Abstract
In the first section of this note, we show that Theorem 1.8.1 of Bayer-Manin can be strengthened in the following way: If the even quantum cohomology of a projective algebraic manifold V is generically semisimple, then V has no odd cohomology and is of Hodge-Tate type. In particular, this answers a question discussed by G. Ciolli. In the second section, we prove that an analytic (or formal ) supermanifold M with a given supercommutative associative \( \mathcal{O}_M \)-bilinear multiplication on its tangent sheaf \( \mathcal{T}_M \) is an F-manifold in the sense of Hertling-Manin if and only if its spectral cover, as an analytic subspace of the cotangent bundle T * M , is coisotropic of maximal dimension. This answers a question of V. Ginzburg. Finally, we discuss these results in the context of mirror symmetry and Landau-Ginzburg models for Fano varieties.
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D. Auroux, L. Katzarkov, and D. Orlov, “Mirror Symmetry for Del Pezzo Surfaces: Vanishing Cycles and Coherent Sheaves,” Invent. Math. 166(3), 537–582 (2006); arXiv:math/0506166.
S. Barannikov, “Semi-infinite Hodge Structures and Mirror Symmetry for Projective Spaces,” arXiv:math/0010157.
S. Barannikov, “Semi-infinite Variations of Hodge Structures and Integrable Hierarchies of KdV Type,” Int. Math. Res. Not., No. 19, 973–990 (2002); arXiv:math/0108148.
A. Bayer, “Semisimple Quantum Cohomology and Blowups,” Int. Math. Res. Not., No. 40, 2069–2083 (2004); arXiv:math/0403260.
A. Bayer and Yu. Manin, “(Semi)simple Exercises in Quantum Cohomology,” in Proc. Fano Conf., Torino, Italy, 2002 (Univ. Torino, Torino, 2004), pp. 143–173; arXiv:math/0103164.
G. Ciolli, “On the Quantum Cohomology of Some Fano Threefolds and a Conjecture of Dubrovin,” Int. J. Math. 16(8), 823–839 (2005); arXiv:math/0403300.
B. Dubrovin, “Geometry and Analytic Theory of Frobenius Manifolds,” in Proc. Int. Congr. Math., Berlin, 1998 (Univ. Bielefeld, Bielefeld, 1998), Vol. 2, pp. 315–326.
A. Givental, “A Mirror Theorem for Toric Complete Intersections,” in Topological Field Theory, Primitive Forms and Related Topics (Birkhäuser, Boston, 1998), Prog. Math. 160, pp. 141–175.
V. V. Golyshev, “Riemann-Roch Variations,” Izv. Ross. Akad. Nauk, Ser. Mat. 65(5), 3–32 (2001) [Izv. Math. 65, 853–881 (2001)].
V. Golyshev, “A Remark on Minimal Fano Threefolds,” arXiv: 0803.0031.
C. Hertling, Frobenius Manifolds and Moduli Spaces for Singularities (Cambridge Univ. Press, Cambridge, 2002).
C. Hertling and Yu. Manin, “Weak Frobenius Manifolds,” Int. Math. Res. Not., No. 6, 277–286 (1999); arXiv:math/9810132.
Yu. I. Manin, Frobenius Manifolds, Quantum Cohomology, and Moduli Spaces (Am. Math. Soc., Providence, RI, 1999), AMS Colloq. Publ. 47.
Yu. I. Manin, “Manifolds with Multiplication on the Tangent Sheaf,” Rend. Mat. Appl., Ser. 7, 26, 69–85 (2006); arXiv:math.AG/0502578.
Yu. I. Manin, “F-Manifolds with Flat Structure and Dubrovin’s Duality,” Adv. Math. 198, 5–26 (2005) (M. Artin’s Fest).
S. A. Merkulov, “Operads, Deformation Theory and F-Manifolds,” in Frobenius Manifolds: Quantum Cohomology and Singularities, Ed. by C. Hertling and M. Marcolli (Vieweg, Wiesbaden, 2004), pp. 213–251; arXiv:math.AG/0210478.
S. A. Merkulov, “PROP Profile of Poisson Geometry,” Commun. Math. Phys. 262(1), 117–135 (2006); arXiv:math.AG/0401034.
D. O. Orlov, “Derived Category of Coherent Sheaves and Motives,” Usp. Mat. Nauk 60(6), 231–232 (2005) [Russ. Math. Surv. 60, 1242–1244 (2005)]; arXiv:math/0512620.
C. Sabbah, Isomonodromic Deformations and Frobenius Manifolds (Springer, Berlin, 2007).
C. Teleman, “The Structure of 2D Semi-simple Field Theories,” arXiv: 0712.0160.
E. Zaslow, “Solitons and Helices: The Search for a Math-Physics Bridge,” Commun. Math. Phys. 175(2), 337–375 (1996).
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To the memory of V.A. Iskovskikh
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Hertling, C., Manin, Y.I. & Teleman, C. An update on semisimple quantum cohomology and F-manifolds. Proc. Steklov Inst. Math. 264, 62–69 (2009). https://doi.org/10.1134/S0081543809010088
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DOI: https://doi.org/10.1134/S0081543809010088