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Rei Murakami

Publications and source records attributed to Rei Murakami.

5 recordsLinked to original sources
arXiv2026

Weak solutions of the generalized Monge-Amp\`ere equation and the supercritical deformed Hermitian-Yang-Mills equation: boundary cases

We prove the existence and uniqueness of weak solutions for the generalized Monge-Amp\`ere equation and the supercritical deformed Hermitian-Yang-Mills equation in cohomology classes lying on the boundary of the solvable region. Moreover, we prove that the associated geometric flows converge to the weak solutions in the sense of currents. The proof combines viscosity-theoretic and pluripotential-theoretic techniques.

math.DG↗
arXiv2025

An analytic proof of Griffiths' conjecture on compact Riemann surfaces

Griffiths' conjecture asserts that a holomorphic vector bundle is ample if and only if it admits a Hermitian metric with positive curvature. In this paper, we present a new proof of this conjecture on compact Riemann surfaces using a system of PDEs introduced by Demailly. Our argument combines techniques developed by Uhlenbeck-Yau for Hermitian-Einstein metrics with Pingali's reduction of the problem to an a priori estimate.

math.DG↗
arXiv2024

Numerical criteria on the complex Hessian quotient equations with the Calabi symmetry

Assuming Calabi symmetry, we prove that a numerical condition ensures the solvability of the complex Hessian quotient equation, as conjectured by Sz\'ekelyhidi. We also propose a conjecture on the existence of a $k$-subharmonic representative in a given cohomology class and confirm it under the assumption of Calabi symmetry or when the class is semiample.

math.DG↗
arXiv2022

$J$-equations and deformed Hermitian-Yang-Mills equations on holomorphic submersions

In this paper, we prove that there exists a solution of the $J$-equation on the total space of a holomorphic submersion if there exist solutions of the $J$-equation on the fibers and the base. The method is an adiabatic limit technique. We also partially prove the converse implication. More precisely, if the total space is $J$-nef, then each fiber is $J$-nef. In addition, if each fiber has a solution of the $J$-equation, then the base is also $J$-nef. Furthermore, we establish similar phenomena for the deformed Hermitian-Yang-Mills equation.

math.DG↗