Stable Pairs and Bps Invariants
نویسندگان
چکیده
0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 1. χ-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 2. BPS rationality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 2.1. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 2.2. Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 2.3. Wall-crossing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 3. Local definition of BPS invariants . . . . . . . . . . . . . . . . . . . . . 276 3.1. Fixed curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 3.2. Chow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 3.3. Nonsingular curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 3.4. Singular curves: Discussion . . . . . . . . . . . . . . . . . . . . . . . 279 3.5. Singular curves: Results . . . . . . . . . . . . . . . . . . . . . . . . . 280 3.6. Nodal curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 Appendix A. Reduced curve classes . . . . . . . . . . . . . . . . . . . . . 287 Appendix B. Stable pairs on Gorenstein curves . . . . . . . . . . . . . . . 289 B.1. Gorenstein curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 B.2. Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 Appendix C. K3 surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 C.1. Nonsingularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 C.2. BPS states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 C.3. Yau-Zaslow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
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