[Paper Review] Siu-Yeung jet differentials on complete intersection surfaces X^2 in P^4(C)

Joël Merker|arXiv (Cornell University)|Dec 19, 2013

Meromorphic and Entire Functions10 references3 citations

TL;DR

This paper constructs explicit holomorphic jet differentials on complete intersection surfaces $X^2 \subset \mathbb{P}^4(\mathbb{C})$ of degrees $d \leq e$, using an extrinsic approach inspired by Siu-Yeung's 1996 method. It proves that for $d \geq 752$ and $e \leq \frac{1}{648}d^2$, the space of global sections of $\text{Sym}^m T_X^*$ has dimension at least $\frac{1}{93312}d^3 - \frac{61}{7776}d^2 - \frac{17}{108}d - \frac{28}{27}$ when $m = \lfloor d/12 \rfloor$, providing the first effective lower bound for such sections on complete intersections beyond hypersurfaces.

ABSTRACT

On a generic complete intersection surface X^2 in P^4(C) having polynomial equations z^d = R(x,y) and t^e = S(x,y) with 752 <= d <= e <= d^2/648, there exist extrinsic meromorphic jet differentials of the form J(x,y,x',y') / [y^d z^{m(d-1)} t^{m(e-1)}] where J(x,y,x',y') = sum_{j+k+p+q=m} A_{j,k,p,q}(x,y) (x')^j (y')^k (R')^p (S')^q (R)^{m-p} (S)^{m-q} with the complex coefficients of the polynomials A_{j,k,p,q}(x,y) satisfying a certain system of linear equations depending explicitly on R, S, the restriction to X^2 of which provides nonzero intrinsic global holomorphic sections of the bundle of symmetric m-differentials Sym^m T_X^*.

Motivation & Objective

• To construct explicit holomorphic jet differentials on complete intersection surfaces $X^2 \subset \mathbb{P}^4(\mathbb{C})$ using an extrinsic, coordinate-based method.
• To provide effective, computable lower bounds on the dimension of global sections of $\text{Sym}^m T_X^*$, overcoming limitations of intrinsic methods.
• To extend Siu-Yeung's extrinsic jet differential technique beyond hypersurfaces to codimension-2 complete intersections.
• To establish a quantitative link between the degrees of defining equations and the existence of nontrivial holomorphic jet differentials.
• To demonstrate that for sufficiently high degrees $d \geq 752$, the space $H^0(X, \text{Sym}^m T_X^*)$ is nontrivial with a concrete dimension estimate.

Proposed method

• Uses an extrinsic construction of meromorphic jet differentials in affine coordinates on $X^2$ defined by $z^d = R(x,y)$, $t^e = S(x,y)$.
• Defines a jet differential $\mathcal{J}(x,y,x',y')$ as a homogeneous polynomial in $x'$, $y'$, with coefficients $A_{j,k,p,q}(x,y)$ of degree $\leq a \leq d - 4m$.
• Imposes divisibility by $y^d$ on the numerator to ensure holomorphicity after restriction to $X^2$, leading to a linear system on coefficients $A_{j,k,p,q}^{h,i}$.
• Applies the Euler characteristic formula from Brückmann (1997) to compute $\chi(\text{Sym}^m T_X^*)$ as a cubic polynomial in $m$ with coefficients depending on $d$ and $e$.
• Derives a sufficient inequality involving $d$ and $e$ to ensure the solution space of the linear system has positive dimension, leading to the final bound.
• Sets $m = \lfloor d/12 \rfloor$ and derives the dimension lower bound by analyzing the degree of freedom in the coefficient system.

Experimental results

Research questions

• RQ1 Can explicit holomorphic jet differentials be constructed on complete intersection surfaces in $\mathbb{P}^4(\mathbb{C})$ using coordinate-based methods?
• RQ2 What is the minimal degree $d$ for which $H^0(X, \text{Sym}^m T_X^*)$ is nontrivial for $m = \lfloor d/12 \rfloor$?
• RQ3 How does the dimension of the space of global jet differentials grow with the degrees $d$ and $e$ of the defining equations?
• RQ4 Can effective lower bounds on $\dim H^0(X, \text{Sym}^m T_X^*)$ be derived for complete intersections beyond hypersurfaces?
• RQ5 What constraints on $d$ and $e$ ensure the existence of nontrivial holomorphic jet differentials via the extrinsic construction?

Key findings

• For $d \geq 752$ and $e \leq \frac{1}{648}d^2$, the space $H^0(X, \text{Sym}^m T_X^*)$ has dimension at least $\frac{1}{93312}d^3 - \frac{61}{7776}d^2 - \frac{17}{108}d - \frac{28}{27}$ when $m = \lfloor d/12 \rfloor$.
• The construction yields a holomorphic jet differential of order $m$ via a linear system of coefficients $A_{j,k,p,q}^{h,i}$ satisfying a system of equations derived from divisibility by $y^d$.
• The dimension bound is derived from a sufficient inequality involving the cubic growth of the solution space, ensuring positivity of the Euler characteristic and existence of nontrivial sections.
• The method provides the first effective, explicit construction of nontrivial holomorphic jet differentials on complete intersection surfaces of codimension 2 in $\mathbb{P}^4$.
• The result establishes a quantitative threshold for hyperbolicity-related phenomena in complete intersections, with $d \geq 752$ being the minimal degree for which the bound is nontrivial.
• The analysis confirms that the extrinsic jet differential construction yields a nonvanishing section of $\text{Sym}^m T_X^*$ for the specified range of degrees.