Yu-Chi Hou (侯侑期）

(Grothendieck's comment on Grothendieck-Riemann-Roch Theorem)

Publication:

Quantization of Semipositive Adjoint Line Bundles. preprint arxiv:2512.11523, (submitted)

Abstract: Let L be a big and semipositive line bundle on a complex projective manifold X, and let θ\in c_1(L) be a smooth semipositive representative. In the adjoint setting H^0(X,L^k\otimes K_X), we prove that Donaldson's quantized Monge--Ampère energy converges to the Monge--Ampère energy for every bounded θ-plurisubharmonic function. This extends the quantization picture from the ample case to the big and semipositive setting, where smooth positive representatives are no longer available and non-pluripolar Monge--Ampère theory is required. The main new input is a comparison theorem between adjoint Bergman kernels and their small ample twists. As a consequence, we prove that the normalized adjoint Bergman measures converge weakly to the corresponding non-pluripolar Monge--Ampère measures. Our result partially answers a question of Berman--Freixas i Montplet concerning the convergence of quantized Monge--Ampère energies in the semipositive setting.

Asymptotic of Bergman Kernel Bull.Inst.Math.Acad.Sin.(N.S.) Vol.17(2022), No.1, pp. 1-51

This is modified from my master thesis at NTU, supervised by Prof. Chin-Yu Hsiao. Thesis version is available at arXiv:2202.03383.Abstract: We give a new proof on the pointwise asymptotic expansion for Bergman kernel associated to k-th tensor power of a hermitian holomorphic line bundle on the points where the curvature of the line bundle is positive and satisfies local spectral gap condition. The main point is to introduce a suitable semi-classical symbol space and related symbolic calculus inspired from recent work of Hsiao and Savale. Particularly, we establish the existence of pointwise asymptotic expansion on the positive part for certain semi-positive line bundles.

Research Notes (Not Intended to Publish):

Localization on P^2

This records a partial progress on a localization calculation on Gromov-Witten invariant, which is a part of MoST Research Assistant Report, supervised by Prof. Chin-Lung Wang.

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