arXiv subjects
Publications and source records attributed to Yuta Watanabe.
In this paper, we generalize the Steenbrink vanishing theorem for ample line bundles on complex projective varieties by extending it to big line bundles on compact complex spaces with multiplier ideal sheaves.
We introduce regularized bivariate affine $q$-Krawtchouk polynomials and show that they give the first eigenmatrix of a translation association scheme on $\operatorname{Mat}_{d\times n}(\operatorname{GR}(p^2,r))$. The relations are defined by Smith type, and the corresponding eigenvalues are written as character sums over Smith type classes. The proof is based on recurrence relations obtained from the transition numbers describing how Smith types change when one row and one column are added to a matrix. These recurrences identify the character sums with the regularized bivariate affine $q$-Krawtchouk polynomials.
In this paper, in order to develop a more general $L^2$-theory for the $\overline{\partial}$-operator on complex spaces, we introduce appropriate notions of singular Griffiths/Nakano positivity on complex spaces and establish various properties of these notions. By applying these results, we provide $L^2$-Dolbeault fine resolutions and cohomological isomorphisms, and $L^2$-existence theorems. As an application, we obtain Nakano-Nadel vanishing theorems on weakly pseudoconvex complex spaces.
The vortex coronagraph is one of the most promising candidates for the Habitable Worlds Observatory (HWO) due to its excellent theoretical performance for an off-axis telescope. A practical realization can be achieved using liquid-crystal polymers to form a vector vortex coronagraph (VVC). Reaching the $10^{-10}$ contrast required for Earth-like planet detection is, however, limited by polarization leakage caused by wavelength-dependent deviations from half-wave retardance. This effect can be mitigated using multi-layer twisted retarders to minimize leakage, and by combining the VVC with multiple polarization gratings (mgVVC) to diffract the polarization leakage out of the science path. We present recent progress within the ESA-funded SUPPPPRESS project, which aims to advance the manufacturing, assembly, and testing of high-performance VVCs. Central singularities of 2 and 6 $\mu$m have been achieved for charge 2 and charge 6 VVCs, respectively, with patterning accuracies better than 1 degree root-mean-square error. Fabrication procedures have been developed to produce individual components with a polarization leakage of $3\times10^{-4}$ over a 10% bandwidth and $8\times10^{-4}$ over a 20% bandwidth. We also report on the development of assembly and alignment procedures for mgVVCs and their metrology. Furthermore, we present initial high-contrast tests at the THD2 bench for both regular VVCs and a double-grating VVC. The double-grating VVC reaches an average contrast between 3 and 10 $\lambda$/D of $2 \times 10^{-8}$ over a small bandwidth and $6\times 10^{-8}$ over a 10% bandwidth. Finally, we report on successful space-environment tests of the assembled liquid-crystal masks.
In this paper, in order to develop a more general $L^2$-theory for the $\overline{\partial}$-operator on complex spaces, we provide $L^2$-Dolbeault fine resolutions and isomorphisms, and $L^2$-estimates, for holomorphic line bundles on complex spaces equipped with singular Hermitian metrics. As applications, we obtain several generalizations of the Nadel vanishing theorem.
We study $T$-designs in the nonbinary Johnson scheme. This scheme generalizes both the Johnson and Hamming schemes and admits a bivariate $Q$-polynomial structure. Zhu (2021) provided a combinatorial characterization of $T$-designs in this scheme for certain index sets $T$, using a relationship between $T$-designs in the nonbinary Johnson scheme and relative designs in the nonbinary Hamming scheme. In this paper, we obtain a characterization that applies to a strictly larger class of index sets $T$, based on a methodological extension of Delsarte's original framework (1973). This new characterization naturally recovers classical block designs and orthogonal arrays as special cases. To describe these designs uniformly, we introduce $(r,s)$-designs, a new family of combinatorial objects that arise naturally from our characterization. We also derive absolute lower bounds on the cardinality of $(r,s)$-designs from the multiplicities of the primitive idempotents of the nonbinary Johnson scheme, and construct examples with index $\lambda=1$ that attain certain natural lower bounds.
Runge-type approximation principles for holomorphic sections of adjoint line bundles are known only for weakly pseudoconvex manifolds. In this paper, we establish a refined form of such principles adapted to the setting of complex spaces and show that they yield global holomorphically embeddings for the regular locus of weakly pseudoconvex complex spaces. The key point is the construction of sequences of singular Hermitian metrics after a canonical resolution of singularities, together with a control of multiplier ideal sheaves via the strong openness property. This refined Runge-type approximation principle enables the globalization of local sections even when singularities persist at infinity. As an application, we solve the Union problem for weakly pseudoconvex complex manifolds.
Let $X$ be a weakly pseudoconvex manifold and $L\longrightarrow X$ be a holomorphic line bundle with a singular positive Hermitian metric $h$. In this article, we provide a points separation theorem and an embedding for the adjoint linear subsystem including the multiplier ideal sheaf $\mathscr{I}(h^m)$, with respect to an appropriate set excluding a singular locus of $h$. We also show that the adjoint bundle of $L$ is big, which constitutes a generalization to weakly pseudoconvex manifolds of Demailly's characterization of positivity in complex and algebraic geometry. To handle analytical methods, an approximation of singular Hermitian metrics is first constructed based on Demailly's approximation, using the strong openness property, preserving the ideal sheaves and compatible with blow-ups. Using the blow-ups obtained from this approximation, the singular holomorphic Morse inequalities and the approximation theorem for holomorphic sections, each twisted by the ideal sheaves, are established. This approximation theorem for sections provides the key to globalization, leading to global embeddings.
The generalized wreath product of symmetric association schemes was introduced by R.A.Bailey in the European Journal of Combinatorics 27 (2006) 428-435. It is recognized as a unification of both the wreath product and the direct product of symmetric association schemes. While its potential applicability to any association scheme had been implied, this paper provides a formal and explicit confirmation of that claim. Moreover, we establish the irreducible representations of its adjacency algebra and Terwilliger algebra.
In this paper, we consider a proper Kähler fibration $f \colon X \to Y$ and a singular Hermitian line bundle $(L, h)$ on $X$ with semi-positive curvature. We prove that the direct image sheaf $f_{*}(\mathcal{O}_{X}(K_{X/Y}+L) \otimes \mathcal{I}(h))$, equipped with the Narasimhan-Simha metric, is singular Nakano semi-positive in the sense that the $\overline{\partial}$-equation can be solved with optimal $L^{2}$-estimate. Our proof does not rely on the theory of Griffiths positivity for the direct image sheaf.
This paper delves into the Terwilliger algebra associated with the ordered Hamming scheme, which extends from the wreath product of one-class association schemes and was initially introduced by Delsarte as a natural expansion of the Hamming schemes. Levstein, Maldonado and Penazzi have shown that the Terwilliger algebra of the Hamming scheme of length $n$ is the $n$-fold symmetric tensor algebra of that of the one-class association scheme. Furthermore, Bhattacharyya, Song and Tanaka have established that the Terwilliger algebra of the wreath product of a one-class association scheme is a direct sum of the ``primary'' subalgebra and commutative subalgebras. This paper extends these findings to encompass both conclusions.
In this paper, we investigate various positivity for singular Hermitian metrics such as Griffiths, $ω$-trace and RC, where $ω$ is a Hermitian metric, and show that these quasi-positivity notions induce $0$-th cohomology vanishing, rational conected-ness, etc. Here, $ω$-trace positivity of smooth Hermitian metrics $h$ on holomorphic vector bundles $E$ represents the positivity of $tr_ωiΘ_{E,h}$.
We investigate the positivity properties of the direct image $f_{\ast}(K_{X/Y} \otimes L)$ of the adjoint line bundle associated with a big and nef line bundle $L$, under a smooth fibration $f: X\to Y$ between projective varieties. We show that the vector bundle $f_{\ast}(K_{X/Y} \otimes L)$ is big.
In this article, we get properties for singular (dual) Nakano semi-positivity and obtain singular type vanishing theorem involving $L^2$-subsheaves on weakly pseudoconvex manifolds by $L^2$-estimates and $L^2$-type Dolbeault isomorphisms. As applications, Fujita's conjecture type theorem with singular Hermitian metrics is presented.
Let $f:X\to Y$ be a surjective projective map and $L$ be a holomorphic line bundle on $X$ equipped with a (singular) semi-positive Hermitian metric $h$. In this article, by studying the canonical metric on the direct image sheaf of the twisted relative canonical bundles $K_{X/Y}\otimes L\otimes\mathscr{I}(h)$, we obtain that this metric has dual Nakano semi-positivity when $h$ is smooth and there is no deformation by $f$ and that this metric has locally Nakano semi-positivity in the singular sense when $h$ is singular.
In this article, we first establish an $L^2$-type Dolbeault isomorphism for the sheaf of logarithmic differential forms twisted by the multiplier ideal sheaf. By using this isomorphism and $L^2$-estimates equipped with a singular Hermitian metric, we obtain logarithmic vanishing theorems involving multiplier ideal sheaves on compact Kähler manifolds with simple normal crossing divisors.
We obtain the Bogomolov-Sommese type vanishing theorem involving multiplier ideal sheaves for big line bundles. We define a dual Nakano semi-positivity of singular Hermitian metrics with L2-estimates and prove the vanishing theorem which is a generalization of the Bogomolov-Sommese type vanishing theorem to holomorphic vector bundles.
We study the positivity properties of the curvature operator for holomorphic Hermitian vector bundles. We obtain new characterization of semi-positive curvature operators for $(n,q)$ and $(p,n)$-forms by L2-estimates. The characterization of Nakano semi-positivity by $L^2$-estimate is already known. Applying our results, we give new characterizations of Nakano semi-negativity.