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Seminar information archive
Seminar information archive ～01/17｜Today's seminar 01/18 | Future seminars 01/19～
Seminar on Geometric Complex Analysis
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Shin-ichi Matsumura (Tohoku University)
A transcendental approach to injectivity theorems for log canonical pairs (JAPANESE)
Shin-ichi Matsumura (Tohoku University)
A transcendental approach to injectivity theorems for log canonical pairs (JAPANESE)
Operator Algebra Seminars
16:45-18:15 Room #118 (Graduate School of Math. Sci. Bldg.)
Yusuke Isono (RIMS, Kyoto Univ.)
Bi-exact groups, strongly ergodic actions and group measure space type III factors with no central sequence
Yusuke Isono (RIMS, Kyoto Univ.)
Bi-exact groups, strongly ergodic actions and group measure space type III factors with no central sequence
2016/06/14
Tuesday Seminar of Analysis
16:50-18:20 Room #126 (Graduate School of Math. Sci. Bldg.)
NIIKUNI, Hiroaki (Maebashi Institute of Technology)
Schr¥"odinger operators on a periodically broken zigzag carbon nanotube (Japanese)
NIIKUNI, Hiroaki (Maebashi Institute of Technology)
Schr¥"odinger operators on a periodically broken zigzag carbon nanotube (Japanese)
Tuesday Seminar on Topology
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Naohiko Kasuya (Aoyama Gakuin University)
Non-Kähler complex structures on R^4 (JAPANESE)
Naohiko Kasuya (Aoyama Gakuin University)
Non-Kähler complex structures on R^4 (JAPANESE)
[ Abstract ]
We consider the following problem. "Is there any non-Kähler complex structure on R^{2n}?" If n=1, the answer is clearly negative. On the other hand, Calabi and Eckmann constructed non-Kähler complex structures on R^{2n} for n ≥ 3. In this talk, I will construct uncountably many non-Kähler complex structures on R^4, and give the affirmative answer to the case where n=2. For the construction, it is important to understand the genus-one achiral Lefschetz fibration S^4 → S^2 found by Yukio Matsumoto and Kenji Fukaya. This is a joint work with Antonio Jose Di Scala and Daniele Zuddas.
We consider the following problem. "Is there any non-Kähler complex structure on R^{2n}?" If n=1, the answer is clearly negative. On the other hand, Calabi and Eckmann constructed non-Kähler complex structures on R^{2n} for n ≥ 3. In this talk, I will construct uncountably many non-Kähler complex structures on R^4, and give the affirmative answer to the case where n=2. For the construction, it is important to understand the genus-one achiral Lefschetz fibration S^4 → S^2 found by Yukio Matsumoto and Kenji Fukaya. This is a joint work with Antonio Jose Di Scala and Daniele Zuddas.
2016/06/13
Numerical Analysis Seminar
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Atsushi Suzuki (Osaka University)
Dissection : A direct solver with kernel detection for finite element matrices
(日本語)
Atsushi Suzuki (Osaka University)
Dissection : A direct solver with kernel detection for finite element matrices
(日本語)
[ Abstract ]
Large-scale sparse matrices are solved in finite element analyses of elasticity and/or flow problems. In some cases, the matrix may be singular, e.g. due to pressure ambiguity of the Navier-Stokes equations, or due to rigid body movements of sub-domain elasticity problems by a domain decomposition method. Therefore, it is better the linear solver understands rank-deficiency of the matrix.
By assuming the matrix is factorized into LDU form with a symmetric partial permutation, and by introducing a threshold to postpone factorization for pseudo null pivots, solvability of the last Schur complement matrix will be examined. Usual procedure for rank-deficiency problem is based on computation of eigenvalues or singular values and an introduced threshold determines the null space. However, developed new algorithm in DOI:10.1002/nme.4729 is based on computation of residuals combined with orthogonal projections onto supposed image spaces and there is no necessary to introduce a threshold for understanding zero value in floating point. The algorithm uses higher precision arithmetic, e.g. quadruple precision, to distinguish numerical round-off errors that occurred during factorization of the whole sparse matrix from ones during the kernel detection procedure itself.
This is joint work with François-Xavier Roux (LJLL, UPMC/ONERA).
Large-scale sparse matrices are solved in finite element analyses of elasticity and/or flow problems. In some cases, the matrix may be singular, e.g. due to pressure ambiguity of the Navier-Stokes equations, or due to rigid body movements of sub-domain elasticity problems by a domain decomposition method. Therefore, it is better the linear solver understands rank-deficiency of the matrix.
By assuming the matrix is factorized into LDU form with a symmetric partial permutation, and by introducing a threshold to postpone factorization for pseudo null pivots, solvability of the last Schur complement matrix will be examined. Usual procedure for rank-deficiency problem is based on computation of eigenvalues or singular values and an introduced threshold determines the null space. However, developed new algorithm in DOI:10.1002/nme.4729 is based on computation of residuals combined with orthogonal projections onto supposed image spaces and there is no necessary to introduce a threshold for understanding zero value in floating point. The algorithm uses higher precision arithmetic, e.g. quadruple precision, to distinguish numerical round-off errors that occurred during factorization of the whole sparse matrix from ones during the kernel detection procedure itself.
This is joint work with François-Xavier Roux (LJLL, UPMC/ONERA).
Tokyo Probability Seminar
16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)
Yuki Tokushige (Research Institute for Mathematical Sciences, Kyoto University)
Jump processes on boudaries of random trees
Yuki Tokushige (Research Institute for Mathematical Sciences, Kyoto University)
Jump processes on boudaries of random trees
Seminar on Geometric Complex Analysis
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Masanori Adachi (Tokyo University of Science)
(JAPANESE)
Masanori Adachi (Tokyo University of Science)
(JAPANESE)
Operator Algebra Seminars
16:45-18:15 Room #118 (Graduate School of Math. Sci. Bldg.)
Liang Kong (Univ. New Hampshire/Harvard Univ.)
Lattices models for topological orders and boundary-bulk duality
Liang Kong (Univ. New Hampshire/Harvard Univ.)
Lattices models for topological orders and boundary-bulk duality
2016/06/08
Number Theory Seminar
16:00-18:30 Room #16:00-17:00は002, 17:30-18:30は056 (Graduate School of Math. Sci. Bldg.)
Bruno Kahn (Institut de mathématiques de Jussieu-Paris Rive Gauche) 16:00-17:00
Torsion order of smooth projective surfaces (English)
Local and global geometric structures of perfectoid Shimura varieties (English)
Bruno Kahn (Institut de mathématiques de Jussieu-Paris Rive Gauche) 16:00-17:00
Torsion order of smooth projective surfaces (English)
[ Abstract ]
To a smooth projective variety $X$ whose Chow group of $0$-cycles is $\mathbb{Q}$-universally trivial one can associate its torsion order ${\mathrm{Tor}}(X)$, the smallest multiple of the diagonal appearing in a cycle-theoretic decomposition à la Bloch-Srinivas. We show that ${\mathrm{Tor}}(X)$ is the exponent of the torsion in the Néron-Severi-group of $X$ when $X$ is a surface over an algebraically closed field $k$, up to a power of the exponential characteristic of $k$.
Xu Shen (Morningside Center of Mathematics) 17:30-18:30To a smooth projective variety $X$ whose Chow group of $0$-cycles is $\mathbb{Q}$-universally trivial one can associate its torsion order ${\mathrm{Tor}}(X)$, the smallest multiple of the diagonal appearing in a cycle-theoretic decomposition à la Bloch-Srinivas. We show that ${\mathrm{Tor}}(X)$ is the exponent of the torsion in the Néron-Severi-group of $X$ when $X$ is a surface over an algebraically closed field $k$, up to a power of the exponential characteristic of $k$.
Local and global geometric structures of perfectoid Shimura varieties (English)
[ Abstract ]
In this talk, we will investigate some geometric structural properties of perfectoid Shimura varieties of abelian type. In the global part, we will construct some minimal and toroidal type compactifications for these spaces, and we will describe explicitly the degeneration of Hodge-Tate period map at the boundaries. In the local part, we will show that each Newton stratum of these perfectoid Shimura varieties can be described by the related (generalized) Rapoport-Zink space and Igusa variety. As a consequence of our local and global constructions, we can compute the stalks of the relative cohomology under the Hodge-Tate period map of the intersection complex (on the minimal compactification), in terms of cohomology of Igusa varieties at the boundary with truncated coefficients.
In this talk, we will investigate some geometric structural properties of perfectoid Shimura varieties of abelian type. In the global part, we will construct some minimal and toroidal type compactifications for these spaces, and we will describe explicitly the degeneration of Hodge-Tate period map at the boundaries. In the local part, we will show that each Newton stratum of these perfectoid Shimura varieties can be described by the related (generalized) Rapoport-Zink space and Igusa variety. As a consequence of our local and global constructions, we can compute the stalks of the relative cohomology under the Hodge-Tate period map of the intersection complex (on the minimal compactification), in terms of cohomology of Igusa varieties at the boundary with truncated coefficients.
2016/06/07
Tuesday Seminar on Topology
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Kenta Hayano (Keio University)
Topology of holomorphic Lefschetz pencils on the four-torus (JAPANESE)
Kenta Hayano (Keio University)
Topology of holomorphic Lefschetz pencils on the four-torus (JAPANESE)
[ Abstract ]
In this talk, we will show that two holomorphic Lefschetz pencils on the four-torus are (smoothly) isomorphic if they have the same genus and divisibility. The proof relies on the theory of moduli spaces of polarized abelian surfaces. We will also give vanishing cycles of some holomorphic pencils of the four-torus explicitly. This is joint work with Noriyuki Hamada (The University of Tokyo).
In this talk, we will show that two holomorphic Lefschetz pencils on the four-torus are (smoothly) isomorphic if they have the same genus and divisibility. The proof relies on the theory of moduli spaces of polarized abelian surfaces. We will also give vanishing cycles of some holomorphic pencils of the four-torus explicitly. This is joint work with Noriyuki Hamada (The University of Tokyo).
2016/06/06
Seminar on Geometric Complex Analysis
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Shin Kikuta (Kogakuin University)
(JAPANESE)
Shin Kikuta (Kogakuin University)
(JAPANESE)
Operator Algebra Seminars
16:45-18:15 Room #118 (Graduate School of Math. Sci. Bldg.)
Yuhei Suzuki (Chiba Univ.)
Minimal ambient nuclear $C^*$-algebras
Yuhei Suzuki (Chiba Univ.)
Minimal ambient nuclear $C^*$-algebras
2016/06/03
Geometry Colloquium
13:00-14:30 Room #118 (Graduate School of Math. Sci. Bldg.)
Takeo Nishinou (Rikkyo University)
On a construction of holomorphic disks (Japanese)
Takeo Nishinou (Rikkyo University)
On a construction of holomorphic disks (Japanese)
[ Abstract ]
Recent study of algebraic and symplectic geometry revealed that holomorphic disks play an important role in several situations, deforming the classical geometry in some sense. In this talk we give a construction of holomorphic disks based on deformation theory, mainly on certain algebraic surfaces.
Recent study of algebraic and symplectic geometry revealed that holomorphic disks play an important role in several situations, deforming the classical geometry in some sense. In this talk we give a construction of holomorphic disks based on deformation theory, mainly on certain algebraic surfaces.
Geometry Colloquium
15:00-16:30 Room #118 (Graduate School of Math. Sci. Bldg.)
Makoto Miura (KIAS)
Caldero's toric degenerations and mirror symmetry (Japanese)
Makoto Miura (KIAS)
Caldero's toric degenerations and mirror symmetry (Japanese)
[ Abstract ]
In this talk, we explain some basic facts on toric degenerations of Fano varieties. In particular, we focus on the toric degenerations of Schubert varieties proposed by Caldero, where we use the string parametrizations of Lusztig--Kashiwara's dual canonical basis. As an application, we introduce a conjectural mirror construction of a linear section Calabi--Yau 3-fold in an orthogonal Grassmannian OG(2,7). This talk is based on joint works with Daisuke Inoue and Atsushi Ito.
In this talk, we explain some basic facts on toric degenerations of Fano varieties. In particular, we focus on the toric degenerations of Schubert varieties proposed by Caldero, where we use the string parametrizations of Lusztig--Kashiwara's dual canonical basis. As an application, we introduce a conjectural mirror construction of a linear section Calabi--Yau 3-fold in an orthogonal Grassmannian OG(2,7). This talk is based on joint works with Daisuke Inoue and Atsushi Ito.
2016/06/01
Mathematical Biology Seminar
16:30-17:30 Room #128演習室 (Graduate School of Math. Sci. Bldg.)
Xiao Dongyuan (Graduate School of Mathematical Sciences, The University of Tokyo)
A variational problem associated with the minimal speed of traveling waves for the spatially
periodic KPP equation (ENGLISH)
Xiao Dongyuan (Graduate School of Mathematical Sciences, The University of Tokyo)
A variational problem associated with the minimal speed of traveling waves for the spatially
periodic KPP equation (ENGLISH)
[ Abstract ]
We consider a spatially periodic KPP equation of the form
$$u_t=u_{xx}+b(x)u(1-u).$$
This equation is motivated by a model in mathematical ecology describing the invasion of an alien species into spatially periodic habitat. We deal with the following variational problem:
$$\underset{b\in A_i}{\mbox{Maximize}}\ \ c^*(b),\ i=1,2,$$
where $c^*(b)$ denotes the minimal speed of the traveling wave of the above equation, and sets $A_1$, $A_2$ are defined by
$$A_1:=\{b\ |\ \int_0^Lb=\alpha L,||b||_{\infty}\le h \},$$
$$A_2:=\{b\ |\ \int_0^Lb^2=\beta L\},$$
with $h>\alpha>0$ and $\beta>0$ being arbitrarily given constants. It is known that $c^*(b)$ is given by the principal eigenvalue $k(\lambda,b)$ associated with the one-dimensional elliptic operator under the periodic boundary condition:
$$-L_{\lambda,b}\psi=-\frac{d^2}{dx^2}\psi-2\lambda\frac{d}{dx}\psi-(b(x)
+\lambda^2)\psi\ \ (x\in\mathbb{R}/L\mathbb{Z}).$$
It is important to note that, in one-dimensional reaction-diffusion equations, the minimal speed $c^*(b)$ coincides with the so-called spreading speed. The notion of spreading speed was introduced in mathematical ecology to describe how fast the invading species expands its territory. In other words, our goal is to find an optimal coefficient $b(x)$ that gives the fastest spreading speed under certain given constraints and to study the properties of such $b(x)$.
We consider a spatially periodic KPP equation of the form
$$u_t=u_{xx}+b(x)u(1-u).$$
This equation is motivated by a model in mathematical ecology describing the invasion of an alien species into spatially periodic habitat. We deal with the following variational problem:
$$\underset{b\in A_i}{\mbox{Maximize}}\ \ c^*(b),\ i=1,2,$$
where $c^*(b)$ denotes the minimal speed of the traveling wave of the above equation, and sets $A_1$, $A_2$ are defined by
$$A_1:=\{b\ |\ \int_0^Lb=\alpha L,||b||_{\infty}\le h \},$$
$$A_2:=\{b\ |\ \int_0^Lb^2=\beta L\},$$
with $h>\alpha>0$ and $\beta>0$ being arbitrarily given constants. It is known that $c^*(b)$ is given by the principal eigenvalue $k(\lambda,b)$ associated with the one-dimensional elliptic operator under the periodic boundary condition:
$$-L_{\lambda,b}\psi=-\frac{d^2}{dx^2}\psi-2\lambda\frac{d}{dx}\psi-(b(x)
+\lambda^2)\psi\ \ (x\in\mathbb{R}/L\mathbb{Z}).$$
It is important to note that, in one-dimensional reaction-diffusion equations, the minimal speed $c^*(b)$ coincides with the so-called spreading speed. The notion of spreading speed was introduced in mathematical ecology to describe how fast the invading species expands its territory. In other words, our goal is to find an optimal coefficient $b(x)$ that gives the fastest spreading speed under certain given constraints and to study the properties of such $b(x)$.
2016/05/31
Tuesday Seminar on Topology
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Benoît Guerville-Ballé (Tokyo Gakugei University)
A linking invariant for algebraic curves (ENGLISH)
Benoît Guerville-Ballé (Tokyo Gakugei University)
A linking invariant for algebraic curves (ENGLISH)
[ Abstract ]
We construct a topological invariant of algebraic plane curves, which is in some sense an adaptation of the linking number of knot theory. As an application, we show that this invariant distinguishes a new Zariski pair of curves (ie a pair of curves having same combinatorics, yet different topology).
We construct a topological invariant of algebraic plane curves, which is in some sense an adaptation of the linking number of knot theory. As an application, we show that this invariant distinguishes a new Zariski pair of curves (ie a pair of curves having same combinatorics, yet different topology).
Algebraic Geometry Seminar
15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Kiwamu Watanabe (Saitama University)
A Characterization of Symplectic Grassmannians (JAPANESE)
Kiwamu Watanabe (Saitama University)
A Characterization of Symplectic Grassmannians (JAPANESE)
[ Abstract ]
In the series of their works, J. M. Hwang and N. Mok have been developing the theory of Varieties of Minimal Rational Tangents (VMRT for short). In this direction, the results of Mok and J. Hong-Hwang allow us to recognize a homogeneous Fano manifold X of Picard number one by looking at its VMRT at a general point. This characterization works for all rational homogeneous manifolds of Picard number one whenever the VMRT is rational homogeneous, which is always the case except for the short root cases; namely for symplectic Grassmannians, and for two varieties of type F*4*.
In this talk we show that, if we impose that the VMRT is the expected one at every point of the variety, we may still characterize symplectic Grassmannians. This is a joint work with G. Occhetta and L. E. Sola Conde (arXiv:1604.06867).
In the series of their works, J. M. Hwang and N. Mok have been developing the theory of Varieties of Minimal Rational Tangents (VMRT for short). In this direction, the results of Mok and J. Hong-Hwang allow us to recognize a homogeneous Fano manifold X of Picard number one by looking at its VMRT at a general point. This characterization works for all rational homogeneous manifolds of Picard number one whenever the VMRT is rational homogeneous, which is always the case except for the short root cases; namely for symplectic Grassmannians, and for two varieties of type F*4*.
In this talk we show that, if we impose that the VMRT is the expected one at every point of the variety, we may still characterize symplectic Grassmannians. This is a joint work with G. Occhetta and L. E. Sola Conde (arXiv:1604.06867).
2016/05/30
Seminar on Geometric Complex Analysis
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Takeo Ohsawa (Nagoya University)
(JAPANESE)
Takeo Ohsawa (Nagoya University)
(JAPANESE)
Tokyo Probability Seminar
16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)
Takafumi Otsuka (Graduate school of science and engineering, Tokyo metropolitan university)
Takafumi Otsuka (Graduate school of science and engineering, Tokyo metropolitan university)
Operator Algebra Seminars
16:45-18:15 Room #118 (Graduate School of Math. Sci. Bldg.)
Yosuke Kubota (Univ. Tokyo)
TBA
Yosuke Kubota (Univ. Tokyo)
TBA
Seminar on Probability and Statistics
13:00-14:10 Room #052 (Graduate School of Math. Sci. Bldg.)
OKADA, Yukinori (Osaka University)
Statistical genetics contributes to elucidation of disease biology and genomic drug discovery
OKADA, Yukinori (Osaka University)
Statistical genetics contributes to elucidation of disease biology and genomic drug discovery
2016/05/27
Colloquium
15:30-16:30 Room #123 (Graduate School of Math. Sci. Bldg.)
Takahiro Kitayama (Graduate School of Mathematical Sciences, University of Tokyo)
Moduli spaces of linear representations and splittings of 3-manifolds
Takahiro Kitayama (Graduate School of Mathematical Sciences, University of Tokyo)
Moduli spaces of linear representations and splittings of 3-manifolds
Geometry Colloquium
10:00-11:30 Room #118 (Graduate School of Math. Sci. Bldg.)
Yohsuke Imagi (Kavli IPMU)
Compact Special Lagrangian T^2-conifolds (Japanese)
Yohsuke Imagi (Kavli IPMU)
Compact Special Lagrangian T^2-conifolds (Japanese)
[ Abstract ]
Special Lagrangian submanifolds may be defined as volume-minimizing Lagrangian submanifolds of Calabi--Yau manifolds. Some interesting but difficult topics are (1) the SYZ conjecture, (2) counting compact special Lagrangian homology spheres, and (3) relation to Fukaya categories---all concerned with singularity of special Lagrangian submanifolds. I first recall some basic facts about these things and then talk about a simple class of singularity modelled on a certain T^2-cone.
Special Lagrangian submanifolds may be defined as volume-minimizing Lagrangian submanifolds of Calabi--Yau manifolds. Some interesting but difficult topics are (1) the SYZ conjecture, (2) counting compact special Lagrangian homology spheres, and (3) relation to Fukaya categories---all concerned with singularity of special Lagrangian submanifolds. I first recall some basic facts about these things and then talk about a simple class of singularity modelled on a certain T^2-cone.
Geometry Colloquium
13:00-14:30 Room #118 (Graduate School of Math. Sci. Bldg.)
Yoshihiko Matsumoto (Osaka University)
Deformation of Einstein metrics and $L^2$ cohomology on strictly pseudoconvex domains (Japanese)
Yoshihiko Matsumoto (Osaka University)
Deformation of Einstein metrics and $L^2$ cohomology on strictly pseudoconvex domains (Japanese)
[ Abstract ]
Any bounded strictly pseudoconvex domain of a Stein manifold carries a complete Kähler-Einstein metric of negative scalar curvature, which is unique up to homothety, as shown by S. Y. Cheng and S. T. Yau. I will discuss the fact that this Cheng-Yau metric deforms into a family of Einstein metrics parametrized by partially integrable CR structures on the boundary under the assumption that the dimension is at least three. The necessary analysis on the linearized Einstein operator can be reduced to a vanishing result of the $L^2$ Dolbeault cohomology with values in the holomorphic tangent bundle.
Any bounded strictly pseudoconvex domain of a Stein manifold carries a complete Kähler-Einstein metric of negative scalar curvature, which is unique up to homothety, as shown by S. Y. Cheng and S. T. Yau. I will discuss the fact that this Cheng-Yau metric deforms into a family of Einstein metrics parametrized by partially integrable CR structures on the boundary under the assumption that the dimension is at least three. The necessary analysis on the linearized Einstein operator can be reduced to a vanishing result of the $L^2$ Dolbeault cohomology with values in the holomorphic tangent bundle.
2016/05/24
Algebraic Geometry Seminar
15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Katsutoshi Yamanoi (Osaka University)
ON PSEUDO KOBAYASHI HYPERBOLICITY OF SUBVARIETIES OF ABELIAN VARIETIES
(tba)
Katsutoshi Yamanoi (Osaka University)
ON PSEUDO KOBAYASHI HYPERBOLICITY OF SUBVARIETIES OF ABELIAN VARIETIES
(tba)
[ Abstract ]
We prove that the Kobayashi pseudo distance of a closed subvariety X of an abelian variety A is a true distance outside the special set Sp(X) of X, where Sp(X) is the union of all positive dimensional translated abelian subvarieties of A which are contained in X. More strongly, we prove that a closed subvariety X of an abelian variety is taut modulo Sp(X); Every sequence fn : D → X of holomorphic mappings from the unit disc D admits a subsequence which converges locally uniformly, unless the image fn(K) of a fixed compact set K of D eventually gets arbitrarily close to Sp(X) as n gets larger. These generalize a classical theorem on algebraic degeneracy of entire holomorphic curves in irregular varieties.
We prove that the Kobayashi pseudo distance of a closed subvariety X of an abelian variety A is a true distance outside the special set Sp(X) of X, where Sp(X) is the union of all positive dimensional translated abelian subvarieties of A which are contained in X. More strongly, we prove that a closed subvariety X of an abelian variety is taut modulo Sp(X); Every sequence fn : D → X of holomorphic mappings from the unit disc D admits a subsequence which converges locally uniformly, unless the image fn(K) of a fixed compact set K of D eventually gets arbitrarily close to Sp(X) as n gets larger. These generalize a classical theorem on algebraic degeneracy of entire holomorphic curves in irregular varieties.
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