendants

We compute stationary gravitational descendants in symplectic ellipsoids of any dimension, and use these to derive a number of new recursive formula for punctured curve counts in symplectic manifolds with ellipsoidal ends. Along the way we develop a framework in which punctured curve counts can be explicitly computed using the standard complex structure on affine space. Finally, we initiate the study of "infinitesimal symplectic cobordisms", which serve as elementary building blocks for symplectic cobordisms between ellipsoids.

math.SG↗

arXiv2022

Singular symplectic spaces and holomorphic membranes

We set up a topological framework for degenerations of symplectic manifolds into singular spaces paying a special attention to the behavior of Lagrangian manifolds and their (holomorphic) membranes. We show that degenerations into singular toric varieties provide a source of exotic Lagrangian tori.

math.SG↗

arXiv2021

Spines for amoebas of rational curves

To every rational complex curve $C \subset (\mathbf{C}^\times)^n$ we associate a rational tropical curve $Γ\subset \mathbf{R}^n$ so that the amoeba $\mathcal{A}(C) \subset \mathbf{R}^n$ of $C$ is within a bounded distance from $Γ$. In accordance with the terminology introduced by Passare and Rullgård, we call $Γ$ the spine of $\mathcal{A}(C)$. We use spines to describe tropical limits of sequences of rational complex curves.

math.AG↗

arXiv2021

Area in real K3-surfaces

For a real K3-surface $X$, one can introduce areas of connected components of the real point set $\mathbb{R} X$ of $X$ using a holomorphic symplectic form of $X$. These areas are defined up to simultaneous multiplication by a positive real number, so the areas of different components can be compared. In particular, it turns out that the area of a non-spherical component of $\mathbb{R} X$ is always greater than the area of any spherical component. In this paper we explore further comparative restrictions on the area for real K3-surfaces admitting a suitable polarization of degree $2g - 2$ (where $g$ is a positive integer) and such that $\mathbb{R} X$ has one non-spherical component and at least $g$ spherical components. For this purpose we introduce and study the notion of simple Harnack curves in real K3-surfaces, generalizing planar simple Harnack curves.

math.AG↗

arXiv2021

Non-commutative amoebas

The group of isometries of the hyperbolic 3-space is one of the simplest non-commutative complex Lie groups. Its quotient by the maximal compact subgroup naturally maps it back to the hyperbolic space. Each fiber of this map is diffeomorphic to the real projective 3-space. The resulting map can be viewed as the simplest non-commutative counterpart of the amoeba map introduced, in the commutative setting, by Gelfand, Kapranov and Zelevinsky. The paper surveys basic properties of the non-commutative amoebas and compares them against their commutative counterparts.

math.CV↗

arXiv2020

Tropical Homology

Given a tropical variety X and two non-negative integers p and q we define homology group $H_{p,q}(X)$. We show that if X is a smooth tropical variety that can be represented as the tropical limit of a 1-parameter family of complex projective varieties, then $\dim H_{p,q}(X)$ coincides with the Hodge number $h^{p,q}$ of a general member of the family.

math.AG↗

arXiv2019

Floor decompositions of tropical curves : the planar case

In a previous paper, we announced a formula to compute Gromov-Witten and Welschinger invariants of some toric varieties, in terms of combinatorial objects called floor diagrams. We give here detailed proofs in the tropical geometry framework, in the case when the ambient variety is a complex surface, and give some examples of computations using floor diagrams. The focusing on dimension 2 is motivated by the special combinatoric of floor diagrams compared to arbitrary dimension. We treat a general toric surface case in this dimension: the curve is given by an arbitrary lattice polygon and include computation of Welschinger invariants with pairs of conjugate points. See also \cite{FM} for combinatorial treatment of floor diagrams in the projective case.

math.AG↗

arXiv2019

On osculating framing of real algebraic links

For a real algebraic link in $RP^3$, we prove that its encomplexed writhe (an invariant introduced by Viro) is maximal for a given degree and genus if and only if its self-linking number with respect to the framing by the osculating planes is maximal for a given degree.

math.AG↗

arXiv2019

Rigid isotopy of maximally writhed links

This is a sequel to the paper \cite{MO-mw} which identified maximally writhed algebraic links in $\rp^3$ and classified them topologically. In this paper we prove that all maximally writhed links of the same topological type are rigidly isotopic, i.e. one can be deformed into another with a family of smooth real algebraic links of the same degree.

math.AG↗

arXiv2019

Examples of tropical-to-Lagrangian correspondence

The paper associates Lagrangian submanifolds in symplectic toric varieties to certain tropical curves inside the convex polyhedral domains of $\R^n$ that appear as the images of the moment map of the toric varieties. We pay a particular attention to the case $n=2$, where we reprove Givental's theorem on Lagrangian embeddability of non-oriented surfaces to $\C^2$, as well as to the case $n=3$, where we see appearance of the graph 3-manifolds studied by Waldhausen as Lagrangian submanifolds. In particular, rational tropical curves in $\R^3$ produce 3-dimensional rational homology spheres. The order of their first homology groups is determined by the multiplicity of tropical curves in the corresponding enumerative problems.

math.SG↗

arXiv2017

Topology of maximally writhed real algebraic knots

Oleg Viro introduced an invariant of rigid isotopy for real algebraic knots in $RP^3$ which can be viewed as a first order Vassiliev invariant. In this paper we look at real algebraic knots of degree $d$ with the maximal possible value of this invariant. We show that for a given $d$ all such knots are topologically isotopic and explicitly identify their knot type.

math.AG↗

arXiv2017

Tropical limit of log-inflection points for planar curves

The paper describes behavior of log-inflection points of curves in $(\mathbb{C}^*)^2$ under passing to the tropical limit. We show that such points accumulate by pairs at the midpoints of bounded edges in the limiting tropical curve. Log-inflection points are points of inflection with respect to the parallelization of $(\mathbb{C}^*)^2$ given by the multiplicative group law.

math.AG↗

arXiv2017

Quantum indices and refined enumeration of real plane curves

We associate a half-integer number, called {\em the quantum index}, to algebraic curves in the real plane satisfying to certain conditions. The area encompassed by the logarithmic image of such curves is equal to $π^2$ times the quantum index of the curve and thus has a discrete spectrum of values. We use the quantum index to refine real enumerative geometry in a way consistent with the Block-Göttsche invariants from tropical enumerative geometry.

math.AG↗

arXiv2017

Maximally writhed real algebraic links

Oleg Viro introduced an invariant of rigid isotopy for real algebraic knots and links in $\Bbb{RP}^3$ which can be viewed as a first order Vassiliev invariant. In this paper we classify real algebraic links of degree $d$ with the maximal value of this invariant in its two versions: $w$ and $w_λ$.

math.AG↗

arXiv2017

Non-existence of torically maximal hypersurfaces

Torically maximal curves (known also as simple Harnack curves) are real algebraic curves in the projective plane such that their logarithmic Gauß map is totally real. In this paper we show that hyperplanes in projective spaces are the only torically maximal hypersurfaces of higher dimensions.

math.AG↗

arXiv2016

Real algebraic knots and links of small degree

The paper gives topological as well as rigid isotopy classification of smooth irreducible algebraic curves in the real projective 3-space for the case when the degree of the curve is at most six and its genus is at most one.

math.AG↗

arXiv2016

Rational quintics in the real plane

From a topological viewpoint, a rational curve in the real projective plane is generically a smoothly immersed circle and a finite collection of isolated points. We give an isotopy classification of generic rational quintics in $\mathbb{RP}^2$ in the spirit of Hilbert's 16th problem.

math.AG↗