arXiv subjects

Yusaku Tiba

Publications and source records attributed to Yusaku Tiba.

6 recordsLinked to original sources

arXiv2026

The First Correction Term in the Asymptotic Expansion of Bohr--Sommerfeld Lagrangian States

Let $Λ$ be a compact Bohr--Sommerfeld Lagrangian submanifold of a compact Kähler manifold equipped with a holomorphic prequantum line bundle. We study the asymptotic expansion of the Lagrangian states associated with $Λ$. In particular, we compute explicitly the first nontrivial correction term and show that it is expressed in terms of geometric invariants of the ambient Kähler manifold and the Lagrangian submanifold, including their scalar curvatures, the second fundamental form, and the mean curvature. As a consequence, we obtain the corresponding second-order asymptotic formula for the $L^2$-norm of the Lagrangian states.

math.SG↗

arXiv2025

Asymptotic estimates of holomorphic sections on Bohr-Sommerfeld Lagrangian submanifolds

Let $M$ be a complex manifold and $L$ be a line bundle over $M$ with a Hermitian metric $h$ whose Chern form is a Kähler form $ω$. Let $X \subset M$ be a Lagrangian submanifold of $(M, ω)$. When $X$ satisfies the Bohr-Sommerfeld condition, we give an asymptotic estimate of the norm $|f|_{h^k}$ on $X$ for $f \in H^0(M, L^k)$.

math.DG↗

arXiv2025

Polarizations and Convergences of holomorphic sections on the tangent bundle of a Bohr-Sommerfeld Lagrangian submanifold

Let $(M, ω)$ be a Kähler manifold and let $(L, \nabla)$ be a prequantum line bundle over $M$. Let $X \subset M$ be a Bohr-Sommerfeld Lagrangian submanifold of $(L, \nabla)$. In this paper, we study an asymptotic behaviour of holomorphic sections of $L^k$ as $k \to \infty$. Our first result shows that the $L^2$-norm of sections of $L^k$ are bounded below around $X$ if these sections converge on $X$ under a suitable trivialization of $L^k$. Since $X$ is a Lagrangian submanifold, we consider that a neighborhood of $X$ is embedded in the tangent bundle $TX$. Let $Ψ_{k}: TX \to TX$ be a multiplication by $\frac{1}{\sqrt{k}}$ in the fibers. The pullback of the Kähler polarization by $Ψ_k$ converges to the real polarization, whose leaves are fibers of $TX$, as $k \to \infty$. Let $(f_k)_{k \in \mathbb{N}}$ be holomorphic sections of $L^k$ near $X$. By trivializing $L^k$, we consider $f_k$ as a function. In our second result, we show that $Ψ^* f_k$ converges to a fiberwise constant function on $TX$ as $k \to \infty$ under some condition on Sobolev norms of $f_k$.

math.CV↗

arXiv2020

Cohomology of vector bundles and non-pluriharmonic loci

In this paper, we study cohomology groups of vector bundles on neighborhoods of a non-pluriharmonic locus in Stein manifolds and in projective manifolds. By using our results, we show variants of the Lefschetz hyperplane theorem.

math.CV↗

arXiv2017

The extension of holomorphic functions on a non-pluriharmonic locus

Let $n \geq 4$ and let $Ω$ be a bounded hyperconvex domain in $\mathbb{C}^{n}$. Let $φ$ be a negative exhaustive smooth plurisubharmonic function on $Ω$. We show that any holomorphic function defined on a connected open neighborhood of the support of $(i\partial \overline{\partial}φ)^{n-3}$ can be extended to the holomorphic function on $Ω$.

math.CV↗

arXiv2014

On a convex level set of a plurisubharmonic function and the support of the Monge-Ampère current

In this paper, we study a geometric property of a continuous plurisubharmonic function which is a solution of the Monge-Ampère equation and has a convex level set. To prove our main theorem, we show a minimum principle of a maximal plurisubharmonic function. By using our results and Lempert's results, we show a relation between the supports of the Monge-Ampère currents and complex $k$-extreme points of closed balls for the Kobayashi distance in a bounded convex domain in $\mathbb{C}^{n}$.

math.CV↗