• Journal of the Korean Mathematical Society
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• v.40 no.3
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• pp.341-371
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• 2003

This article discusses in detail how the study of proper holomorphic rational mappings between balls in different dimensions relates to positivity conditions and to isometric imbedding of holomorphic bundles. The first chapter discusses rational proper mappings between balls; the second chapter discusses seven distinct positivity conditions for real-valued polynomials in several complex variables; the third chapter reveals how these issues relate to an isometric imbedding theorem for holomorphic vector bundles proved by the author and Catlin.

• Honam Mathematical Journal
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• v.5 no.1
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• pp.25-43
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• 1983

• East Asian mathematical journal
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• v.6 no.2
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• pp.155-158
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• 1990

• Korean Journal of Mathematics
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• v.7 no.1
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• pp.79-84
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• 1999

• Journal of the Korean Mathematical Society
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• v.52 no.2
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• pp.269-331
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• 2015

For a fixed positive integer g, we let $\mathcal{P}_g=\{Y{\in}\mathbb{R}^{(g,g)}{\mid}Y=^tY>0\}$ be the open convex cone in the Euclidean space $\mathbb{R}^{g(g+1)/2}$. Then the general linear group GL(g, $\mathbb{R}$) acts naturally on $\mathcal{P}_g$ by $A{\star}Y=AY^tA(A{\in}GL(g,\mathbb{R}),\;Y{\in}\mathcal{P}_g)$. We introduce a notion of polarized real tori. We show that the open cone $\mathcal{P}_g$ parametrizes principally polarized real tori of dimension g and that the Minkowski modular space 𝔗g = $GL(g,\mathbb{Z}){\backslash}\mathcal{P}_g$ may be regarded as a moduli space of principally polarized real tori of dimension g. We also study smooth line bundles on a polarized real torus by relating them to holomorphic line bundles on its associated polarized real abelian variety.

• East Asian mathematical journal
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• v.14 no.2
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• pp.371-375
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• 1998

• Journal of the Korean Mathematical Society
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• v.61 no.1
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• pp.91-107
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• 2024

We determine the N → ∞ asymptotics of the expected value of entanglement entropy for pure states in H_1,N ⊗ H_2,N, where H_1,N and H_2,N are the spaces of holomorphic sections of the N-th tensor powers of hermitian ample line bundles on compact complex manifolds.