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Search Results: 1 - 10 of 620 matches for " Takeo Nishinou "
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Convergence of Hermitian-Yang-Mills Connections on K?hler Surfaces and mirror symmetry
Takeo Nishinou
Mathematics , 2003,
Abstract: 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 K\"ahler 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 $\pi:\hat M_{\ep} \to B$ 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.
Correspondence theorems for tropical curves
Takeo Nishinou
Mathematics , 2009,
Abstract: In this paper, we study the correspondence between tropical curves and holomorphic curves. The main subjects in this paper are superabundant tropical curves. First we give an effective combinatorial description of these curves. Based on this description, we calculate the obstructions for appropriate deformation theory, describe the Kuranishi map, and study the solution space of it. The genus one case is solved completely, and the theory works for many of the higher genus cases, too.
Counting curves via degeneration
Takeo Nishinou
Mathematics , 2011,
Abstract: 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.
Rational curves in Fano hypersurfaces and tropical curves
Takeo Nishinou
Mathematics , 2011,
Abstract: Using ideas from the theory of tropical curves and degeneration, we prove that any Fano hypersurface (and more generally Fano complete intersections) is swept by at most quadratic rational curves.
Degeneration and curves on K3 surfac
Takeo Nishinou
Mathematics , 2015,
Abstract: 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.
Disc counting on toric varieties via tropical curves
Takeo Nishinou
Mathematics , 2006,
Abstract: 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.
Toric degenerations, tropical curves and Gromov-Witten invariants of Fano manifolds
Takeo Nishinou
Mathematics , 2010,
Abstract: 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.
Describing tropical curves via algebraic geometry
Takeo Nishinou
Mathematics , 2015,
Abstract: In this paper, we give a general deformation theoretical set up for the problem of the correspondence between tropical curves and holomorphic curves. Using this formulation, the correspondence theorem for non-superabundant tropical curves is naturally solved. This formulation also gives an effective combinatorial description of the superabundant tropical curves, which will give the basis for the study of correspondence between tropical curves and holomorphic curves for superabundant cases [8].
Toric degenerations of toric varieties and tropical curves
Takeo Nishinou,Bernd Siebert
Mathematics , 2004,
Abstract: 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.
Toric degenerations of Gelfand-Cetlin systems and potential functions
Takeo Nishinou,Yuichi Nohara,Kazushi Ueda
Mathematics , 2008,
Abstract: 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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