Soutenance de thèse Naoufal BOUCHAREB
Date(s) : 20/06/2025 iCal
14h00 - 15h00
We study the classification of affine holomorphic bundles over a compact complex manifold $X$.
The thesis has two goals: first we prove classification theorems for affine bundles over Riemann surfaces, notably for affine bundles over the projective line. In particular we study the moduli space of framed, non-degenerate rank 2 affine bundles over $\mathbb{P}^1_\mathbb{C}$ whose linearisation, viewed as locally free sheaf, is isomorphic to $\mathcal{O}(n_1)\oplus \mathcal{O}(n_2)$ where $n_1>n_2$. We show that this moduli space can be identified with the « topological cokernel » of a morphism of linear spaces over the projective space $\mathbb{P}(\mathbb{C}[X_0,X_1]_{l})$ of binary forms of degree $l:= -2-n_2$, in particular it fibres over this projective space with vector spaces as fibres. We show that the stratification of $\mathbb{P}(\mathbb{C}[X_0,X_1]_{l})$ defined by the level sets of the fibre dimension map is determined explicitly by $d:= n_1-n_2$ and the cactus rank stratification of $\mathbb{P}((\mathbb{C}[X_0,X_1]_{l})$.
The second goal has a broader scope: the classification of affine bundles with fixed linearisation type over a compact complex manifold $X$. More precisely, let $E$ be a fixed holomorphic vector bundle over $X$ and $\mathcal{E}$ the associated locally free sheaf. We will assume that there exists an effective divisor $D$ on $X$ such that $H^1(X,\mathcal{E}(D))=0$. This condition is always satisfied if $X$ is a projective complex manifold.
We define in a natural way the moduli stack of affine bundles on $X$ whose linearisation is isomorphic to $E$, and we prove that this moduli stack is isomorphic to the quotient stack $[H^0(X,\mathcal{E}(D)_D)/H^0(X,\mathcal{E}(D))\ltimes \mathrm{Aut}(E)]$ of the complex vector space $H^0(X,\mathcal{E}(D)_D)$ by the semi-direct product of the automorphism group $\mathrm{Aut}(E)$ (which is always an affine algebraic group) by the complex vector space $H^0(X,\mathcal{E}(D))$.
Emplacement
I2M Saint-Charles - Salle de séminaire
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