Let $X\rightarrow D^*$ be fiberation of projective manifold, here $D^*$ means a punctured disk. Then it induces a variation of hodge structure, i.e a holomorphic vector bundle $H_{\mathbb C}$ over $D^*$(the fiber is kth DeRham cohomology group of $X_t$ with coefficient $\mathbb C$). If the monodromy transformation is unipotent, we have isomorphism

$$\mathcal O_{D^*}(H_{\mathbb C})\cong \mathcal O_{D^*}\otimes_{\mathbb C} H$$

The above set-up is my understanding of one of the statement from the paper "variation of hodge structure and singularity of period map" by Wilfried Schmid.

Then I'm a little bit confused about the statement. For me, it seems that the above isomorphism implies that $\mathcal O_{D^*}(H_{\mathbb C})$ is a holomorphic trivial vector bundle, since the RHS is trivial. On the other hand, Kodaira's $I_b$ type singular fiber in elliptic surface has unipotent monodromy but not trivial $H^1$ bundle. So what's wrong with my understanding.

Any comments will be helpful.

  • 4
    $\begingroup$ The vector bundle extends to $D$ which makes it trivial. The connection is non-trivial (and has log poles). The fibre over the closed point has an MHS (this is what Schmid proves) but that MHS is not the MHS of $H^1$ of the closed fibre but there is more complicated relation. $\endgroup$
    – Kapil
    Jun 16, 2022 at 2:47
  • $\begingroup$ Thanks @Kapil. Firstly you confirm that is is trivial. But how about the elliptic surface which has a $I_b$ type singular fiber. In this case, the monodromy is unipotent. But it is hard for me to see that the $H^1$ bundle on the punctured disk. $\endgroup$
    – xin fu
    Jun 16, 2022 at 3:16
  • $\begingroup$ VHS on the punctured disk only determines an irreducible component of $I_b$ in an elliptic fibration (in general, the notion is Neron model). Take the logarithmic extension of VHS to $D$, the flat sections are linear combinations of the holomorphic sections with coefficients involving $log(t)$, and go to $\infty$ at $t=0$. Therefore, the lattice degenerate at $0$, and the quotient is a copy of $\mathbb C^*$. It is identified with the smooth locus of one component of $I_b$. $\endgroup$
    – AG learner
    Jun 24, 2022 at 21:33


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