We know that a mixed Hodge structure on a complex vector space $H$ with an integral lattice $H_{\mathbb Z}$ consists of the weight filtration and the Hodge filtration. For an isolated hypersurface singularity represented by a function germ $f: ({\mathbb C}^n, 0) \to ({\mathbb C}, 0)$, we have the vanishing cohomology bundle

$${\mathscr H}^* = \bigcup_{t \in {\mathbb D}^*} H^{n-1}(V_t; {\mathbb C}) \to {\mathbb D}^*.$$ Here $V_t = f^{-1}(t)$. A mixed Hodge structure can be defined on the fibres of ${\mathscr H}^*$. Let $\{{\mathscr F}^p\}_{p=0}^n$ be the Hodge filtration. The integral lattice is spanned by the parallel integral cocycles with respect to the Gauss-Manin connection.

A simple case is that when $f$ is a quasi-homogeneous polynomial. Suppose $d_1,\ldots, d_n, D$ are integers such that $$f( t^{d_1} z_1, \ldots, t^{d_n} z_n) = t^D f(z_1, \ldots, z_n).$$ Denote $\nu_i = d_i / D$. Then for the holomorphic $n$-form $\omega = z_1^{k_1}\cdots z_n^{k_n} dz_1 \wedge \cdots \wedge dz_n$, we have the so-called ``geometric sections'' of the cohomology bundle, whose value at $t$ is the residue form


This section belongs to ${\mathscr F}^p$ if and only if the charge of $\omega$, given by $l = \sum_{i=1}^n (k_i+1) \nu_i$, is less than or equal to $n-p$. (One can refer to the book ``Singularity I'' of Arnold etc.)

Here is the question: for quasi-homogeneous singularity, what is the complex conjugation of a geometric section (which can be given explicitly)? In the paper "${\mathcal N}=2$ Landau-Ginzburg vs. Calabi-Yau $\sigma$-models: non-perturbative aspects" by S. Cecotti, there is a physics description of this complex conjugation on page 1766 which I don't quite understand (especially the term of ``spectral flow''). Is there any mathematical literature which contains such discussion?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.