地点：海纳苑2幢204室

时间：5月18日，16:15 - 17:35

报告人：谢松晏 Song-Yan Xie（中国科学院大学）

摘要:  

Pretalk:  Introduction to Nevanlinna theory and complex hyperbolicity

Abstract: How many values can a nonconstant entire function miss? In 1879, Picard gave a stunning answer: at most one. This seemingly simple result opened a century-long journey. We trace the milestones: Hadamard's famous regret over Jensen's formula, Nevanlinna's quantitative First and Second Main Theorems (1925), Cartan's generalization to higher dimensions (1933), and the geometric synthesis achieved by Kobayashi (1967/70), whose hyperbolicity conjectures now guide the search for entire curves in projective varieties. Along the way, we encounter a rich interplay between complex analysis, algebraic geometry, andnumber theory — culminating in Lang's conjecture and the Vojta dictionary.

Research talk:  Vanishing of Invariant 2-Jet Differentials and Applications

Abstract: We present two recent advances in Nevanlinna theory and complex hyperbolicity, both powered by a new method of proving vanishing for invariant \(2\)-jet differentials via algebraic reduction and computer algebra.

First, we establish an effective Second Main Theorem in Nevanlinna theory for three generic smooth conics in \(\mathbb{P}^2\). For any algebraically nondegenerate entire curve, we obtain\[T_f(r) \;\leqslant\; 5 \sum_{i=1}^3 N_f^{[1]}(r,\mathcal{C}_i) + o(T_f(r)) \quad\parallel,

\]with the optimal truncated counting function. This settles a long-standing open problem.

Second, we prove new \emph{key vanishing lemmas} for negatively twisted invariant logarithmic and compact \(2\)-jet differentials. Together with the P\u{a}un--Rousseau strategy, these yield improved degree bounds for the Kobayashi hyperbolicity conjecture in dimension two:\[d \geqslant 17 \quad\text{(very generic surfaces in }\mathbb{P}^3\text{)}, \qquadd \geqslant 12 \quad\text{(generic curve complements in }\mathbb{P}^2\text{)}.\]

Both results rely on the same core framework: an algebraic reduction that turns the geometric problem into a finite linear system, which is then solved by optimized Maple computations. The method also illuminates the path toward the 2-jet approach thresholds \(d=15\) and \(d=11\).