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A relative Nash-Tognoli theorem over $\mathbb{Q}$ and application to the $\mathbb{Q}$-algebraicity problem

Enrico Savi

TL;DR

This work develops a relative version over ${\\mathbb Q}$ of the Nash–Tognoli theorem, enabling compact nonsingular real algebraic sets with specified submanifolds to be modeled by polynomial equations with rational coefficients.The approach combines ${\\mathbb Q}$-regularity, Bott–Samelson desingularizations of Schubert varieties, and relative bordism to produce ${\\mathbb Q}$-nonsingular models and a Nash diffeomorphism that matches the submanifolds exactly.It proves that real Grassmannians have projectively ${\\mathbb Q}$-algebraic homology by constructing explicit ${\\mathbb Q}$-desingularizations of embedded Schubert varieties, giving algebraic representatives for homology classes.These results open pathways for the ${\\mathbb Q}$-algebraicity problem in low dimensions and provide a framework for rational-coefficient descriptions with potential algorithmic applications.

Abstract

We prove a relative version over $\mathbb{Q}$ of Nash-Tognoli theorem, that is: Let $M$ be a compact smooth manifold with closed smooth submanifolds $M_1,\dots,M_\ell$ in general position, then there exists a nonsingular real algebraic set $M'\subset\mathbb{R}^n$ with nonsingular algebraic subsets $M_1',\dots,M_\ell'$ and a diffeomorphism $h:M\to M'$ such that $h(M_i)=M_i'$ for all $i=1,\dots,\ell$ such that $M',M_1',\dots,M_\ell'$ are described, both globally and locally, by polynomial equations with rational coefficients. In addition, if $M,M_1,\dots,M_\ell$ are nonsingular algebraic sets, then we prove the diffeomorphism $h:M\to M'$ can be chosen semialgebraic and the result can be extended to the noncompact case. In the proof we describe also the $\mathbb{Z}/2\mathbb{Z}$-homological cycles of real embedded Grassmannian manifolds by nonsingular algebraic representatives over $\mathbb{Q}$ via the Bott-Samelson resolution of Schubert varieties.

A relative Nash-Tognoli theorem over $\mathbb{Q}$ and application to the $\mathbb{Q}$-algebraicity problem

TL;DR

This work develops a relative version over ${\\mathbb Q}$ of the Nash–Tognoli theorem, enabling compact nonsingular real algebraic sets with specified submanifolds to be modeled by polynomial equations with rational coefficients.The approach combines ${\\mathbb Q}$-regularity, Bott–Samelson desingularizations of Schubert varieties, and relative bordism to produce ${\\mathbb Q}$-nonsingular models and a Nash diffeomorphism that matches the submanifolds exactly.It proves that real Grassmannians have projectively ${\\mathbb Q}$-algebraic homology by constructing explicit ${\\mathbb Q}$-desingularizations of embedded Schubert varieties, giving algebraic representatives for homology classes.These results open pathways for the ${\\mathbb Q}$-algebraicity problem in low dimensions and provide a framework for rational-coefficient descriptions with potential algorithmic applications.

Abstract

We prove a relative version over of Nash-Tognoli theorem, that is: Let be a compact smooth manifold with closed smooth submanifolds in general position, then there exists a nonsingular real algebraic set with nonsingular algebraic subsets and a diffeomorphism such that for all such that are described, both globally and locally, by polynomial equations with rational coefficients. In addition, if are nonsingular algebraic sets, then we prove the diffeomorphism can be chosen semialgebraic and the result can be extended to the noncompact case. In the proof we describe also the -homological cycles of real embedded Grassmannian manifolds by nonsingular algebraic representatives over via the Bott-Samelson resolution of Schubert varieties.
Paper Structure (9 sections, 18 theorems, 71 equations, 4 figures)

This paper contains 9 sections, 18 theorems, 71 equations, 4 figures.

Table of Contents

  1. Review on (real) ${\mathbb Q}$-algebraic geometry
  2. Fundamental examples
  3. Homology of real embedded Grassmannians
  4. ${\mathbb Q}$-Desingularization of real embedded Schubert varieties
  5. Real embedded Grassmannians have projectively ${\mathbb Q}$-algebraic homology
  6. Relative ${\mathbb Q}$-algebraic constructions
  7. ${\mathbb Q}$-Algebraic bordism classes and unoriented relative bordisms
  8. A review on ${\mathbb Q}$-stable ${\mathbb Q}$-algebraic sets
  9. Nash-Tognoli theorem over ${\mathbb Q}$ & the Relative ${\mathbb Q}$-algebraicity problem

Key Result

Proposition 1.6

Let $V\subset{\mathbb R}^{n}$ and $Z\subset{\mathbb R}^{n}$ be two ${\mathbb Q}$-nonsingular ${\mathbb Q}$-algebraic sets of the same dimension $d$ such that $Z\subsetneq V$. Then $V\setminus Z\subset{\mathbb R}^{n}$ is a ${\mathbb Q}$-nonsingular ${\mathbb Q}$-algebraic set of dimension $d$ as well

Figures (4)

  • Figure 2.1: Disposition of the $a_i$'s and $b_i$'s with respect to the partition $\lambda$.
  • Figure 3.1: Inductive step constructing a relative bordism.
  • Figure 4.1: Starting situation after the application of Theorem \ref{['thm:Q-spine-cobordism']}.
  • Figure 4.2: Topological construction of $N_i$ with $i\in\alpha$ and $i\notin\beta$.

Theorems & Definitions (46)

  • Definition 1.1
  • Definition 1.2: ${\mathbb R}|{\mathbb Q}$-regular points
  • Example 1.3
  • Definition 1.4
  • Proposition 1.6: GSa
  • Lemma 1.7
  • proof
  • Lemma 1.8
  • proof
  • Lemma 2.1
  • ...and 36 more