[Paper Review] Positivity in convergence of the inverse $\sigma_{n-1}$-flow

Jian Xiao|arXiv (Cornell University)|Oct 29, 2016

Geometry and complex manifolds22 references3 citations

TL;DR

This paper verifies a weakened version of Lejmi and Székelyhidi's conjecture on the solvability of the inverse $σ_{n-1}$-flow on compact Kähler manifolds by establishing pointwise positivity of the $(n-1,n-1)$-form $ω^{n-1} - \frac{(n-1)!}{(n-1)!1!}\omega^{0}\wedge\alpha^{n-1}$ under global numerical positivity conditions. The key result confirms the existence of a strictly positive representative in the cohomology class, extending to Kähler 3-folds and providing a sufficient condition for differences of movable curve classes to lie in the interior of the movable cone.

ABSTRACT

We study positivity in the conjecture proposed by Lejmi and Sz\\'{e}kelyhidi on finding effective necessary and sufficient conditions for solvability of the inverse $\\sigma_k$ equation, or equivalently, for convergence of the inverse $\\sigma_k$-flow. In particular, for the inverse $\\sigma_{n-1}$-flow we partially verify their conjecture by obtaining the desired positivity for $(n-1, n-1)$ cohomology classes. As an application, we also partially verify their conjecture for 3-folds.

Motivation & Objective

• To verify a weakened form of Lejmi and Székelyhidi's conjecture on solvability of the inverse $σ_k$-equation for $k = n-1$.
• To establish pointwise positivity of the $(n-1,n-1)$-form associated with the inverse $σ_{n-1}$-flow under global numerical positivity conditions.
• To extend the result to Kähler 3-folds via prior results on $k=1$ and $k=n-1$ cases.
• To provide a sufficient condition for the difference of two movable curve classes to lie in the interior of the movable cone of curves.

Proposed method

• Utilizes divisorial Zariski decomposition for pseudo-effective $(1,1)$ classes, as established by Boucksom.
• Applies a Morse-type bigness criterion for movable $(n-1,n-1)$ classes, drawing from Xia (2014) and LX (2016a).
• Employs a restricted version of reverse Khovanskii-Teissier inequalities, derived from Popovici (2016) and Wang-Ning (2016).
• Applies complex Monge-Ampère type equations to construct $(k+l)$-subharmonic representatives, leveraging results from [DK12] and [Sun16].
• Reduces pointwise inequalities to intersection number inequalities on complex tori via cohomological reduction.
• Uses the method of [Pop16] to handle $(k+l)$-subharmonic forms and derive volume form inequalities.

Experimental results

Research questions

• RQ1 Does the numerical positivity condition on all subvarieties of dimension $p$ with $k \leq p \leq n-1$ imply pointwise positivity of the $(n-1,n-1)$-form in the inverse $\sigma_{n-1}$-flow?
• RQ2 Can the weak conjecture be verified for $k = n-1$ on general compact Kähler manifolds?
• RQ3 Does the difference of two movable curve classes lie in the interior of the movable cone under suitable positivity assumptions?
• RQ4 Can the positivity of the $(n-1,n-1)$-form be established using $(k+l)$-subharmonic forms and volume form inequalities?
• RQ5 Is the reverse Khovanskii-Teissier inequality valid for $(k+l)$-subharmonic forms in the context of movable cohomology classes?

Key findings

• The paper proves that under the global numerical positivity condition (3), the cohomology class $\{\omega^{n-1} - \frac{(n-1)!}{(n-1)!1!}\omega^{0}\wedge\alpha^{n-1}\}$ contains a smooth strictly positive $(n-1,n-1)$-form, verifying the weak conjecture for $k=n-1$.
• As a consequence, the conjecture is fully verified for Kähler 3-folds, extending previous results on $k=1$ and $k=n-1$ cases.
• A sufficient condition is established under which the difference of two movable curve classes lies in the interior of the movable cone, using refined structure of the movable cone of curves.
• The proof relies on the divisorial Zariski decomposition and Morse-type bigness criterion for movable $(n-1,n-1)$ classes.
• A restricted reverse Khovanskii-Teissier inequality is shown to hold for $(k+l)$-subharmonic forms under suitable assumptions, enabling volume form comparisons.
• The method reduces pointwise inequalities to intersection number inequalities on complex tori, leveraging cohomological and geometric positivity tools.