Abstract
In this chapter we outline some applications of Homological Mirror Symmetry to classical problems in Algebraic Geometry, like rationality of algebraic varieties and the study of algebraic cycles. Several examples are studied in detail.
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References
M. Abouzaid, On the Fukaya categories of higher genus surfaces. Adv. Math. 217 (2008), no. 3, 1192–1235.
D. Auroux, L. Katzarkov, S. Donaldson, and M. Yotov. Fundamental groups of complements of plane curves and symplectic invariants. Topology 43 (2004), no. 6, 1285–1318.
D. Auroux, L. Katzarkov, and D. Orlov. Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves 6. Invent. Math. 166 (2006), no. 3, 537–582.
D. Auroux, L. Katzarkov, T. Pantev and D. Orlov. Mirror symmetry for Del Pezzo surfaces II.in prep.
F. Bogomolov. The Brauer group of quotient spaces of linear representations., Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 3, 485–516, 688.
A. Beauville and R. Donagi. La variété des droites d’une hypersurface cubique de dimension 4. C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 14, 703–706.
H. Clemens. Cohomology and obstructions II. (2002) AG 0206219.
H. Clemens and Ph. Griffiths. The intermediate Jacobian of the cubic threefold. Ann. of Math. 95 (1972), no. 2, 281–356.
M. Green and Ph. Griffiths. Algebraic Cycles I,II. Preprints (2006).
D. Cox and S. Katz. \textit Mirror symmetry and algebraic geometry, volume 68 of \textit MathematicalSurveys and Monographs. American Mathematical Society, Providence, RI, (1999).
T. deFernex. Adjunction beyond thresholds and birationally rigid hypersurfaces (2006). AG/0604213.
L. Ein and M. Musta. Mircea Invariants of singularities of pairs. International Congress of Mathematicians. Vol. II, 583–602, Eur. Math. Soc., Zrich, 2006. (Reviewer: Tommaso De Fernex) 14J17 (14E15).
M. Gross and B. Siebert. Mirror Symmetry via logarithmic degeneration data. I. J. Differential Geom. 72 (2006), no. 2, 169–338.
M. Gross and L. Katzarkov. Mirror Symmetry and vanishing cycles. in preparation.
B. Hassett. Some rational cubic fourfolds. J. Algebraic Geom. 8 (1999), no. 1, 103–114.
V.A. Iskovskikh, Y.I. Manin. Three-dimensional quartics and counterexamples to the Lüroth problem., Mat. Sb. (N.S.) 86 (1971), no. 128 140–166.
V.A. Iskovskikh, On Rationality Criteria for Connic Bundles, Mat. Sb. 7, (1996), 75–92.
A. Iliev and L. Manivel. Cubic hypersurfaces and integrable systems., preprint 0606211.
K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, and E. Zaslow. Mirror symmetry, volume 1 of Clay Mathematics Monographs. American Mathematical Society, Providence, RI, 2003. With a preface by Vafa.
K. Hori and C. Vafa. Mirror symmetry, (2000), hep-th/0002222.
A. Kapustin, L. Katzarkov, D. Orlov, and M. Yotov. Homological mirror symmetry for manifolds of general type, 2004. preprint.
L. Katzarkov. Algebraic cycles and Homological mirror symmetry, in preparation.
L. Katzarkov. Mirror symmetry and nonrationality, in preparation.
L. Katzarkov. Tropical Geometry and Algebraic cycles, in preparation.
L. Katzarkov, M. Kontsevich, and T. Pantev. Hodge theoretic aspects of mirror symmetry. arXiv:0806.0107.
J. Kollár. Singularities of pairs, Algebraic geometry—Santa Cruz 1995, 221–287, Proc. Sympos. Pure Math., 62, Part 1, Amer. Math. Soc., Providence, RI, (1997).
A. Kuznetsov. Derived categories of cubic fourfolds. 0808.3351.
G. D. Kerr. Weighted Blowups and Mirror symmetry for Toric Surfaces, 0609162v1, To appear in Adv in Math.
G. Mikhalkin. Tropical geometry and its applications., 0601041, preprint. 132, 141
D. Orlov. Derived categories of icoherent sheaves and equivalences between them. Uspekhi Mat. Nauk 58 (2003), no. 3(351), 89–172; translation in Russian Math. Surveys 58 (2003), no. 3, 511–591.
D. Orlov. Triangulated categories of singularities and D-branes in Landau-Ginzburg models. \textit Tr. Mat. Inst. Steklova, 246(Algebr. Geom. Metody, Svyazi i Prilozh.):240–262, 2004.
A. Pukhlikov. Birationally rigid Fano varieties. The Fano Conference, 659–681, Univ. Torino, Turin, 2004.
P. Seidel. Vanishing cycles and mutation., European Congress of Mathematics, Vol. II (Barcelona, 2000), 65–85, Progr. Math., 202, Birkhäuser, Basel, 2001.
P. Seidel. Fukaya categories and deformations., Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 351–360, Higher Ed. Press, Beijing, (2002).
J. Wlodarczyk. Birational cobordisms and factorization of birational maps, J. Algebraic Geom. 9 (2000), no. 3, 425–449.
C. Voisin. Torelli theorems for cubics in CP 5, Inv. Math. 86, (1986), no. 3, 577–601.
S. Zucker. The Hodge conjecture for four dimensional cubic. Comp. Math. 34, (1977), 199–209.
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Katzarkov, L. (2008). Homological Mirror Symmetry and Algebraic Cycles. In: Homological Mirror Symmetry. Lecture Notes in Physics, vol 757. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68030-7_5
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DOI: https://doi.org/10.1007/978-3-540-68030-7_5
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