# Extending vector bundles from subvarieties

Let $X$ be a smooth projective variety and let $Y\subset X$ be a smooth subvariety. Given a vector bundle $E$ on $Y$, when can $E$ be extended to a vector bundle $\tilde E$ on $X$? I.e., are there cohomology groups containing the obstructions to extending $E$?

• And preceding SGA2 there is Grauert's theorem that, in the case (over $\mathbb{C}$) that $\dim{X} > 2$ and $Y$ is an ample divisor, $E$ will so extend as soon as it extends to a vector bundle on a complex neighborhood of $Y$. A generalization in formal geometry appears in SGA2. – Vesselin Dimitrov Jul 29 '14 at 13:03
• Let we have a complex manifold $X$, containing an analytic subset $A$ of complex codimension at least two. Let $(E,h) \to K\setminus A$ be a Hermitian-holomorphic vector bundle such that $F_h \in L^n(X\setminus A)$, then there exists a unique vector bundle $\hat E \to X$ such that $\hat E |_{X\setminus A}\cong E$. See Harris, A., and Tonegawa, Y. Analytic continuation of vector bundles with Lp-curvature, Int. J. Math. 11 No.l, (2000), 29-40. – user21574 Nov 14 '17 at 14:00
• For a relative version of my previous comment on a surjective holomorphic fibre space see Theorem 2.2. of onlinelibrary.wiley.com/doi/10.1002/mana.19992040103/full – user21574 Nov 14 '17 at 14:50
• For topological condition to get extension theorems on bundles see Lemma 5 and Lemma 6 of the paper V. V. Shevchishin , The Ock–Grawert principle for the extension of holomorphic line bundles with integrable curvature V. V. Shevchishin , Mat. Zametki, 50:5 (1991), 109–119 link.springer.com/article/10.1007%2FBF01157706 – user21574 Nov 15 '17 at 14:50
• A theorem of Siu: Let $X$ is a Stein manifold of dimension at least $3$, $K$ is a holomorphically convex compact subset of $X$ with connected complement, and $E$ is a holomorphic vector bundle on $X\setminus K$, then there is a finite subset $P$ of $K$ such that $E$ extends to a holomorphic vector bundle on $X\setminus P$: See A Hartogs type extension theorem for coherent analytic sheaves. Ann. of Math. (2) 93 1971 166–188. 32.50 and also for details of proof see projecteuclid.org/download/pdf_1/euclid.afm/1485802742(Since Siu didn't prove it) – user21574 Nov 16 '17 at 13:18

Without hypotheses on $Y$ there is no hope to define such obstructions, already for line bundles. A natural hypothesis is to take for $Y$ a (smooth) ample divisor in $X$, of dimension $\geq 2$. In this paper, Fujita gives some cohomological conditions which imply that $E$ extends : $H^2(Y, \mathcal{E}nd(E)(-tY))=0$ for all $t\geq 1$ and $H^p(Y, E(tY))=0$ for all $t\in\mathbb{Z}$ and $0<p<\dim Y$. Be aware that these conditions are extremely strong.
• This has been a while. You mention that $Y$ needs to be a smooth ample divisor. Fujita only mentions non singularity of $X$. Is the assumption of smoothness of the divisor implicit throughout the paper? – user127776 Dec 11 '20 at 16:35
• No, I guess I put "smooth" because the OP did so. You are right, Fujita makes no smoothness assumption on $Y$. – abx Dec 11 '20 at 17:27
• I think he is implicitly assuming that $X$ is a projective variety. His Lemma 3.2 is obviously false in the affine case. – abx Dec 11 '20 at 20:43
An obvious obstruction comes from topology: the Chern classes of your bundle should be obtained from restriction of Hodge classes on an ambient variety. This is (more or less) enough to extend a smooth bundle $B$ from $Y$ to $X$. To be precise, you need the classifying map from $Y$ to the space $BU(r)$ to be extendable to a continuous map from $X\supset Y$ to $BU(r)$, where $r$ is rank $B$. This is not the only obstruction, because a way to find a holomorphic bundle with prescribed $(p,p)$-Chern classes on $X$ amounts to a result which is much stronger than the Hodge conjecture (and false, generally speaking). The easiest obstruction to finding a bundle with prescribed $(p,p)$-Chern classes comes from the Bogomolov inequality: for any stable $B$, one has $$\int_M [2rc_2(B) - (r - 1)c_1(B)^2]\wedge \omega^{n-2}> 0,$$ where $n$ is dimension of your manifold $M$, and $\omega$ its Kahler form (case of unstable bundles is considered separately using the Jordan-Holder filtration). Also, if this inequality is non-strict, $B$ admits a projectively flat connection, and therefore $c_2$ and the rest of Chern classes are powers of $c_1$.