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14
Symplectic QuasiStates and SemiSimplicity of Quantum Homology
, 2007
"... We review and streamline our previous results and the results of Y. Ostrover on the existence of Calabi quasimorphisms and symplectic quasistates on symplectic manifolds with semisimple quantum homology. As an illustration, we discuss the case of symplectic toric Fano 4manifolds. We present also ..."
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Cited by 22 (3 self)
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We review and streamline our previous results and the results of Y. Ostrover on the existence of Calabi quasimorphisms and symplectic quasistates on symplectic manifolds with semisimple quantum homology. As an illustration, we discuss the case of symplectic toric Fano 4manifolds. We present also new results due to D. McDuff: she observed that for the existence of quasimorphisms/quasistates it suffices to assume that the quantum homology contains a field as a direct summand, and she showed that this weaker condition holds true for one point blowups of nonuniruled symplectic manifolds.
FROM EXCEPTIONAL COLLECTIONS TO MOTIVIC DECOMPOSITIONS Via Noncommutative Motives
, 2012
"... Making use of noncommutative motives we relate exceptional collections (and more generally semiorthogonal decompositions) to motivic decompositions. On one hand we prove that the Chow motive M(X)Q of every smooth and proper DeligneMumford stack X, whose bounded derived category D b (X) of cohere ..."
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Cited by 9 (4 self)
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Making use of noncommutative motives we relate exceptional collections (and more generally semiorthogonal decompositions) to motivic decompositions. On one hand we prove that the Chow motive M(X)Q of every smooth and proper DeligneMumford stack X, whose bounded derived category D b (X) of coherent schemes admits a full exceptional collection, decomposes into a direct sum of tensor powers of the Lefschetz motive. Examples include projective spaces, quadrics, toric varieties, homogeneous spaces, Fano threefolds, and moduli spaces. On the other hand we prove that if M(X)Q decomposes into a direct direct sum of tensor powers of the Lefschetz motive and moreover D b (X) admits a semiorthogonal decomposition, then the noncommutative motive of each one of the pieces of the semiorthogonal decomposition is a direct sum of ⊗units. As an application we obtain a simplification of Dubrovin’s conjecture.
Derived categories of coherent sheaves on rational homogeneous manifolds
 Doc. Math. 11 (2006), 261–331 (electronic). MR 2262935 (2008f:14032
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AN UPDATE ON SEMISIMPLE QUANTUM COHOMOLOGY AND F–MANIFOLDS
, 803
"... Abstract. In the first section of this note we show that the Theorem 1.8.1 of Bayer–Manin ([BaMa]) can be strengthened in the following way: if the even quantum cohomology of a projective algebraic manifold V is generically semi–simple, then V has no odd cohomology and is of Hodge–Tate type. In part ..."
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Cited by 3 (0 self)
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Abstract. In the first section of this note we show that the Theorem 1.8.1 of Bayer–Manin ([BaMa]) can be strengthened in the following way: if the even quantum cohomology of a projective algebraic manifold V is generically semi–simple, then V has no odd cohomology and is of Hodge–Tate type. In particular, this addressess a question in [Ci]. In the second section, we prove that an analytic (or formal) supermanifold M with a given supercommutative associative OM–bilinear multiplication on its tangent sheaf TM is an F–manifold in the sense of [HeMa], iff 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.
RESEARCH STATEMENT
"... My research has been devoted to algebraic geometry motivated by physics. Since the prediction of mirror symmetry by physicists around 1990 as a spinoff of studying string theory, there has been a continuos exchange of new ideas and conjectures between algebraic geometry and string theory. The first ..."
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Cited by 1 (0 self)
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My research has been devoted to algebraic geometry motivated by physics. Since the prediction of mirror symmetry by physicists around 1990 as a spinoff of studying string theory, there has been a continuos exchange of new ideas and conjectures between algebraic geometry and string theory. The first conjecture of mirror symmetry (proven for many cases by Givental [Giv96, Giv98]) relates the number of rational curves on a projective variety V to period integrals on the space of deformations of the complex structure of its mirror partner ˜ V. The introduction of quantum cohomology and Frobenius manifolds [Dub93, Dub96, KM94, Man99] gave a conceptional framework to this conjecture. Some of my research ([BM04, Bay04], see section 2) has been devoted to the study of socalled semisimple Frobenius manifolds. Their study leads to very explicit mirror symmetry statements, and raises questions related to quantum cohomology, unfoldings of singularities and the derived category of coherent sheaves. For example, I have proven that semisimplicity is stable under blowups at points (Theorem 2.4.1), which was suggested by corresponding results on the
Derived Categories of Coherent Sheaves on Rational Homogeneous Manifolds
 DOCUMENTA MATH.
, 2006
"... One way to reformulate the celebrated theorem of Beilinson is that (O(−n),...,O) and (Ω n (n),...,Ω 1 (1), O) are strong complete exceptional sequences in D b (Coh P n), the bounded derived category of coherent sheaves on P n. In a series of papers ([Ka1], [Ka2], [Ka3]) M. M. Kapranov generalized t ..."
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One way to reformulate the celebrated theorem of Beilinson is that (O(−n),...,O) and (Ω n (n),...,Ω 1 (1), O) are strong complete exceptional sequences in D b (Coh P n), the bounded derived category of coherent sheaves on P n. In a series of papers ([Ka1], [Ka2], [Ka3]) M. M. Kapranov generalized this result to flag manifolds of type An and quadrics. In another direction, Y. Kawamata has recently proven existence of complete exceptional sequences on toric varieties ([Kaw]). Starting point of the present work is a conjecture of F. Catanese which says that on every rational homogeneous manifold X = G/P, where G is a connected complex semisimple Lie group and P ⊂ G a parabolic subgroup, there should exist a complete strong exceptional poset (cf. def. 2.1.7 (B)) and a bijection of the elements of the poset with the Schubert varieties in X such that the partial order on the poset is the order induced by the BruhatChevalley order (cf. conjecture 2.2.1