Duals of log abelian varieties
We prove the existence of the dual of a polarizable log abelian variety.
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arXiv subjects
Chikara Nakayama
Publications and source records attributed to Chikara Nakayama.
Tangential morphisms via log arithmetic geometry
We give a reformulation of tangential morphisms (which is a generalization of Deligne's tangential base point) via log geometry.
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Logarithmic geometry and Frobenius, II
Based on the strong analogy between the category of log mixed Hodge structures and the category ${\cal A}_X$ of $\ell$-adic nature, which we have introduced in the previous part and is closely related to the weight-monodromy conjecture, we prove the $\ell$-adic analogues of some theorems in Hodge theory related to the SL(2)-orbit theorem.
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Universality of almost periodic orbits in certain composite functions
We consider composite functions in the elementary algebraic framework. Without any use of the Fourier transform, we find almost periodic orbits which suitably characterizes certain composite functions. In particular, we provide special composite functions that tend asymptotically to almost periodic orbits.
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Logarithmic Tate conjectures over finite fields
We formulate an analogue of Tate conjecture on algebraic cycles, for the log geometry over a finite field. We show that the weight-monodromy conjecture follows from this conjecture and from the semi-simplicity of the Frobenius action. This conjecture suggests the existence of the monodromy cycle which gives the monodromy operator and an action of ${\frak{sl}}(2)$ on the cohomology, and which lives in the world of log motives.
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Explicit construction of recurrent neural networks effectively approximating discrete dynamical systems
We consider arbitrary bounded discrete time series originating from dynamical system with recursivity. More precisely, we provide an explicit construction of recurrent neural networks which effectively approximate the corresponding discrete dynamical systems.
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Logarithmic geometry and Frobenius
Based on the logarithmic algebraic geometry and the theory of Deligne systems, we define an abelian category of $\ell$-adic sheaves with weight filtrations on a logarithmic scheme over a finite field, which is similar to the category of variations of mixed Hodge structure. We consider asymptotic behaviors and simple cases of higher direct images of objects of this category. This category is closely related to the monodromy-weight conjecture.
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Geometric polarized log Hodge structures for semistable families
We prove that a projective semistable morphism of fs log analytic spaces yields polarized log Hodge structures in the canonical way.
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Classifying spaces of degenerating mixed Hodge structures, VI: Log real analytic functions and log $C^{\infty}$ functions
To advance our log Hodge theory, we introduce log real analytic functions and log $C^{\infty}$ functions, define how to integrate them, and prove the log Poincaré lemma. We give better understandings of the degeneration of Hodge structure, including a geometric interpretation of the theory of $SL(2)$-orbits.
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Mixed objects are embedded into log pure objects
We prove that a variation of mixed Hodge structure is embedded in a logarithmic variation of pure Hodge structure, and a generalized version of this result. These results suggest some simple construction of the category of mixed motives by using log pure motives.
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Deligne--Beilinson cohomology and log Hodge theory
We show that the description of Deligne--Beilinson cohomology is improved by using log Hodge theory. We consider the log relative version of it, and also present a fundamental conjecture in log Hodge theory.
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Moduli of logarithmic abelian varieties with PEL structure
We construct the fine moduli space of log abelian varieties with PEL structure, which gives a toroidal compactification of the moduli space of abelian varieties with PEL structure.
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Toroidal compactifications and Borel--Serre compactifications
We discuss connections of toroidal compactifications and Borel--Serre compactifications in view of the fundamental diagram of extended period domains. We give a complement to a work of Goresky--Tai.
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Classifying spaces of degenerating mixed Hodge structures, V: Extended period domains and algebraic groups
For a linear algebraic group $G$ over $\bf Q$, we consider the period domains $D$ classifying $G$-mixed Hodge structures, and construct the extended period domains $D_{\mathrm{BS}}$, $D_{\mathrm{SL}(2)}$, and $Γ\backslash D_Σ$. In particular, we give toroidal partial compactifications of mixed Mumford--Tate domains.
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Logarithmic abelian varieties, Part VII: Moduli
We construct the fine moduli space of log abelian varieties, which gives a compactification of the moduli space of abelian varieties.
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Logarithmic abelian varieties, Part VI: Local moduli and GAGF
This is Part VI of our series of papers on log abelian varieties. In this part, we study local moduli and GAGF of log abelian varieties.
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Geometric polarized log Hodge structures with a base of log rank one
We prove that a projective vertical exact log smooth morphism of fs log analytic spaces with a base of log rank one yields polarized log Hodge structures in the canonical way.
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On log motives
We define the categories of log motives and log mixed motives. The latter gives a new formulation for the category of mixed motives. We prove that the former is a semisimple abelian category if and only if the numerical equivalence and homological equivalence coincide, and that it is also equivalent to that the latter is a Tannakian category. We discuss various realizations, formulate Tate and Hodge conjectures, and verify them in curve case.
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