Searcharxiv⌕ Search

arXiv subjects

Song-Yan Xie

Publications and source records attributed to Song-Yan Xie.

At least 19 recordsLinked to original sources
arXiv2026

Analytic Construction of Rational Curves on Fano Manifolds

Inspired by methods for constructing entire curves in Oka geometry, we give an analytic construction of rational curves on a complex Fano manifold $X$. Yau's theorem provides a K\"ahler metric with positive Ricci curvature. Using this curvature to guide deformations of holomorphic discs, we construct maps from discs of radii tending to infinity with uniformly bounded area. A central point is to preserve the derivative normalization through the limiting process. This yields a nonconstant entire map $f:\mathbb C\rightarrow X$ of finite area. This map extends across infinity to a nonconstant holomorphic map $\mathbb P^1\to X$. Combined with algebraic arguments in characteristic zero, the construction yields proofs of the rational connectedness of Fano manifolds and of Hartshorne's conjecture on ample tangent bundles.

math.CV↗
arXiv2026

Analytic and Algebraic Oka-1 Approximation for Smooth Projective Morphisms with Rationally Connected Fibers

Let $\pi:Z\rightarrow Y$ be a smooth projective morphism of complex manifolds with connected rationally connected fibers. We prove holomorphic approximation on arbitrary compact sets and finite-jet interpolation on arbitrary closed discrete sets for continuous liftings defined on open Riemann surfaces and holomorphic near those sets. For smooth projective morphisms of smooth complex algebraic varieties and algebraic base maps from smooth affine curves, the approximating liftings can be chosen algebraic, with interpolation on any finite set. In both cases the resulting lifting is homotopic to the initial one through continuous liftings of the fixed base map. For connected smooth projective complex manifolds, this gives the equivalence between the algebraic Oka-1 property and rational connectedness. Every rationally connected smooth projective complex manifold is also Oka-1.

math.CV↗
arXiv2026

Kobayashi Hyperbolicity of General Surfaces via the Poincar\'e Problem

We prove that a general surface in $\mathbb{P}^3$ of degree at least $18$ contains no rational or elliptic curves, strengthening the classical result of Clemens by replacing the original very general assumption by a genuine Zariski-open condition. Previously, nonexistence results in the ``general'' setting were known only in much higher degrees. Combining this with established algebraic degeneracy results for entire curves, we deduce the Kobayashi hyperbolicity of a general surface in $\mathbb{P}^3$ of degree at least $18$, thereby resolving a question asked by Demailly--El Goul. Our proof uses foliations induced by $2$-jet differentials. Two independent such differentials give rise to a multi-foliation tangent to all rational and elliptic curves. We establish a Poincar\'e-type bound for its algebraic leaves, yielding a mechanism to upgrade very general statements to general ones. Our method also applies to complements of plane curves. In particular, we prove that the complement of two general cubic curves in $\mathbb{P}^2$ is hyperbolically embedded.

math.AG↗
arXiv2026

Periods of Oka K3 surfaces are dense

We give the first known examples of Oka K3 surfaces: every smooth hypersurface of multidegree \((2,2,2)\) in \((\mathbb{P}^1)^3\) is Oka. Building on this example, we prove that in every local Kuranishi family of K3 surfaces the points with Oka fibers form a dense $G_\delta$ subset of the base, and the sublocus of Oka fibers with vanishing N\'eron--Severi group is likewise a dense $G_\delta$ subset. We also obtain projective and nonprojective Oka Kummer surfaces and prove an analogous density statement in the Kummer period domain.

math.CV↗
arXiv2026

Invariant two-jets and effective hyperbolicity for complements of two plane curves

Let $D=C_1+C_2\subset\mathbb{P}^2$ be a simple normal crossing union of smooth plane curves of degrees $1\leqslant d_1\leqslant d_2$. We prove an effective Second Main Theorem for a general ordered pair whenever \[ d_1,d_2\geqslant3, \qquad\text{or}\qquad d_1=2,\ d_2\geqslant5, \qquad\text{or}\qquad d_1=1,\ d_2\geqslant8. \] For each admissible degree pair, there is a nonempty Zariski-open set of ordered pairs $(C_1,C_2)$ for which every algebraically nondegenerate entire curve $f:\mathbb{C}\to\mathbb{P}^2$ whose image is not contained in $D=C_1+C_2$ satisfies \[ T_f(r)\leqslant \mathcal{A}_{d_1,d_2}N_f^{[1]}(r,D)+o(T_f(r)) \ \|. \] For two cubics one may take $\mathcal{A}_{3,3}=57$; for a conic and a quintic, $\mathcal{A}_{2,5}=45$; and for a line and an octic, $\mathcal{A}_{1,8}=69$. Intersecting the resulting Zariski-open parameter locus with Xi Chen's very-general algebraic-hyperbolicity locus yields Kobayashi hyperbolicity and hyperbolic embedding of the complement. The proof first constructs one negatively twisted invariant two-jet differential. It then obtains a second equation either from a Demailly--El Goul zero-locus argument or by differentiating with mixed $\mathcal{O}_{\mathbb{P}^2}(3)$ slanted vector fields. A finite calculation is needed only for a short list of low twists. In those cases, exact rank certificates over finite fields prove the required Key Vanishing Lemma.

math.AG↗
arXiv2026

Sharp Summability of Nevanlinna Defects for Finite-Lower-Order Holomorphic Curves

For a countable family of hyperplanes $H_j\subset \mathbb{P}^m$, $j\in\mathbb{N}$, in general position and a linearly nondegenerate holomorphic curve $f\colon \mathbb{C}\to \mathbb{P}^m$ of finite lower order, we prove that the Nevanlinna defects $\delta_f(H_j)$ satisfy $$ \sum_{j=1}^{\infty}\delta_f(H_j)^{1/3}<\infty. $$ This resolves a long-standing open problem in Nevanlinna theory and extends Weitsman's celebrated scalar endpoint theorem (the case $m=1$) as well as Krutin's results for exponents strictly greater than $1/3$. The same uniform finite-family estimate yields the corresponding endpoint theorem for divisors cut out on a projective variety by ambient hypersurfaces of uniformly bounded degree, assuming that the divisors are in general position with respect to the variety and that the curve is not contained in the support of any divisor.

math.CV↗
arXiv2026

Elliptic complements of cubic hypersurfaces

Let $D\subset\mathbb{P}^n$, $n\geqslant2$, be an arbitrary cubic hypersurface, and let $D_{\mathrm{red}}$ denote its reduced support. We prove that $\mathbb{P}^n\setminus D$ is holomorphically elliptic, and hence Oka, unless $D_{\mathrm{red}}$ is the union of three distinct hyperplanes containing a common codimension-two linear subspace. In the exceptional case, $\mathbb{P}^n\setminus D\cong(\mathbb{C}\setminus\{0,1\})\times\mathbb{C}^{n-1}$, so the complement is not Oka. As applications, we prove that, for every elliptic curve $E$, the space of degree-three holomorphic maps $E\to\mathbb{P}^1$, and the space of degree-three holomorphic self-maps of $\mathbb{P}^1$, are both holomorphically elliptic, and hence Oka. The second application is connected with the classification through an irreducible cubic hypersurface in $\mathbb{P}^4$.

math.AG↗
arXiv2026

A Symmetry Method for Key Vanishing Lemmas and Optimal $2$-Jet Thresholds for Hyperbolicity

This manuscript reports on an ongoing project. The finite full-column-rank computations underlying the Key Vanishing Lemmas isolated below are currently undergoing exact computer-assisted verification. Conditional on successful completion of that verification, the arguments presented here would establish the two positivity endpoints of Demailly's invariant two-jet Riemann--Roch calculation: every very general surface in $\mathbb{P}^3$ of degree $15$ would be Kobayashi hyperbolic, and the complement of every general smooth plane curve of degree $11$ would be Kobayashi hyperbolic. At the logarithmic endpoint the same conditional argument would also give a Second Main Theorem with coefficient $1422$; the preceding work treated all degrees $d\geqslant12$. The argument separates an infinite geometric range from a finite algebraic obstruction.An $\mathcal{O}(3)$ slanted-vector-field package combines a quadratic acceleration with a classical matrix lift. Their incompatible actions on the invariant fibre ring, together with the common-twist, prime-ideal, and relative-family interfaces, exclude the dependence alternative for every weight $m\geqslant3$. This improves the differentiation range from $t\geqslant8$ to $t\geqslant4$ and leaves, beyond the twist-one ranges, only five compact and thirteen logarithmic systems. The residual systems are reduced by symmetry. A two-term Fermat perturbation carries an involution, so a hypothetical negatively twisted two-jet differential has an invariant or anti-invariant component. Imposing this eigencondition before elimination adds exact linear constraints without adding unknowns. The required full-column-rank statements are isolated in the Key Vanishing Lemmas as finite computational targets. This paper records their finite reduction and proof-to-certificate interface. Their exact computer-assisted verification is in progress.

math.AG↗
arXiv2026

Vanishing of Invariant 2-Jet Differentials and Improved Hyperbolicity Degree Bounds in Dimension Two

This paper establishes new degree bounds for Kobayashi hyperbolicity in dimension two. Our main results are: -- A very generic surface in $\mathbb{P}^3$ of degree at least $17$ is Kobayashi hyperbolic. -- The complement of a generic curve in $\mathbb{P}^2$ of degree at least $12$ is Kobayashi hyperbolic. These bounds improve the long-standing records in the field, lowering the threshold from $18$ to $17$ for surfaces (P\u{a}un) and from $14$ to $12$ for complements (Rousseau). Central to the proofs are new vanishing results for certain negatively twisted invariant $2$-jet differentials, obtained through a novel combination of algebraic reduction and computer algebra. Since Demailly's Santa Cruz lectures in 1995, the thresholds for the existence of such differentials---and consequently the limits of what $2$-jet techniques can accomplish toward the Kobayashi conjecture in dimension two---have been recognized as $d = 15$ in the compact case and $d = 11$ in the logarithmic case. While previous approaches were unable to reach these targets, the present work provides both the theoretical foundations and the algorithmic framework required to access them, and has already improved the known bounds to $d = 17$ and $d = 12, 13$, respectively. As an unexpected byproduct, our computational method reveals the existence of nonzero negatively twisted invariant $2$-jet differentials with weighted degree $3$ for hyperelliptic-type equations of degree at least $11$ in the logarithmic case and degree at least $15$ in the compact case. Moreover, in the logarithmic setting, we establish an effective quantitative refinement via a Second Main Theorem in Nevanlinna theory.

math.CV↗
arXiv2026

Frequently hypercyclic meromorphic curves with slow growth

We construct entire curves in projective spaces that exhibit frequent hypercyclicity under translations along countably many prescribed directions while maintaining optimal slow growth rates. Furthermore, we establish a fundamental dichotomy by proving the impossibility of such curves simultaneously preserving frequent hypercyclicity for uncountably many directions under equivalent growth constraints. This result reveals a striking contrast with classical hypercyclicity phenomena, where entire functions can achieve hypercyclicity over some uncountable direction set without growth rate compromise. Our methodology is rooted in Nevanlinna theory and guided by the Oka principle, offering new insights into the relationship between dynamical properties and growth rates of entire curves in projective spaces.

math.CV↗
arXiv2025

A Second Main Theorem for Entire Curves Intersecting Three Conics

We establish a Second Main Theorem for entire holomorphic curves \( f: \mathbb{C} \to \mathbb{P}^2 \) intersecting a generic configuration of three conics \(\mathcal{C}= \mathcal{C}_1+ \mathcal{C}_2+ \mathcal{C}_3 \) in the complex projective plane $\mathbb{P}^2$. Using invariant logarithmic $2$-jet differentials with negative twists, we prove the estimate \[ T_f(r) \leqslant 5 \sum_{i=1}^3 N_f^{[1]}(r, \mathcal{C}_i) + o\big(T_f(r)\big)\quad\parallel, \] where \( T_f(r) \) is the Nevanlinna characteristic function, and \( N_f^{[1]}(r, \mathcal{C}_i) \) is the $1$-truncated counting function. The key innovation of our approach is establishing new vanishing lemmas of the form \[ H^0\bigl(\mathbb{P}^2,\, E_{2,m}T_{\mathbb{P}^2}^*(\log \mathcal{C}) \otimes \mathcal{O}_{\mathbb{P}^2}(-t)\bigr) = 0 \] for specific pairs \((m, t)\), achieved by combining algebro-geometric arguments with computer-assisted computations through a mod-\(p\) reduction technique. This provides a systematic method for proving vanishing results for negatively twisted jet differentials--a key component in complex hyperbolic geometry--thereby filling a missing piece in two-dimensional complex hyperbolicity theory.

math.CV↗
arXiv2025

Generating all Ahlfors currents by a single entire curve

Let \(X\) be a compact complex manifold possessing the \emph{Runge approximation property on discs}, meaning that every holomorphic map from a closed disc into \(X\) is approximable by a global holomorphic map from \(\mathbb{C}\). We construct an entire curve \(F : \mathbb{C} \to X\) such that the associated family of concentric holomorphic discs \(\{F|_{\overline{\mathbb{D}}_r}\}_{r>0}\) generates all Ahlfors currents on \(X\), thereby settling a conjecture of Sibony.

math.CV↗
arXiv2024

Universal holomorphic maps with slow growth II. functional analysis methods

By means of hypercyclic operator theory, we complement our previous results on hypercyclic holomorphic maps between complex Euclidean spaces having slow growth rates,by showing {\it abstract abundance} rather than {\it explicit existence}. Next, we establish that, in the space of holomorphic maps from $\mathbb{C}^n$ to any connected Oka manifold $Y$, equipped with the compact-open topology, there exists a {\em dense} subset consisting of common {\em frequently hypercyclic} elements for all nontrivial translation operators. To our knowledge, this is new even for $n=1$ and $Y=\mathbb{C}$.

math.CV↗
arXiv2023

Entire holomorphic curves into $\mathbb{P}^n(\mathbb{C})$ intersecting $n+1$ general hypersurfaces

Let $\{D_i\}_{i=1}^{n+1}$ be $n+1$ hypersurfaces in $\mathbb{P}^n(\mathbb{C})$ with total degrees $\sum_{i=1}^{n+1} °D_i\geqslant n+2$, in general position and satisfying a generic geometric condition: every $n$ hypersurfaces intersect only at smooth points and the intersection is transversal. Then, for every algebraically nondegenerate entire holomorphic curve $f\colon\mathbb{C}\rightarrow\mathbb{P}^n(\mathbb{C})$, we show a Second Main Theorem: $$ \sum_{i=1}^{n+1} δ_f(D_i) < n+1 $$ in terms of defect inequality in Nevanlinna theory. This is the first result in the literature on Second Main Theorem for $n+1$ general hypersurfaces in $\mathbb{P}^n(\mathbb{C})$ with optimal total degrees.

math.CV↗
arXiv2023

Entire curves generating all shapes of Nevanlinna currents

First, we show that every complex torus $\mathbb{T}$ contains some entire curve $g: \mathbb{C}\rightarrow \mathbb{T}$ such that the concentric holomorphic discs $\{g\restriction_{\overline{\mathbb D}_{r}}\}_{r>0}$ can generate all the Nevanlinna/Ahlfors currents on $\mathbb T$ at cohomological level. This confirms an anticipation of Sibony. Developing further our new method, we can construct some twisted entire curve $f: \mathbb{C}\rightarrow \mathbb{CP}^1\times E$ in the product of the rational curve $\mathbb{CP}^1$ and an elliptic curve $E$, such that, concerning Siu's decomposition, demanding any cardinality $|J|\in \mathbb{Z}_{\geqslant 0}\cup \{\infty\}$ and that $\mathcal{T}_{\mathrm{diff}}$ is trivial ($|J|\geqslant 1$) or not ($|J|\geqslant 0$), we can always find a sequence of concentric holomorphic discs $\{f\restriction_{\overline{\mathbb D}_{r_j}}\}_{j \geqslant 1}$ to generate a Nevanlinna/Ahlfors current $\mathcal{T}=\mathcal{T}_{\mathrm{alg}}+\mathcal{T}_{\mathrm{diff}}$ with the singular part $\mathcal{T}_{\mathrm{alg}}=\sum_{j\in J} \,λ_j\cdot[\mathsf C_j]$ in the desired shape. This fulfills the missing case where $|J|=0$ in the previous work of Huynh-Xie. By a result of Duval, each $\mathsf C_j$ must be rational or elliptic. We will show that there is no a priori restriction on the numbers of rational and elliptic components in the support of $\mathcal{T}_{\mathrm{alg}}$, thus answering a question of Yau and Zhou. Moreover, we will show that the positive coefficients $\{λ_j\}_{j\in J}$ can be arbitrary as long as the total mass of $\mathcal{T}_{\mathrm{alg}}$ is less than or equal to $1$. Our results foreshadow striking holomorphic flexibility of entire curves in Oka geometry, which deserves further exploration.

math.CV↗
arXiv2023

Entire curves producing distinct Nevanlinna currents

First, inspired by a question of Sibony, we show that in every compact complex manifold $Y$ with certain Oka property, there exists some entire curve $f: \mathbb{C}\rightarrow Y$ generating all Nevanlinna/Ahlfors currents on $Y$, by holomorphic discs $\{f\restriction_{\mathbb{D}(c, r)}\}_{c\in \mathbb{C}, r>0}$. Next, we answer positively a question of Yau, by constructing some entire curve $g: \mathbb{C}\rightarrow X$ in the product $X:=E_1\times E_2$ of two elliptic curves $E_1$ and $E_2$, such that by using concentric holomorphic discs $\{g\restriction_{\mathbb{D}_{ r}}\}_{r>0}$ we can obtain infinitely many distinct Nevanlinna/Ahlfors currents proportional to the extremal currents of integration along curves $[\{e_1\}\times E_2]$, $[E_1\times \{e_2\}]$ for all $e_1\in E_1, e_2\in E_2$ simultaneously. This phenomenon is new, and it shows tremendous holomorphic flexibility of entire curves in large scale geometry.

math.CV↗
arXiv2023

Universal holomorphic maps with slow growth I. An Algorithm

We design an Algorithm to fabricate universal holomorphic maps between any two complex Euclidean spaces, within preassigned transcendental growth rate. As by-products, universal holomorphic maps from $\mathbb{C}^n$ to $\mathbb{CP}^m$ ($n\leqslant m$) and to complex tori having slow growth are obtained. We take inspiration from Oka manifolds theory, Nevanlinna theory, and hypercyclic operators theory.

math.CV↗