A degeneration formula of Gromov-Witten invariants with respect to a curve class for degenerations from blow-ups
| dc.creator | Liu, Chien-Hao | |
| dc.creator | Yau, Shing-Tung | |
| dc.date | 2004-08-11 | |
| dc.date.accessioned | 2026-07-07T05:11:12Z | |
| dc.date.available | 2026-07-07T05:11:12Z | |
| dc.description | In two very detailed, technical, and fundamental works, Jun Li constructed a theory of Gromov-Witten invariants for a singular scheme of the gluing form $Y_1\cup_D Y_2$ that arises from a degeneration $W/{\Bbb A}^1$ and a theory of relative Gromov-Witten invariants for a codimension-1 relative pair $(Y,D)$. As a summit, he derived a degeneration formula that relates a finite summation of the usual Gromov-Witten invariants of a general smooth fiber $W_t$ of $W/{\Bbb A}^1$ to the Gromov-Witten invariants of the singular fiber $W_0=Y_1\cup_D Y_2$ via gluing the relative pairs $(Y_1,D)$ and $(Y_2,D)$. The finite sum mentioned above depends on a relative ample line bundle $H$ on $W/{\Bbb A}^1$. His theory has already applications to string theory and mathematics alike. For other new applications of Jun Li's theory, one needs a refined degeneration formula that depends on a curve class $β$ in $A_{\ast}(W_t)$ or $H_2(W_t;{\Bbb Z})$, rather than on the line bundle $H$. Some monodromy effect has to be taken care of to deal with this. For the simple but useful case of a degeneration $W/{\Bbb A}^1$ that arises from blowing up a trivial family $X\times{\Bbb A}^1$, we explain how the details of Jun Li's work can be employed to reach such a desired degeneration formula. The related set $Ω_{(g,k;β)}$ of admissible triples adapted to $(g,k;β)$ that appears in the formula can be obtained via an analysis on the intersection numbers of relevant cycles and a study of Mori cones that appear in the problem. This set is intrinsically determined by $(g,k;β)$ and the normal bundle ${\cal N}_{Z/X}$ of the smooth subscheme $Z$ in $X$ to be blown up. | |
| dc.description | 13 pages, 2 figures | |
| dc.identifier | https://arxiv.org/abs/math/0408147 | |
| dc.identifier | http://arxiv.org/abs/math/0408147 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/72162 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | High Energy Physics - Theory | |
| dc.subject | 14N35; 81T30 | |
| dc.title | A degeneration formula of Gromov-Witten invariants with respect to a curve class for degenerations from blow-ups | |
| dc.type | text |