Refinement of the Infinitesimal Variation of Hodge Structure: the case of canonical curves

Abstract

Let $C$ be a smooth complex projective curve with canonical divisor $K_C$ very ample. We explore the relation between the cup-product $$ H^1 (\Theta_C ) \longrightarrow (H^0({\cal O}_C (K_C))^{\ast} \otimes H^1 ({\cal O}_C) $$ where $\Theta_C ={\cal O}_C (-K_C)$ is the holomorphic tangent bundle of $C$, and the geometry of the canonical embedding of $C$. The cup-product, following Griffiths, stratifies ${\mathbb P}(H^1 (\Theta_C ))$ by the subvarieties $\Sigma_r$, according to the rank $r$ of $\xi \in H^1 (\Theta_C )$ viewed as the linear map $$ \xi:H^0({\cal O}_C (K_C)) \longrightarrow H^1 ({\cal O}_C) $$ or, equivalently, by the dimension of the kernel of $\xi$ $$ W_{\xi}=ker(\xi). $$ The refinement consists of the filtration of $W^{\bullet}_{\xi} ([\phi])$ of $W_{\xi}$, varying with $[\phi] \in {\mathbb P}(W_{\xi})$. This filtration has geometric meaning: 1) it is related to special divisors on $C$, 2) it `counts' certain rational normal curves in the canonical embedding of $C$. As an illustration, the results about the strata $\Sigma_0$ and $\Sigma_1$ are recovered and as corollaries one obtains the classical theorems of Max Noether on projective normality of the canonical embedding and Babbage-Enriques-Petri about the canonical curve being cut out by quadrics. The refinement brings out new aspects: quiver representations, Fano toric varieties with a distinguished anti-canonical divisor, dimer models. The quiver emerges from the construction and properties of the refinement; the Fano variety arises from the graph underlying the quiver and related to the Higgs structures. The graph underlying the refinement becomes an important part of the theory: it connects to topics such as the Topological Quantum field theory, moduli of elliptic curves with marked points, modular curves, higher categorical structures.