Birational geometry of higher-dimensional algebraic varieties.
Biography
Graduated from Lomonosov Moscow State University (Department of Mathematics and Mechanics, Chair of Algebra) in 1984. Ph.D. (Cand. Sci.) — 1987 from Moscow State University. D.Sci. (Mathematics) — 1998 from Steklov Institute of Mathematics. The lists of my publications contains about 80 items. In 1998 I was an invited speaker at the International Congress of Mathematicians in Berlin (45-minute sectional address).
Main publications:
A. Pukhlikov, Birationally Rigid Varieties, Mathematical Surveys and Monographs, 190, Amer. Math. Soc., Providence, RI, 2013, 367 pp.
A. V. Pukhlikov, “Birationally rigid Fano fibrations”, Izv. Math., 64:3 (2000), 563–581
A. V. Pukhlikov, “Birationally rigid Fano double hypersurfaces”, Sb. Math., 191:6 (2000), 883–908
A. V. Pukhlikov, “Birational automorphisms of Fano hypersurfaces”, Invent. Math., 134:2 (1998), 401–426
A. V. Pukhlikov, “Birational automorphisms of algebraic threefolds with a pencil of Del Pezzo surfaces”, Izv. Math., 62:1 (1998), 115–155
A. V. Pukhlikov, “Rationally connected Galois covers of Fano-Mori fibre spaces”, Math.Zeitschrift, 312:1 (2026), 27, 15 pp., arXiv: 2311.11803
2025
3.
A. V. Pukhlikov, “Affine Complements in the Projective Space with Trivial Automorphism Group”, Proc. Steklov Inst. Math., 329 (2025), 207–216
4.
A. V. Pukhlikov, “Birational rigidity of quartic three-folds with a double point of rank 3”, Advances in Geometry, 25:4 (2025), 447–461, arXiv: 2410.15193
2024
5.
A. V. Pukhlikov, “Birationally rigid hypersurfaces with quadratic singularities of low rank”, Sb. Math., 215:6 (2024), 823–840
6.
A. V. Pukhlikov, “Birationally rigid Fano-Mori fibre spaces”, Forum of Mathematics Sigma, 12 (2024), e84, 63 pp., arXiv: 2305.16219
Aleksandr V. Pukhlikov, “The Gabrielov-Khovanskii problem for polynomials”, Arnold Math. J., 2:1 (2016), 29–68
2015
24.
A. V. Pukhlikov, “Two Inequalities for a Sequence of Blow-Ups”, Math. Notes, 97:6 (2015), 970–973
25.
A. V. Pukhlikov, “Birationally rigid Fano fibre spaces. II”, Izv. Math., 79:4 (2015), 809–837
2014
26.
Th. Eckl, A. Pukhlikov, “On the locus of nonrigid hypersurfaces”, Automorphisms in Birational and Affine Geometry (Levico Terme, Italy, October 2012), Springer Proceedings in Mathematics & Statistics, 79, Springer, 2014, 121–139
A. V. Pukhlikov, “Birational rigidity of Fano complete intersections”, Dokl. Math., 87:1 (2013), 34–35
2012
32.
A. V. Pukhlikov, “$K$-Trivial Structures on Fano Complete Intersections”, Math. Notes, 91:4 (2012), 568–574
33.
A. V. Pukhlikov, “On the multiplicity of solutions of a system of algebraic equations”, Proc. Steklov Inst. Math., 276 (2012), 234–249
2010
34.
A. V. Pukhlikov, “Birational geometry of Fano double spaces of index two”, Izv. Math., 74:5 (2010), 925–991
35.
A. V. Pukhlikov, “Existence of the Kähler–Einstein Metric on Certain Fano Complete Intersections”, Math. Notes, 88:4 (2010), 552–558
36.
A. V. Pukhlikov, “Birationally rigid varieties. II. Fano fibre spaces”, Russian Math. Surveys, 65:6 (2010), 1083–1171
2009
37.
A. V. Pukhlikov, “Rationality problem and birational rigidity”, Cohomological and geometric approaches to rationality problems, Progr. Math., 282, Birkhäuser, Boston, 2009, 275–311
A. V. Pukhlikov, “Birational geometry of algebraic varieties with a pencil of Fano cyclic covers”, Pure Appl. Math. Q., 5:2, Special Issue: In honor of Friedrich Herzebruch (2009), 641–700
A. V. Pukhlikov, “Singularities of Algebraic Subvarieties and Problems of Birational Geometry”, Proc. Steklov Inst. Math., 267 (2009), 235–247
40.
A. V. Pukhlikov, “Birational Geometry of Singular Fano Varieties”, Proc. Steklov Inst. Math., 264 (2009), 159–177
41.
F. A. Bogomolov, Vik. S. Kulikov, Yu. I. Manin, V. V. Nikulin, D. O. Orlov, A. N. Parshin, Yu. G. Prokhorov, A. V. Pukhlikov, M. Reid, I. R. Shafarevich, V. V. Shokurov, “Vasilii Alekseevich Iskovskikh (obituary)”, Russian Math. Surveys, 64:5 (2009), 939–946
2010
42.
A. V. Pukhlikov, “On the self-intersection of a movable linear system”, J. Math. Sci., 164:1 (2010), 119–130
2008
43.
A. V. Pukhlikov, “Birational geometry of Fano double covers”, Sb. Math., 199:8 (2008), 1225–1250
2007
44.
A. V. Pukhlikov, “Explicit examples of birationally rigid Fano varieties”, Mosc. Math. J., 7:3 (2007), 543–560
A. V. Pukhlikov, “Birationally rigid varieties. I. Fano varieties”, Russian Math. Surveys, 62:5 (2007), 857–942
46.
A. V. Pukhlikov, “Kvadratichnye metody v teorii optimalnogo upravleniya i teorii sistem”, Nelineinaya dinamika i upravlenie, 5, Fizmatlit, M., 2007, 69–98
2006
47.
A. V. Pukhlikov, “Birational geometry of algebraic varieties with a pencil of Fano complete intersections”, Manuscripta Math., 121:4 (2006), 491–526
A. V. Pukhlikov, “Birationally rigid varieties with a pencil of Fano double covers. III”, Sb. Math., 197:3 (2006), 335–368
2005
49.
A. V. Pukhlikov, “Geometry of discontinuous systems near a manifold of singularities of third order”, Differ. Equ., 41:12 (2005), 1710–1716
50.
A. V. Pukhlikov, “Biratsionalno zhestkie pryamye proizvedeniya Fano”, Dokl. RAN, 401:2 (2005), 163–165
51.
A. V. Pukhlikov, “Birational geometry of Fano direct products”, Izv. Math., 69:6 (2005), 1225–1255
2004
52.
A. V. Pukhlikov, “Birationally rigid Fano varieties”, The Fano Conference, Univ. Torino, Turin, 2004, 659–681
53.
A. V. Pukhlikov, “Optimal control of distributions”, Nonlinear dynamics and control. 2, Comput. Math. Model., 15:3 (2004), 223–256
2007
54.
A. V. Pukhlikov, “Integral representations in the theory of controlled systems”, Nonlinear dynamics and control, Comput. Math. Model., 18, no. 4, 2007, 344–368
55.
A. V. Pukhlikov, “Fano direct products as rationally connected fibrations”, J. Math. Sci., 140:2 (2007), 303–320
2004
56.
A. V. Pukhlikov, “Birationally rigid varieties with a pencil of Fano double covers. II”, Sb. Math., 195:11 (2004), 1665–1702
57.
A. V. Pukhlikov, “Birationally rigid varieties with a pencil of double Fano covers. I”, Sb. Math., 195:7 (2004), 1039–1071
2003
58.
A. V. Pukhlikov, “Birationally rigid singular Fano hypersurfaces”, Algebraic geometry, 12, J. Math. Sci. (N. Y.), 115:3 (2003), 2428–2436
A. V. Pukhlikov, “Birationally rigid iterated Fano double covers”, Izv. Math., 67:3 (2003), 555–596
2002
60.
A. V. Pukhlikov, “Birational geometry of higher-dimensional Fano hypersurfaces”, Dokl. Akad. Nauk, 385:1 (2002), 28–30
61.
A. V. Pukhlikov, “Discontinuous dynamical systems in optimal control theory”, Nonlinear dynamics and control, Comput. Math. Model., 13:1 (2002), 81–105
62.
A. V. Pukhlikov, “Birational Automorphisms of Algebraic Varieties with a Pencil of Fano Double Covers”, Funct. Anal. Appl., 36:1 (2002), 74–75
63.
A. V. Pukhlikov, “Birationally rigid Fano hypersurfaces”, Izv. Math., 66:6 (2002), 1243–1269
64.
A. V. Pukhlikov, “Birational automorphisms of Fano hypersurfaces with singularities”, Russian Math. Surveys, 57:1 (2002), 157–158
65.
F. Campana, Th. Peternell, A. V. Pukhlikov, “Generalized Tsen's theorem and rationally connected Fano fibrations”, Sb. Math., 193:10 (2002), 1443–1468
66.
A. V. Pukhlikov, “Birationally rigid Fano hypersurfaces with isolated singularities”, Sb. Math., 193:3 (2002), 445–471
2001
67.
F. Kampana, T. Peternell, A. V. Pukhlikov, “Obobschennaya teorema Tzena i ratsionalno svyaznye mnogoobraziya”, Dokl. RAN, 381:6 (2001), 742–744
68.
A. V. Pukhlikov, “Birationally rigid Fano complete intersections”, J. Reine Angew. Math., 541 (2001), 55–79
A. V. Pukhlikov, “Singularities in discontinuous dynamical systems”, Dokl. Akad. Nauk, 378:4 (2001), 456–458
70.
A. V. Pukhlikov, “The geometry of discontinuous systems near a hyperbolic singular point of a sliding field”, Differ. Equ., 37:3 (2001), 377–400
1997
71.
A. V. Pukhlikov, “Birational automorphisms of double spaces with singularities”, J. Math. Sci. (New York), 85:4 (1997), 2128–2141
1996
72.
V. A. Iskovskikh, A. V. Pukhlikov, “Birational automorphisms of multidimensional algebraic manifolds”, J. Math. Sci., 82:4 (1996), 3528–3613
2000
73.
A. Corti, A. Pukhlikov, M. Reid, “Fano 3-fold hypersurfaces”, Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser., 281, Cambridge Univ. Press, Cambridge, 2000, 175–258
74.
A. V. Pukhlikov, “Essentials of the method of maximal singularities”, Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser., 281, Cambridge Univ. Press, Cambridge, 2000, 73–100
75.
A. V. Pukhlikov, “Biratsionalnye avtomorfizmy polnykh peresechenii Fano”, Doklady Akademii nauk, 374:2 (2000), 160–163
76.
A. V. Pukhlikov, “A Gauss–Ostrogradskii theorem for integral transforms over the Euler characteristic”, Integral Transform. Spec. Funct., 9:4 (2000), 299–312
A. V. Pukhlikov, “Birationally rigid Fano fibrations”, Izv. Math., 64:3 (2000), 563–581
78.
A. V. Pukhlikov, “Fiber-wise birational correspondences”, Math. Notes, 68:1 (2000), 103–112
79.
A. V. Pukhlikov, “Birational automorphisms of algebraic varieties with a pencil of double quadrics”, Math. Notes, 67:2 (2000), 192–199
80.
A. V. Pukhlikov, “Birationally rigid Fano double hypersurfaces”, Sb. Math., 191:6 (2000), 883–908
1999
81.
A. V. Pukhlikov, “Geometry of discontinuous systems near an elliptic singularity”, Differential Equations, 35:11 (1999), 1516–1529 (2000)
82.
A. V. Pukhlikov, “Structurally stable singularities of discontinuous dynamical systems”, Differential Equations, 35:5 (1999), 675–682
83.
A. V. Pukhlikov, “Certain examples of birationally rigid varieties with a pencil of double quadrics”, Algebraic geometry, 9, J. Math. Sci. (New York), 94:1 (1999), 986–995
A. V. Pukhlikov, “Birational Automorphisms of Fano Double Covers”, Funct. Anal. Appl., 33:4 (1999), 313–314
85.
A. V. Pukhlikov, “Birationally rigid varieties with a sheaf of Fano hypersurfaces”, Russian Math. Surveys, 54:1 (1999), 260–261
86.
A. I. Kostrikin, V. S. Kulikov, Yu. I. Manin, V. V. Nikulin, A. N. Parshin, Yu. G. Prokhorov, A. V. Pukhlikov, M. Reid, A. N. Tyurin, I. R. Shafarevich, V. V. Shokurov, “Vasilii Alekseevich Iskovskikh (on his 60th birthday)”, Russian Math. Surveys, 54:4 (1999), 863–868
1998
87.
A. V. Pukhlikov, “Optimal bilinear systems on a plane”, Differential Equations, 34:11 (1998), 1516–1521 (1999)
88.
A. V. Pukhlikov, “Birational automorphisms of three-dimensional algebraic varieties fibered by Del Pezzo surfaces”, Dokl. Akad. Nauk, 361:1 (1998), 14–16
89.
A. V. Pukhlikov, “The discrete maximum principle”, Dokl. Akad. Nauk, 360:6 (1998), 747–749
A. V. Pukhlikov, “Birational automorphisms of higher-dimensional algebraic varieties”, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), Doc. Math., Extra Vol. II, 1998, 97–107 (electronic)
92.
A. V. Pukhlikov, “Birational automorphisms of algebraic threefolds with a pencil of Del Pezzo surfaces”, Izv. Math., 62:1 (1998), 115–155
1997
93.
A. V. Pukhlikov, “Integral transformations in the theory of controllable dynamic systems”, Dokl. Math., 55:2 (1997), 315–317
94.
A. V. Pukhlikov, “Integral representations in the theory of discrete controllable systems”, Dokl. Math., 55:2 (1997), 312–314
95.
A. V. Pukhlikov, “Birational automorphisms of algebraic threefolds fibred by cubic surfaces”, Russian Math. Surveys, 52:1 (1997), 230–231
96.
A. V. Pukhlikov, Mat. Pros., 1, MCCME, Moscow, 1997, 85–89
1996
97.
A. V. Pukhlikov, “Fredholm determinants and the Newton method in problems of the control of distributions”, Automat. Remote Control, 57:6 (1996), 814–818
98.
A. V. Pukhlikov, “Singular regimes in problems of the control of distributions. I. Scalar controls”, Automat. Remote Control, 57:5 (1996), 634–643
1995
99.
A. V. Pukhlikov, “Degenerations of Hamiltonian structures in optimal control problems”, Differential Equations, 31:11 (1995), 1841–1847 (1996)
100.
A. V. Pukhlikov, “Analytic representations in problems of the control of distributions”, Automat. Remote Control, 56:12 (1995), 1698–1708 (1996)
101.
A. V. Pukhlikov, “Rational curves with prescribed branches”, Moscow Univ. Math. Bull., 50:1 (1995), 20–22
102.
A. V. Pukhlikov, “Hamiltonian structures in optimal control theory”, J. Dynam. Control Systems, 1:3 (1995), 379–401
A. V. Pukhlikov, “Problems of the control of distributions in dynamical systems”, Automat. Remote Control, 56:4 (1995), 517–525
104.
A. V. Pukhlikov, “A remark on the theorem of V. A. Iskovskikh and Yu. I. Manin on a three-dimensional quartic”, Proc. Steklov Inst. Math., 208 (1995), 244–254
1994
105.
A. V. Pukhlikov, “Liouville's theorem in optimal control problems”, Differential Equations, 30:11 (1994), 1808–1814 (1995)
1993
106.
A. G. Khovanskii, A. V. Pukhlikov, “Integral transforms based on Euler characteristic and their applications”, Integral Transform. Spec. Funct., 1:1 (1993), 19–26
A. V. Pukhlikov, “Poisson's theorem in optimal control problems”, Differential Equations, 29:11 (1993), 1685–1690 (1994)
108.
A. I. Propoĭ, A. V. Pukhlikov, “Quadratic approximation in nonlinear extremal problems”, Phys. Dokl., 38:8 (1993), 320–322 (1994)
109.
V. A. Iskovskih, A. V. Pukhlikov, “Birational automorphisms of Fano varieties”, Geometry of complex projective varieties (Cetraro, 1990), Sem. Conf., 9, Mediterranean, Rende, 1993, 191–201
110.
A. V. Pukhlikov, “Asymptotically periodic systems. II”, Autom. Remote Control, 54:10 (1993), 1493–1503
111.
A. V. Pukhlikov, “Asymptotically periodic systems. I”, Autom. Remote Control, 54:5 (1993), 747–754
112.
A. I. Propoǐ, A. V. Pukhlikov, “The stochastic Newton method in nonlinear extremal problems”, Autom. Remote Control, 54:4 (1993), 605–613
113.
A. V. Pukhlikov, A. G. Khovanskii, “Finitely additive measures of virtual polyhedra”, St. Petersburg Math. J., 4:2 (1993), 337–356
114.
A. V. Pukhlikov, A. G. Khovanskii, “The Riemann–Roch theorem for integrals and sums of quasipolynomials on virtual polytopes”, St. Petersburg Math. J., 4:4 (1993), 789–812
1989
115.
A. V. Pukhlikov, “Maximal singularities on the Fano variety $V_6^3$”, Moscow Univ. Math. Bull., 44:2 (1989), 70–75
116.
A. V. Pukhlikov, “Birational automorphisms of a double space and double quadric”, Math. USSR-Izv., 32:1 (1989), 233–243
117.
A. V. Pukhlikov, “Birational automorphisms of a three-dimensional quartic with a quadratic singularity”, Math. USSR-Sb., 63:2 (1989), 457–482
1987
118.
A. V. Pukhlikov, “Birational isomorphisms of four-dimensional quintics”, Invent. Math., 87:2 (1987), 303–329
Бирационально жесткие расслоения Фано Alexander Pukhlikov Annual memorial conference dedicated to the memory of Andrei Nikolaevich Tyurin October 28, 2014 11:40