# Tsuchiya-Ueno-Yamada's proof that sheaves of conformal blocks are locally free

I'm referring to Tsuchiya-Ueno-Yamada's (TUY hereafter) celebrated paper Conformal Field Theory on Universal Family of Stable Curves with Gauge Symmetries. One of the main goals of their paper is to prove that sheaves of covacua (whose dual are the sheaves of conformal blocks) for WZW-models are locally free, i.e., can be viewed as vector bundles. One important consequence is that the dimensions of the spaces of conformal blocks do not rely on the complex structures of the complex curves. Now, my question can be summarized as follows:

Is their proof of local freeness (still) valid in the analytic setting?

(I'm NOT asking about the algebraic setting, which I'm sure is correct.)

Background: Let $$\mathfrak g$$ be a complex simple Lie algebra, and $$\hat{\mathfrak g}_l$$ the level $$l\in\mathbb N$$ affine Lie algebra. Let $$\mathfrak F:=(\pi:\mathcal C\rightarrow\mathcal B)$$ be a family of $$N$$-pointed compact Riemann surfaces (let's first ignore nodal curves), associated with $$N$$ irreducible highest weight integrable representations $$W_1,\dots,W_N$$ of $$\hat{\mathfrak g}_l$$. Set $$W_\bullet=W_1\otimes\cdots\otimes W_N$$. Then the sheaf of covacua $$\mathcal V$$ is a sheaf of $$\mathcal O_{\mathcal B}$$-module defined to be the quotient of $$W_\bullet\otimes\mathcal O_{\mathcal B}$$ by a certain subsheaf. Its dual $$\mathcal V^*$$ is the sheaf of conformal blocks. (Cf. [TUY] Def. 4.1.2.)

The way TUY proved that $$\mathcal V$$ (and hence $$\mathcal V^*$$) is locally free is as follows. In Sec. 5.1-5.3, they defined a (projectively flat) connection on $$\mathcal V$$. A standard result says that an (analytic or algebraic) coherent sheaf equipped with a connection is locally free. Thus, it reduces to proving the coherence of $$\mathcal V$$. In [TUY], the coherene is proved in Thm. 4.2.4. However, if you check their proof, you will see (cf. Lemma 4.2.2 and the paragraphs before that) that they actually only proved that $$\mathcal V$$ is finitely-generated, but not proved that the kernal of any morphism $$\mathcal O_{\mathcal B}^n|_U\rightarrow\mathcal V|_U$$ (where $$n\in\mathbb N$$ and $$U\subset\mathcal B$$ is open) is finitely generated, which is also part of the definition of coherent sheaves. This is certainly OK if we assume $$\mathfrak F$$ is an algebraic family. In the algebraic setting, $$\mathcal V$$ is clearly a quasi-coherent algebraic sheaf (since it is the cokernal of a morphism between two (infinite rank) free sheaves). So "finitely generated" implies algebraic coherence. However, in the analytic setting, which seems to be the setting TUY is working in, I want to argue that:

• "$$\mathcal V$$ is a finitely generated $$\mathcal O_{\mathcal B}$$-module" is not enough to show that "$$\mathcal V$$ is coherent". Indeed, $$\mathcal V$$ is not a quasi-coherent analytic sheaf. To my knowledge, in the literature of complex geometry, only "quasi-coherent analytic Frechet sheaves" are defined. (See the book of Eschmeier & Putinar.) But $$\mathcal V$$ clearly does not admit a Frechet structure since $$W_1,\dots,W_N$$ do not. Also, I'm wondering if there is a proof that "quasi-coherent analytic Frechet sheaf + finitely generated ==> analytic coherent"? See also a related question about quasi-coherent analytic sheaves.

• "Finitely generated + existence of connection" is, I believe, not enough to show that $$\mathcal V$$ is a locally free $$\mathcal O_{\mathcal B}$$-module. It will only imply that for each point $$b\in\mathcal B$$, the stalk $$\mathcal V_b$$ is a free $$\mathcal O_{\mathcal B,b}$$-module.

Thus, my opion is that TUY's proof that $$\mathcal V$$ is locally free is OK in the algebraic setting, but needs adjustment in the analytic setting. Am I right or wrong?

• I haven't thought about their paper in a while, but I think Oka's coherence theorem can be used on the dual module ... maybe? Aug 23, 2020 at 1:18
• @S.Carnahan Hi Scott, I think for the dual module, the difficulty is to show that it is finitely generated. Its sheaves of relations are finitely generated by Oka's theorem, as you said. Aug 23, 2020 at 1:54