A finite dimensional proof of Verlinde Formula

Abstract

We prove two recurrence relations among dimensions $$D_g(r,d,\omega ):=\dim {\mathrm{H}}^0({\mathcal{U}}_{C,\omega },{\Theta }_{{\mathcal{U}}_{C,\omega }})$$ of spaces of generalized theta functions on moduli spaces ${\mathcal{U}}_{C,\omega }$. By using of these recurrence relations, an explicit formula (Verlinde formula) of $D_g(r,d,\omega )$ is proved (See Theorem 4.3).

This is the second part of arXiv:1802.08392 [math.AG], some new results are added (See Lemma 4.6, Lemma 4.7 and Lemma 4.8), which compute the dimension of generalized theta functions on moduli spaces of parabolic bundles on projective line with at most 3 parabolic points. Two references [5] and [6] are added in the list of references. arXiv admin note: text overlap with arXiv:2001.11640