Retract rational varieties are uniformly retract rational

Technical Summary: "Retract Rational Varieties Are Uniformly Retract Rational"

Problem Statement
The paper addresses a fundamental question in the theory of algebraic varieties regarding the local structure of rational and retract rational varieties. Specifically, it investigates whether the property of being "retract rational" implies "uniform retract rationality."

• Retract Rationality: An algebraic variety $X$ over an infinite field $K$ is retract rational if there exists a Zariski open dense subset $V \subset X$, a natural number $m$, a Zariski open set $U \subset K^m$, and regular mappings $V \to U \to V$ whose composition is the identity on $V$.
• Uniform Retract Rationality: $X$ is uniformly retract rational if for every point $x \in X$, there exists a Zariski open neighborhood $V$ of $x$ (not necessarily dense in the global sense, but local to $x$) satisfying the same retraction condition.

The central question (Question 1.7) asks: Are all nonsingular retract rational algebraic varieties uniformly retract rational?

This inquiry is motivated by the broader context of Gromov ellipticity. Gromov introduced "algebraic ellipticity" for complex nonsingular varieties, a property implying strong homotopy and approximation features. It was known that uniform rationality implies algebraic ellipticity. While rational varieties are a subset of retract rational varieties, it was an open problem whether all nonsingular rational complex projective varieties are algebraically elliptic (Question 1.3). A positive answer to the uniform retract rationality question would provide a partial resolution to this, as Observation 1.6 establishes that uniform retract rationality implies algebraic ellipticity (in the complex case) or malleability (in the real case).

Methodology
The proof relies on a combination of local algebraic geometry, commutative algebra, and the manipulation of polynomial mappings. The core strategy involves extending local rational mappings to regular germs.

1. Local Extension of Mappings (Proposition 3.1): The pivotal technical tool is Proposition 3.1, which asserts that if a rational mapping $F: K^n \dashrightarrow Y$ is regular at a point $x_0$ on a subvariety $X \subset K^n$, then there exists a regular germ $G: (K^n, x_0) \to Y$ that agrees with $F$ on $X$ near $x_0$.
2. Construction of the Extension: The proof of Proposition 3.1 utilizes Proposition 2.2. This proposition constructs a specific polynomial mapping $\sigma: X \times K^{n-m} \to K^n$ (where $m = \dim X$) with the following properties:
   • It acts as the identity on $X$ when the second variable is zero.
   • Its derivative at the base point is an isomorphism.
   • The induced homomorphism on local rings is surjective modulo a specific ideal generated by the denominators of the rational map.
     This construction allows the author to lift the rational map defined on $X$ to a regular map on the ambient space $K^n$ locally.
3. Genericity Arguments: The construction of $\sigma$ relies on Lemma 2.3, which uses dimension counting on Grassmannians to ensure that a generic linear subspace intersects the zero set of the relevant ideal only at the origin. This ensures the necessary transversality and algebraic independence for the extension to work.
4. Application of Lemma 2.4: This lemma ensures that the regularity of a rational function is preserved under small perturbations of the mapping, allowing the transition from the constructed germ on the product space back to the ambient space.

Key Contributions and Results
The primary result of the paper is Theorem 1.8:

Let $X$ be a nonsingular retract rational algebraic variety over an infinite field $K$. Then it is uniformly retract rational.

The proof proceeds by taking an arbitrary point $x_0 \in X$. Since $X$ is retract rational, there exist open sets and retractions globally. The author extends the retraction map $r: U \to V$ (where $V \subset X$) to a rational map on the ambient space. Using Proposition 3.1, they construct a regular germ $G$ defined on a neighborhood of $x_0$ in the ambient space that restricts to the identity on $X$ near $x_0$. This germ effectively provides the local retraction required for uniform retract rationality.

Corollaries and Significance
The paper derives Corollary 1.9 as a direct consequence:

Nonsingular rational complex projective varieties are algebraically elliptic.

Significance Claims:

• Resolution of a Specific Question: The paper provides an affirmative answer to Question 1.7, establishing that the local property (uniformity) follows from the global property (retract rationality) for nonsingular varieties over infinite fields.
• Partial Answer to Gromov's Question: By combining Theorem 1.8 with Observation 1.6, the paper confirms that all nonsingular rational complex projective varieties are algebraically elliptic. This resolves the second part of Question 1.3 for the class of rational varieties.
• Clarification of Hierarchy: The results clarify the hierarchy of properties for irreducible nonsingular complex projective varieties:
  $$\text{Uniformly Rational} \implies \text{Rational} \implies \text{Retract Rational} \implies \text{Uniformly Retract Rational} \implies \text{Algebraically Elliptic}$$
  The paper notes that the reverse implications do not hold in general (citing examples of elliptic varieties that are not rational or not retract rational).

The paper does not claim to solve Question 1.3 in its entirety (i.e., for all nonsingular varieties, not just rational ones), nor does it propose new experimental applications. Its contribution is strictly theoretical, bridging the gap between retract rationality and uniform retract rationality using algebraic extension techniques.