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Takeo Nishinou

Publications and source records attributed to Takeo Nishinou.

16 recordsLinked to original sources
arXiv2023

Deformation of singular curves on surfaces

In this paper, we consider deformations of singular complex curves on complex surfaces. Despite the fundamental nature of the problem, little seems to be known for curves on general surfaces. Let $C\subset S$ be a complete integral curve on a smooth surface. Let $\tilde C$ be a partial normalization of $C$, and $φ\colon \tilde C\to S$ be the induced map. In this paper, we consider deformations of $φ$. The problem of the existence of deformations will be reduced to solving a certain explicit system of polynomial equations. This system is universal in the sense that it is determined solely by simple local data of the singularity of $C$, and does not depend on the global geometry of $C$ or $S$. Under a relatively mild assumption on the properties of these equations, we will show that the map $φ$ has virtually optimal deformation property.

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arXiv2021

Integration of vector fields on cell complexes and Morse theory

In this paper, we investigate vector fields on polyhedral complexes and their associated trajectories. We study vector fields which are analogue of the gradient vector field of a function in the smooth case. Our goal is to define a nice theory of trajectories of such vector fields, so that the set of them captures the topology of the polyhedral complex, as in classical Morse theory. Since we do not assume the polyhedral complex to be a manifold, the definition of vector fields on it is very different from the smooth case. Nevertheless, we will show that there exist nice classes of functions and metrics which give gradient vector fields with desired properties. Our construction relies on Forman's discrete Morse theory. In particular, the class of functions we use is an improvement of Forman's discrete Morse functions. A notable feature of our theory is that our gradient vector fields are defined purely from functions and metrics as in the smooth case, contrary to the case of discrete Morse theory where we need the data of dimension of cells. This allows us to implement several useful constructions which were not available in the discrete case.

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arXiv2020

Deformation of pairs and semiregularity

In this paper, we study relative deformations of maps into a family of Kähler manifolds whose images are divisors. We show that if the map satisfies a condition called semiregularity, then it allows relative deformations if and only if the cycle class of the image remains Hodge in the family. This gives a refinement of the so-called variational Hodge conjecture. We also show that the semiregularity of maps is related to classical notions such as Cayley-Bacharach conditions and d-semistability.

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arXiv2020

Obstructions to deforming maps from curves to surfaces

This paper studies the obstructions to deforming a map from a complex variety to another variety which is an immersion of codimension one. We extend the classical notion of semiregularity of subvarieties to maps between varieties, and show that it largely extends the applicability. Then we apply the main result to several situations. First, we give deformations of non-reduced curves on surfaces in a geometrically controlled way. Also, we give a simple but effective criterion for the vanishing of the obstructions to equisingular deformations of nodal curves on surfaces. Finally, we construct nodal curves with very small geometric genus on surfaces of general type.

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arXiv2020

Realization of tropical curves in abelian surfaces

We construct algebraic curves in abelian surfaces by utilizing tropical curves in real tori. We give a necessary and sufficient condition for a tropical curve in a real torus to be realizable by an algebraic curve in an abelian surface. When this condition is satisfied, the number of algebraic curves can be computed by a combinatorial formula. This provides the algebraic-tropical correspondence theorem for abelian surfaces analogous to Mikhalkin's correspondence theorem for toric surfaces. Thus, the number of algebraic curves passing through generic points in an abelian surface can be computed purely combinatorially via tropical curves.

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arXiv2019

Degeneration and curves on K3 surfac

This paper studies curves on quartic K3 surfaces, or more generally K3 surfaces which are complete intersection in weighted projective spaces. A folklore conjecture concerning rational curves on K3 surfaces states that all K3 surfaces contain infinite number of irreducible rational curves. It is known that all K3 surfaces, except those contained in the countable union of hypersurfaces in the moduli space of K3 surfaces satisfy this property. In this paper we present a new approach for constructing curves on varieties which admit nice degenerations. We apply this technique to the above problem and prove that there is a Zariski open dense subset in the moduli space of quartic K3 surfaces whose members satisfy the conjecture. Various other curves of positive genus can be also constructed.

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arXiv2017

Correspondence theorems for tropical curves I

In this paper, we study the deformation theory of degenerate algebraic curves on singular varieties which appear as the degenerate limit of families of varieties. For this purpose, we systematically develop a new method to calculate the obstruction cohomology class of degenerate algebraic curves. This enables us to judge whether a given degenerate curve can be deformed to a smooth curve or not in variety of situations. In this paper, we apply it to curves of genus one on degeneration of toric varieties. In particular, we obtain the necessary and sufficient condition for the realizability of tropical curves of genus one, extending various results obtained so far.

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arXiv2017

Graphs and obstruction theory for algebraic curves

In this paper we study a construction of algebraic curves from combinatorial data. In the study of algebraic curves through degeneration, graphs usually appear as the dual intersection graph of the central fiber. Properties of such graphs can be encoded in so-called tropical curves. Our main concern is the relation between algebraic curves and tropical curves where the deformation problem is obstructed. Particular emphasis is put on the role of higher valent vertices of tropical curves, which has not been developed well so far in spite of its importance in this area of study. We will give a general formula describing the obstruction, a new criterion for the vanishing of the obstruction, and a relation between the number of algebraic curves and the number of integral points in certain polytopes. We also prove the optimal version of the correspondence between tropical curves and algebraic curves when the tropical curves are regular, generalizing previous results.

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arXiv2015

Counting curves via degeneration

We develop a technique to study curves in a variety which has a degeneration into some union of varieties. The class of such varieties is very broad, but the theory becomes particularly useful when the variety has a degeneration into a union of toric varieties. Hypersurfaces are typical examples, and we study lines on K3 surfaces and quintic Calabi-Yau hypersurfaces in detail. In particular, we combinatorially prove the existence of 2875 lines in a generic quintic Calabi-Yau 3-fold. Also, we give a geometric construction of walls in the Gross-Siebert construction of Calabi-Yau varieties.

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arXiv2010

Potential functions via toric degenerations

This is a short companion paper to arXiv:0810.3470. We construct an integrable system on an open subset of a Fano manifold equipped with a toric degeneration, and compute the potential function for its Lagrangian torus fibers if the central fiber is a toric Fano variety admitting a small resolution.

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arXiv2010

Toric degenerations, tropical curves and Gromov-Witten invariants of Fano manifolds

In this paper, we give a tropical method for computing Gromov-Witten type invariants of Fano manifolds of special type. This method applies to those Fano manifolds which admit toric degenerations to toric Fano varieties with singularities allowing small resolutions. Examples include (generalized) flag manifolds of type A, and some moduli space of rank two bundles on a genus two curve.

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arXiv2009

Toric degenerations of Gelfand-Cetlin systems and potential functions

We define a toric degeneration of an integrable system on a projective manifold, and prove the existence of a toric degeneration of the Gelfand-Cetlin system on the flag manifold of type A. As an application, we calculate the potential function for a Lagrangian torus fiber of the Gelfand-Cetlin system.

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arXiv2006

Disc counting on toric varieties via tropical curves

In this paper, we define two numbers. One comes from counting tropical curves with a stop and the other is the number of holomorphic discs in toric varieties with Lagrangian boundary condition. Both of these curves should satisfy some matching conditions. We show that these numbers coincide. These numbers can be considered as Gromov-Witten type invariants for holomorphic discs, and they have both similarities and differences to the counting numbers of closed holomorphic curves. We study several aspects of them.

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arXiv2006

Toric degenerations of toric varieties and tropical curves

We show that the counting of rational curves on a complete toric variety that are in general position to the toric prime divisors coincides with the counting of certain tropical curves. The proof is algebraic-geometric and relies on degeneration techniques and log deformation theory. This generalizes results of Mikhalkin obtained by different methods in the surface case to arbitrary dimensions.

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arXiv2004

Convergence of Hermitian-Yang-Mills Connections on Kähler Surfaces and mirror symmetry

The purpose of this paper is to exhibit a natural construction between complex geometry and symplectic geometry following the idea of mirror symmetry. Suppose we are given a family of pairs of 2-dimensional Kähler tori and stable holomorphic vector bundles on them $(\hat M_{\ep}, E_{\ep})$, \ep \in (0, 1]$, and each has structure of a Lagrangian torus fibration $π:\hat M_{\ep} \to B$ whose fibers are of diameter $O(\ep)$, and let $A_{\ep}$ be a family of hermitian Yang-Mills(HYM) connections on $E_{\ep}$. As $\ep$ goes to zero, $A_{\ep}$ will, modulo possible bubbles, converge to a connection which is flat on each fiber. Since each fiber is a torus, limit connection will determine elements of the dual torus, which are points of the fiber of the mirror $M_1$. These points gather to make up (special) Lagrangian variety.

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