In [16] the fundamental relationship between stable quotient invariants and the B-model for local P 2 in all genera was studied under some specialization of equivariant variables. We generalize the argument of [16] to full equivariant settings without the specialization. Our main results are the proof of holomorphic anomaly equations for the equivariant Gromov-Witten theories of local P 2 and local P 3 . We also state the generalization to full equivariant formal quintic theory of the result in [17].

We propose and prove a mirror theorem for the elliptic quasimap invariants of smooth Calabi-Yau complete intersections in projective spaces. The theorem combined with the wall-crossing formula appeared in [3] implies mirror theorems of Zinger and Popa for the elliptic Gromov-Witten invariants of those varieties. This paper and the wall-crossing formula provide a unified framework for the mirror theory of rational and elliptic Gromov-Witten invariants.

We define a formal Gromov-Witten theory of the quintic 3-fold via localization on P 4 . Our main result is a direct geometric proof of holomorphic anomaly equations for the formal quintic in precisely the same form as predicted by B-model physics for the true Gromov-Witten theory of the quintic 3-fold. The results suggest that the formal quintic and the true quintic theories should be related by transformations which respect the holomorphic anomaly equations. Such a relationship has been recently found by Q. Chen, S. Guo, F. Janda, and Y. Ruan via the geometry of new moduli spaces.

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We give a short direct proof for the degeneration formula of Gromov-Witten invariants including its cycle version for degenerations with smooth singular locus in the setting of minimal/basic stable log maps of Abramovich-Chen, Chen, Gross-Siebert. Contents 14 6. Gluing stable log maps 17 7. The splitting stack 25 8. Decomposing moduli spaces of curves 27 9. Comparing perfect obstruction theories 29 References 36