On a Bers Theorem for the SL(3,R)-Hitchin component and on its pseudo-Kähler structure

Let S be a closed oriented surface, denote with Teich(S) its Teichmuller space, and with Hit_3(S) the Hitchin component of the SL(3,R)-character variety, endowed with the Labourie-Loftin complex structure. The well-known Bers Theorem provides a biholomorphism between Teich(S)xTeich($\overline{S}$) and an open subset of the SL(2,C)-character variety corresponding to quasi-Fuchsian representations. In this talk, we present an analog result for Hit_3(S). In fact, we introduce a notion of complex affine spheres in $C^3$, which extend the classical (real) affine spheres in $R^3$, and use them to prove that the natural inclusion of the diagonal of Hit_3(S)x{Hit_3($\overline{S}$)} inside the character variety of SL(3,C) extends to a unique local biholomorphism on a "big" open subset of Hit_3(S)xHit_3($\overline{S}$) which in particular contains Hit_3(S)xTeich($\overline{S}$) and Teich(S) x Hit_3($\overline{S}$). As a consequence of this approach, we also show that the Goldman symplectic form is compatible with the Labourie-Loftin complex structure, determining a pseudo-Kahler structure on Hit_3(S). This is joint work with Nathaniel Sagman.