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Samuel Lerbet

Publications and source records attributed to Samuel Lerbet.

6 recordsLinked to original sources
arXiv2026

Witt groups of smooth real curves and surfaces

We study the $\mathbf{I}^*$-cohomology of a smooth real algebraic curve in terms of its real locus and its geometric genus. We notably extend results of Monnier to the twisted case, which is crucial to the understanding of proper pushforwards of Witt groups. We also perform some computations related to transfers along the finite \'etale extension $\mathbb{C}/\mathbb{R}$. We further describe how to compute twisted Witt groups of surfaces, extending work of Sujatha, and the image of the global signature homomorphism following Monnier. As an application of the main methods of the paper, we describe the shifted and twisted Witt groups of smooth anisotropic quadrics over $\mathbb{R}$ of dimension $\leq 3$.

math.AG↗
arXiv2026

On the cohomological classification of vector bundles on smooth real affine surfaces and threefolds

We study the cohomological classification of vector bundles on smooth real affine surfaces and threefolds. We show that, as was observed in joint work in A. Asok and J. Fasel and with S. Banerjee and J. Fasel, under suitable cohomological assumptions on the real locus of such varieties, this classification mirrors the one obtained on algebraically closed base fields by Mohan Kumar and Murthy and by Asok and Fasel. Using an argument due to Fasel, we also give an efficient proof of a theorem of Kucharz characterising the triples of algebraic cycles that can be realised as the Chern classes of a rank $3$ bundle on a smooth real affine threefold. We further answer the questions left open by Kucharz; to our knowledge, we give the first instance of a projective module over a smooth affine $\mathbb{R}$-algebra of dimension $3$ with trivial Chern classes which is not stably free.

math.AG↗
arXiv2025

Splitting vector bundles over real algebraic varieties

Suppose $X$ is a smooth affine real variety and $\mathscr{E}$ is a vector bundle over $X$. We analyze the problem of splitting off a free rank one summand from $\mathscr{E}$ in corank $0$ and $1$. The problem in corank $0$ can be viewed as the search for a real analog of Murthy's celebrating splitting theorem in the algebraically closed case: to wit, beyond the vanishing of the top Chern class in Chow theory, are the obstructions to splitting ``purely topological''? In a sense, the answer in this case is yes, and we give a proof, using motivic techniques, of a mild extension of the results of Bhatwadekar-Sridharan and Bhatwadekar-Das-Mandal. In corank $1$, in the algebraically closed situation, Murthy's splitting conjecture (now a theorem in characteristic $0$) predicts that the vanishing of the top Chern class in Chow theory is the only obstruction to splitting off a free rank $1$ summand, and we can search for a suitable ``real'' analog of this assertion. We observe that several natural guesses for a ``real'' analog of Murthy's splitting conjecture cannot be true, i.e., that the situation over the real numbers is rather complicated.

math.AG↗
arXiv2024

On the image of higher signature maps

Given a smooth variety $X$ over the field $\mathbb{R}$ of real numbers and a line bundle $\mathcal{L}$ on $X$ with associated topological line bundle $L=\mathcal{L}(\mathbb{R})$, we study the quadratic real cycle class map $\widetilde{\gamma}_{\mathbb{R}}^c:\widetilde{\mathrm{CH}}^c(X,\mathcal{L})\rightarrow\mathrm{H}^c(X(\mathbb{R}),\mathbb{Z}(L))$ from the $c$-th Chow-Witt group of $X$ to the $c$-th cohomology group of its real locus $X(\mathbb{R})$ with coefficients in the local system $\mathbb{Z}(L)$ associated with $L$. We focus on the cases $c\in\{0,d-2,d-1,d\}$ where $d$ is the dimension of $X$ and we formulate a precise conjecture on the image of $\widetilde{\gamma}_{\mathbb{R}}$ in terms of the exponents of its cokernel that is corroborated by the results obtained in those codimensions.

math.AG↗
arXiv2023

Motivic stable cohomotopy and unimodular rows

We relate the group structure of van der Kallen on orbit sets of unimodular rows with values in a smooth algebra $A$ over a field $k$ with the motivic cohomotopy groups of the spectrum of $A$ with coefficients in $\mathbb{A}^n\setminus 0$ in the sense of Asok and Fasel. In the last section, we compare the motivic cohomotopy theory studied in this paper and defined by $\mathbb{A}^{n+1}\setminus 0$ or, equivalently, by an $\mathbb{A}^1$-weakly equivalent quadric $Q_{2n+1}$ to that considered by Asok and Fasel, defined by a quadric $Q_{2n}$, by means of explicit morphisms $Q_{2n+1}\rightarrow Q_{2n}$, $Q_{2n}\times\mathbb{G}_m\rightarrow Q_{2n+1}$ of quadrics.

math.KT↗