Pluri-Hodge Decomposition and Associated Jacobian
- Prof. Y-T. SIU
The deformational invariance of the m-genus, the dimension of H⁰(X;mK_X), is known for the case of a compact complex algebraic manifold X and is conjectured for the case of a compact Kähler manifold. For m=1, the deformational invariance in the Kähler case follows from the Hodge decomposition. The question arises whether H⁰(X;mK_X)) for m≥2 is naturally a direct summand of the cohomology group of some flat bundle so that the deformational invariance of the m-genus can be explained in terms of such a "pluri-Hodge decomposition". The talk will discuss the question, starting with the case of a compact Riemann surface, and study the construction of jacobians associated to such a "pluri-Hodge decomposition" for a compact Riemann surface.