?
New examples of cylindrical Fano fourfolds
P. 443–463.
Prokhorov Y., Zaidenberg M.
We produce new families of smooth Fano fourfolds with Picard rank 1, which contain cylinders, i.e., Zariski open subsets of form Z × A 1 , where Z is a quasiprojective variety. The affine cones over such a fourfold admit effective G a -actions. Similar constructions of cylindrical Fano threefolds and fourfolds were done previously in [KPZ11, KPZ14, PZ16].
Publication based on the results of:
In book
Vol. 75: Algebraic Varieties and Automorphism Groups. , Tokyo: American Mathematical Society, World Scientific, 2017.
Loginov K., Przyjalkowski V., Trepalin A., Труды Математического института им. В.А. Стеклова РАН 2025 Т. 329 С. 132–164
We introduce and study the notion of G-coregularity of algebraic varieties endowed with an action of a finite group G. We compute the G-coregularity of smooth del Pezzo surfaces of degree at least 6, and give a characterization of groups that can act on conic bundles with G-coregularity 0. We describe the relations between the notions of G-coregularity, G-log ...
Added: September 4, 2025
Przyjalkowski V., Shramov K., Communications in Number Theory and Physics 2020 Vol. 14 No. 3 P. 511–553
We prove that if a smooth variety with non-positive canonical class can be embedded into a weighted projective space of dimension n as a well formed complete intersection and it is not an intersection with a linear cone therein, then the weights of the weighted projective space do not exceed n+1. Based on this bound ...
Added: October 13, 2020
Kuznetsov A., Известия РАН. Серия математическая 2019 Т. 83 № 3 С. 127–132
Мы показываем, что ограниченная производная категория когерентных пучков на общей поверхности Энриквеса может быть реализована как полуортогональная компонента в производной категории гладкого многообразия Фано с диагональным ромбом Ходжа. ...
Added: June 4, 2019
Galkin S., Iritani H., , in: Primitive Forms and Related Subjects — Kavli IPMU 2014.: Tokyo: Mathematical Society of Japan, 2019. P. 55–115.
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 associated to Euler's Gamma-function.
We illustrate in the case of ...
Added: September 1, 2018
Coates T., Galkin S., Kasprzyk A. et al., Experimental Mathematics 2020 Vol. 29 No. 2 P. 183–221
We collect a list of known four-dimensional Fano manifolds and compute their quantum periods. This list includes all four-dimensional Fano manifolds of index greater than one, all four-dimensional toric Fano manifolds, all four-dimensional products of lower-dimensional Fano manifolds, and certain complete intersections in projective bundles. ...
Added: September 1, 2018
Prokhorov Y., Kuznetsov A., Shramov K., Japanese Journal of Mathematics 2018 Vol. 13 No. 1 P. 109–185
We discuss various results on Hilbert schemes of lines and conics and automorphism groups of smooth Fano threefolds of Picard rank 1. Besides a general review of facts well known to experts, the paper contains some new results, for instance, we give a description of the Hilbert scheme of conics on any smooth Fano threefold ...
Added: November 22, 2017
Fonarev A., Kuznetsov A., / Series arXiv "math". 2016.
We prove that the derived category D(C) of a generic curve of genus greater than one embeds into the derived category D(M) of the moduli space M of rank two stable bundles on C with fixed determinant of odd degree. ...
Added: April 10, 2017
Prokhorov Y., Reid M., , in: Advanced Studies in Pure MathematicsVol. 70: Minimal Models and Extremal Rays.: Kyoto: Mathematical Society of Japan, 2016.
Let X be a Q-Fano 3-fold of rank 1 with qQ(X)=qF(X)=2 and KX not Cartier. Let A be a Weil divisor on X such that −KX = 2A. Then dim |A| ≤ 4. Moreover, if dim |A| = 4, then X belongs to the single irreducible family. ...
Added: October 13, 2016
Galkin S., / Series math "arxiv.org". 2016. No. 1604.04652.
For Fano manifolds we define Ap\'ery constants and Ap\'ery class as particular limits of ratios of coefficients of solutions of the quantum differential equation. We do numerical computations in case of homogeneous varieties. These numbers are identified to be polynomials in the values of Riemann zeta-function with natural arguments. ...
Added: April 19, 2016
Prokhorov Y., Zaidenberg M., European Journal of Mathematics 2016 Vol. 2 No. 1 P. 262–282
We construct four different families of smooth Fano fourfolds with Picard rank 1, which contain cylinders, i.e., Zariski open subsets of the form Z ×A1, where Z is a quasiprojective variety. The affine cones over such a fourfold admit effective Ga-actions. Similar constructions of cylindrical Fano threefolds were done previously in the papers by Kishimoto ...
Added: November 27, 2015
Kuznetsov A., Debarre O., / Series math "arxiv.org". 2015.
This paper performs a systematic study of Gushel–Mukai varieties—Fano manifolds with Picard number 1, coindex 3, and degree 10 (higher-dimensional analogues of prime Fano threefolds of genus 6). We introduce a new approach to the classification of these varieties which includes mildly singular varieties, gives a criterion for an isomorphism of such varieties, and describes ...
Added: November 15, 2015
Prokhorov Y., Zaidenberg M., / Series math "arxiv.org". 2015. No. 1507.01748.
We construct new families of smooth Fano fourfolds with Picard rank 1, which contain cylinders, i.e., Zariski open subsets of form Z×, where Z is a quasiprojective variety. The affine cones over such a fourfold admit effective G_a-actions. Similar constructions of cylindrical Fano threefolds and fourfolds were done previously in [KPZ11, KPZ14, PZ15]. ...
Added: October 13, 2015
Galkin S., Iritani H., / 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
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., 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
Coates T., Galkin S., Kasprzyk A. et al., / Series math "arxiv.org". 2014. No. 1406.4891.
We collect a list of known four-dimensional Fano manifolds and compute their quantum periods. This list includes all four-dimensional Fano manifolds of index greater than one, all four-dimensional toric Fano manifolds, all four-dimensional products of lower-dimensional Fano manifolds, and certain complete intersections in projective bundles. ...
Added: June 20, 2014
Galkin S., Shinder E., / Series math "arxiv.org". 2014. No. 1405.5154.
We find a relation between a cubic hypersurface Y and its Fano variety of lines F(Y) in the Grothendieck ring of varieties. We prove that if the class of an affine line is not a zero-divisor in the Grothendieck ring of varieties, then Fano variety of lines on a smooth rational cubic fourfold is birational ...
Added: May 21, 2014
Galkin S., / 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
Galkin S., Golyshev V., Iritani H., / Series math "arxiv.org". 2014. No. 1404.6407.
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: May 4, 2014
Coates T., Corti A., Galkin S. et al., , in: European Congress of Mathematics Kraków, 2 – 7 July, 2012.: Zürich: European Mathematical Society Publishing house, 2014. Ch. 16 P. 285–300.
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: February 19, 2014
Cheltsov Ivan, Shramov Constantin, Experimental Mathematics 2013 Vol. 22 No. 3 P. 313–326
We study del Pezzo surfaces that are quasismooth and well-formed weighted hypersurfaces. In particular, we find all such surfaces whose α-invariant of Tian is greater than 2/3. ...
Added: January 27, 2014
Prokhorov Y., Springer Proceedings in Mathematics & Statistics 2014 Vol. 79 P. 215–229
We give a sharp bound for orders of elementary abelian 2-groups of birational automorphisms of rationally connected threefolds. ...
Added: January 24, 2014