Degree formula for the discriminant divisor of lagrangian fibrations of irreducible symplectic manifolds

Abstract

Let \(M\) be an irreducible holomorphic symplectic manifold with a Lagrangian fibration \(f: M \to \mathbb P^n\), whose discriminant locus is \(\Delta = \bigcup_i \Delta_i \subset \mathbb P^n\). This thesis defines weights \(w_i \in \mathbb Q\) such that \[ 24 \Biggl(\frac{n! \int_M \sqrt{\hat A}(M)}{d_1 \dotsm d_n}\Biggr)^{\frac 1 n} = \sum_i w_i \deg(\Delta_i), \] where \((d_1, \dotsc, d_n)\) is the polarization type of \(f\).

The definition of the \(w_i\) involves the cohomology sheaves of the \(\Omega T\) complex \[ f^* \Omega_{\mathbb P^n} \to \Omega_M \cong T_M \to f^* T_{\mathbb P^n}, \] and this thesis gives an in-depth analysis of those sheaves.

Furthermore, the definition involves the choice of a Kähler form \(\omega\) on \(M\), which induces a polarization of type \((d_1, \dotsc, d_n)\) on the smooth fibers of \(f\). To show that the \(w_i\) do not depend on the choice of \(\omega\) is the main undertaking of this thesis.

If the characteristic cycle \(\Theta_i\) over \(\Delta_i\) is compact, then one can define the weights as \[ w_i = \frac{\chi(\Theta_i)}{\deg_{\Theta_i}(\omega)}. \] In case of non-compact characteristic cycles one can choose an appropriate compact subcycle \(\overline{\Theta}_i \subset \Theta_i\) to compute \(w_i\) in the same way.