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## Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures

Cornell University
,
2014.
No. 1404.6407.

Galkin S., Golyshev V., Iritani H.

We propose Gamma Conjectures for Fano manifolds which can be thought of as a square root of the index theorem. Studying the exponential asymptotics of solutions to the quantum differential equation, we associate a principal asymptotic class A_F to a Fano manifold F. We say that F satisfies Gamma Conjecture I if A_F equals the Gamma class Γ_F. When the quantum cohomology of F is semisimple, we say that F satisfies Gamma Conjecture II if the columns of the central connection matrix of the quantum cohomology are formed by Γ_F Ch(E_i) for an exceptional collection {E_i} in the derived category of coherent sheaves D^b_{coh}(F). Gamma Conjecture II refines part (3) of Dubrovin's conjecture. We prove Gamma Conjectures for projective spaces, toric manifolds, certain toric complete intersections and Grassmannians.

Galkin S., Golyshev V., Iritani H., Duke Mathematical Journal 2016 Vol. 165 No. 11 P. 2005-2077

We propose Gamma Conjectures for Fano manifolds which can be thought of as a square root of the index theorem. Studying the exponential asymptotics of solutions to the quantum differential equation, we associate a principal asymptotic class A_F to a Fano manifold F. We say that F satisfies Gamma Conjecture I if A_F equals the ...

Added: November 18, 2014

Entov M., Verbitsky M., / Cornell University. Series math "arxiv.org". 2014.

Let M be a closed symplectic manifold of volume V. We say that M admits a full symplectic packing by balls if any collection of symplectic balls of total volume less than V admits a symplectic embedding to M. In 1994 McDuff and Polterovich proved that symplectic packings of Kahler manifolds can be characterized in ...

Added: February 5, 2015

Galkin S., / Cornell University. Series math "arxiv.org". 2014. No. 1404.7388.

Consider a Laurent polynomial with real positive coefficients such that the origin is strictly inside its Newton polytope. Then it is strongly convex as a function of real positive argument. So it has a distinguished Morse critical point --- the unique critical point with real positive coordinates. As a consequence we obtain a positive answer ...

Added: May 4, 2014

Abouzaid M., Auroux D., Efimov Alexander I. et al., Journal of the American Mathematical Society 2013 Vol. 26 No. 4 P. 1051-1083

We prove that the wrapped Fukaya category of a punctured sphere ($ S^{2}$ with an arbitrary number of points removed) is equivalent to the triangulated category of singularities of a mirror Landau-Ginzburg model, proving one side of the homological mirror symmetry conjecture in this case. By investigating fractional gradings on these categories, we conclude that ...

Added: October 31, 2013

Coates T., Corti A., Galkin S. et al., / Cornell University. Series math "arxiv.org". 2012. No. 1212.1722.

We consider mirror symmetry for Fano manifolds, and describe how one can recover the classification of 3-dimensional Fano manifolds from the study of their mirrors. We sketch a program to classify 4-dimensional Fano manifolds using these ideas. ...

Added: September 14, 2013

Coates T., Corti A., Galkin S. et al., Geometry and Topology 2016 Vol. 20 No. 1 P. 103-256

The quantum period of a variety X is a generating function for certain Gromov-Witten invariants of X which plays an important role in mirror symmetry. In this paper we compute the quantum periods of all 3-dimensional Fano manifolds. In particular we show that 3-dimensional Fano manifolds with very ample anticanonical bundle have mirrors given by ...

Added: November 18, 2014

Galkin S., Iritani H., / Cornell University. Series math "arxiv.org". 2015. No. 1508.00719.

The asymptotic behaviour of solutions to the quantum differential equation of a Fano manifold F defines a characteristic class A_F of F, called the principal asymptotic class. Gamma conjecture of Vasily Golyshev and the present authors claims that the principal asymptotic class A_F equals the Gamma class G_F associated to Euler's Γ-function. We illustrate in ...

Added: August 5, 2015

Positselski L., / Cornell University. Series math "arxiv.org". 2014. No. 1209.2995.

Contraherent cosheaves are globalizations of cotorsion (or similar) modules over commutative rings obtained by gluing together over a scheme. The category of contraherent cosheaves over a scheme is a Quillen exact category with exact functors of infinite product. Over a quasi-compact semi-separated scheme or a Noetherian scheme of finite Krull dimension (in a different version ...

Added: February 6, 2013

Positselski L., Efimov A., / Cornell University. Series math "arxiv.org". 2013. No. arXiv:1102.0261.

We define the triangulated category of relative singularities of a closed subscheme in a scheme. When the closed subscheme is a Cartier divisor, we consider matrix factorizations of the related section of a line bundle, and their analogues with locally free sheaves replaced by coherent ones. The appropriate exotic derived category of coherent matrix factorizations ...

Added: December 22, 2013

V. V. Shevchishin, Izvestiya. Mathematics 2009 Vol. 73 No. 4 P. 797-859

In this paper we prove the non-existence of Lagrangian embeddings of the Klein bottle K in R4 and CP2. We exploit the existence of a special embedding of K in a symplectic Lefschetz pencil pr:X→S2 and study its monodromy. As the main technical tool, we develop the combinatorial theory of mapping class groups. The results ...

Added: March 18, 2013

Bezrukavnikov R., Finkelberg M. V., / Cornell University. Series math "arxiv.org". 2012. No. 1208.3696.

Mark Haiman has reduced Macdonald positivity conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjectures where the symmetric group is replaced by the wreath product $S_n\ltimes (Z/r Z)^n$. He has proven the original conjecture by establishing the geometric statement about the Hilbert ...

Added: February 6, 2013

Michael Finkelberg, Leonid Rybnikov, / Cornell University. Series math "arxiv.org". 2013.

Drinfeld zastava is a certain closure of the moduli space of maps from the projective line to the Kashiwara flag scheme of an affine Lie algebra g^. In case g is the symplectic Lie algebra spN, we introduce an affine, reduced, irreducible, normal quiver variety Z which maps to the zastava space isomorphically in characteristic ...

Added: December 27, 2013

Fedor Bogomolov, Yuri Prokhorov, / Cornell University. Series math "arxiv.org". 2013.

We discuss the problem of stable conjugacy of finite subgroups of Cremona
groups. We show that the group $H^1(G,Pic(X))$ is a stable birational invariant
and compute this group in some cases. ...

Added: November 21, 2014

Lee K., Shabalin T., / Cornell University. Series math "arxiv.org". 2014.

We construct exceptional collections of maximal length on four families of
surfaces of general type with $p_g=q=0$ which are isogenous to a product of
curves. From these constructions we obtain new examples of quasiphantom
categories as their orthogonal complements. ...

Added: October 17, 2014

Efimov A., / Cornell University. Series math "arxiv.org". 2013.

In this paper, we show that bounded derived categories of coherent sheaves (considered as DG categories) on separated schemes of finite type over a field of characteristic zero are homotopically finitely presented. This confirms a conjecture of Kontsevich. The proof uses categorical resolution of singularities of Kuznetsov and Lunts, which is based on the ordinary ...

Added: October 31, 2013

Lev Soukhanov, / Cornell University. Series math "arxiv.org". 2014.

We consider the systems of diffusion-orthogonal polynomials, defined in the
work [1] of D. Bakry, S. Orevkov and M. Zani and (particularly) explain why
these systems with boundary of maximal possible degree should always come from
the group, generated by reflections. Our proof works for the dimensions $2$ (on
which this phenomena was discovered) and $3$, and fails in ...

Added: September 19, 2014

Fedor Bogomolov, Böhning C., / Cornell University. Series math "arxiv.org". 2012.

In this article we determine the stable cohomology groups H^i_s (A_n, Z/p) of the alternating groups A_n for all integers n and i, and all primes p. ...

Added: December 4, 2013

Campana F., Demailly J., Misha Verbitsky, / Cornell University. Series math "arxiv.org". 2013.

We prove that any compact K\"ahler 3-dimensional manifold which has no non-trivial complex subvarieties is a torus. This is a very special case of a general conjecture on the structure of 'simple manifolds', central in the bimeromorphic classification of compact K\"ahler manifolds. The proof follows from the Brunella pseudo-effectivity theorem, combined with fundamental results of ...

Added: May 13, 2013

Ivan Cheltsov, Park J., Won J., / Cornell University. Series math "arxiv.org". 2013.

For each del Pezzo surface $S$ with du Val singularities, we determine
whether it admits a $(-K_S)$-polar cylinder or not. If it allows one, then we
present an effective divisor $D$ that is $\mathbb{Q}$-linearly equivalent to
$-K_S$ and such that the open set $S\setminus\mathrm{Supp}(D)$ is a cylinder.
As a corollary, we classify all the del Pezzo surfaces with du ...

Added: December 27, 2013

Fedor Bogomolov, De Oliveira B., / Cornell University. Series math "arxiv.org". 2014.

In the authors's previous work on symmetric differentials and their
connection to the topological properties of the ambient manifold, a class of
symmetric differentials was introduced: closed symmetric differentials
([BoDeO11] and [BoDeO13]). In this article we give a description of the local
structure of closed symmetric 2-differentials on complex surfaces, with an
emphasis towards the local decompositions as products of ...

Added: November 21, 2014

Victor Kulikov, Shustin E., / Cornell University. Series math "arxiv.org". 2014.

We study the geometry of equiclassical strata of the discriminant in the space of plane curves of a given degree, which are families of curves of given degree, genus and class (degree of the dual curve). Our main observation is that the use of duality transformation leads to a series of new sufficient conditions for ...

Added: February 2, 2015

Kharlamov V., Viktor Kulikov, / Cornell University. Series math "arxiv.org". 2013.

In this article, we investigate some properties of cyclic coverings of complex surfaces of general type branched along smooth curves that are numerically equivalent to a multiple of the canonical class. The main results concern coverings of surfaces of general type with p_g=0 and Miyaoka--Yau surfaces; in particular, they provide new examples of multicomponent moduli ...

Added: December 27, 2013

Cheltsov Ivan, Wilson A., Journal of Geometric Analysis 2013 Vol. 23 No. 3 P. 1257-1289

We classify smooth del Pezzo surfaces whose α-invariant of Tian is bigger than 1. ...

Added: November 14, 2013

Brav C. I., Thomas H., Mathematische Annalen 2011 Vol. 351 No. 4 P. 1005-1017

We establish faithfulness of braid group actions generated by twists along an ADE configuration of 22-spherical objects in a derived category. Our major tool is the Garside structure on braid groups of type ADE. This faithfulness result provides the missing ingredient in Bridgeland's description of a space of stability conditions associated to a Kleinian singularity. ...

Added: September 29, 2014