AI
Mathematical work with AI
Research manuscripts, grouped by topic. Each entry includes a summary and a disclosure of AI involvement.
Green functions & theta functions
2 papersCounterexamples to Hosono's conjecture on Azukawa pseudometrics and indicatrix volumes
Constructs a bounded pseudoconvex family of hyperconvex domains in complex dimension two that disproves both parts of Hosono's conjecture on variation of Azukawa pseudometrics and indicatrix volumes.
Indefinite theta Hessians at Green-function parameters
Constructs triply connected circle domains for which the logarithmic theta Hessian at the Green-function parameter has one positive and one negative eigenvalue, disproving Saitoh's negative-definiteness conjecture in Yamada's theta-function formulation. The proof computes both leading eigenvalue coefficients using harmonic measures and a second-order theta expansion.
Valuations & Lelong numbers
5 papersCritical divisorial detectors and equisingular analytic approximation
Characterizes decreasing equisingular analytic approximation for log canonical plurisubharmonic germs in arbitrary dimension and for germs with finite-colength multiplier ideals on smooth surfaces. For toric weights, an analytic sandwich construction removes the toric-approximant restriction from the An–Seo criterion and recovers Guan’s diagonal theorem.
Equality in the Demailly–Pham inequality and monomial valuations
Presents a candidate analytic proof of the Demailly–Pham equality classification for plurisubharmonic germs in the local Cegrell class with positive Lelong number, in terms of diagonal logarithmic greenifications. It derives, conditional on that classification, the finite-discrepancy case of Baily's Conjecture 6.3 for arbitrary centers, using multiplier-ideal restriction and discrepancy rigidity to obtain monomial parameters in the original formal ring.
Ideals attaining equality in the Demailly–Pham inequality
Presents an algebraic candidate proof of Baily's Conjecture 6.2: in arbitrary codimension, ideals attaining equality in the Demailly–Pham inequality are precisely those whose integral closures are diagonal in formal regular parameters. The proof combines hyperplane descent and Rees degeneration in the primary case with transverse primary truncations and compatible coordinate jets. The diagonal exponents are determined by the extended mixed multiplicities.
Flat tame weights and valuative separation
We prove that a tame maximal weight is flat if and only if its holomorphic relative types define a centered real valuation. This yields a bijection between flat tame maximal weights modulo bounded functions and centered real valuations of finite log discrepancy. Zhou weights are flat as a consequence. Applying the criterion to the quasi-monomial weights constructed by Boucksom–Favre–Jonsson proves the remaining implication (4) ⇒ (1) in their Theorem A.
Generalized Lelong numbers determine valuative transforms
Proves that generalized Lelong numbers for primary-ideal analytic weights determine valuative transforms. The revised proof uses ample perturbations on a fixed model and a negative-definite exceptional intersection matrix.
Bergman kernels & minimal L² integrals
5 papersCounterexamples from Green sublevel families for higher order p-Bergman kernels
Constructs explicit Green sublevel families that disprove convexity, monotonicity for p > 2, and the corresponding Lᵖ concavity statement.
Counterexamples to ξ-reduction and minimal-integral duality below one
Increasing the derivative order on the unit disk yields counterexamples to ξ-reduction and minimal-integral duality for every 0 < p < 1. The paper also proves minimal-integral duality on arbitrary pseudoconvex domains for p ≥ 1.
Failure of fiberwise log-plurisubharmonicity for classical p-Bergman kernels
Gives counterexamples to fiberwise log-plurisubharmonicity for every p > 2, with an explicit kernel formula and strictly negative Levi form along a holomorphic section. Exhaustion also yields bounded pseudoconvex counterexamples, establishing the sharp threshold p = 2.
A four-connected domain with no point of equality in the higher order Suita conjecture
Constructs a planar domain where equality never occurs at any point or derivative order, with examples for every connectivity n ≥ 4.
A ξ-Bergman kernel proof of the concavity of minimal L² integrals
Derives Guan’s concavity theorem from fiberwise ξ-Bergman kernel variation, matrix curvature, and finite-dimensional approximation.
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