## Mariusz Koras

**Mariusz Koras** (born November 10^{th}, 1951, Piaseczno, Poland, died September 15^{th}, 2017, Betina, Murter Island, Croatia) was a Polish mathematician, specializing in algebraic geometry, mainly in complex affine geometry. He was also a high-altitude climber with many achievements in Tatra Mountains and Alps. Mariusz was married with Krystyna Stańkowska-Koras and had two children, Maria and Krzysztof. He loved nature and animals (at home too: the number of their cats reached 10 at some point). He liked Pink Floyd, crime stories, and paintings by W.Kandinsky.

**Career **

Mariusz Koras was a student of Andrzej Białynicki-Birula. He obtained a doctorate in 1981 for a thesis *Meromorphic actions of reductive groups*. In 2000 he received his habilitation at the University of Warsaw. The thesis entitled *Linearization of algebraic actions of C** summarizes results of him and his long-time collaborator, Peter Russell (McGill University), which costitute a key step in the solution of the Linearization Conjecture for algebraic

**C***-actions on the 3-dimensional affine space

**C**

^{3}. He became a Professor of Mathematics in 2014. For most of his life he was working at the University of Warsaw, Institute of Mathematics, Faculty of Mathematics, Informatics and Mechanics, where he taught numerous courses. He was very liked by students. Mariusz Koras was a supervisor of one PhD. student, Karol Palka.

He published in journals like *Duke Mathematical Journal*, *Journal of Algebraic Geometry*, *Journal of Algebra*, *Transformation Groups* or *Mathematische Annalen*. He was a co-author and a co-editor of the book *Affine Algebraic Geometry. The Russell Festschrift. A *member of the Polish Mathematical Society. Among his collaborators are R. V. Gurjar, K. Masuda, M. Miyanishi, K. Palka and P. Russell.

**Selected mathematical results**

**The Linearization Conjecture**

It states that every algebraic action of **C*** on an affine space **C**^{n} becomes linear after a suitable choice of coordinates, that is, in those coordinates it is given by t · (x_{1 }, ... , x_{n}) = (t^{a1}x_{1 , ... , }t^{an}x_{n}) for some integers a_{1},…, a_{n. }Koras and Russel made the key step in its proof in dimension three [17]. They have created a list of specific 3-dimensional complex varieties diffeomorphic with **C**^{3} (nowadays called *Koras-Russel threefolds*) and they showed that the Linearization Conjecture in dimension three holds if and only if each of these varieties is an exotic affine space, that is, is not isomorphic to **C**^{3} as an algebraic variety. Afterwards these varieties were distinguished from **C**^{3} by S. Kaliman and L. Makar-Limanov using a new invariant (ML-invariant) of an affine variety. It is still not known whether Koras-Russell threefolds are counterexamples to the Zariski Cancellation Conjecture. For example, it is not known whether R×**C**^{1} is isomorphic to **C**^{4} as an algebraic variety, where R = {x+x^{2}y+z^{2}+t^{3} = 0} is the Koras-Russell cubic. Recently, they became an object of interest of **A**^{1}-homotypy theory, as some of them were proved to be **A**^{1}-contractible. Articles related to the problem of linearization are [5,15-18,20-25].

**The Coolidge-Nagata conjecture**

It states that every algebraic complex planar curve which topologically is the same as the projective line is in fact Cremona-equivalent to a line, that is, it can be mapped onto a line by a birational automorphism of the projective plane. There is no classification of such curves, but many infinite, rather exotic families have been constructed. Note that, in contrast, general rational curves of degree at least 6 are known to be non-Cremona-equivalent to a line. The conjecture was proven together with Palka, based on a generalization of the theory of almost minimal models and results from the logarithmic Minimal Model Program, see [2]. They announced also a proof of another conjecture, which says that a curve as above may have at most four singular points [26].

**Classification results for open surfaces**

Mariusz Koras was an expert in the theory of algebraic surfaces. He obtained many significant results using the theory of the logarithmic Kodaira-Iitaka dimension and developing the theory of almost minimal models and tools to study singularities of surfaces. The results include characterizations of various classes of topologically contractible or **Q**-acyclic surfaces and their singularities, see [4,8,9,12,14,19].

**Classification of planar C*-embeddings**

A theorem of Abhyankar-Moh and Suzuki says that the affine line **C**^{1} can be embedded into the affine plane **C**^{2} in a unique way up to a choice of algebraic coordinates. An analogous problem of describing possible embeddings of **C***=**C**^{1}\{0} into **C**^{2} is surprisingly subtle. A conditional list have been obtained by Borodzik and Żołądek. Koras and Russell started a project of unconditional classification. In a collaboration with Cassou-Nogues, Palka and Russell, Koras obtained a series of results toward the classification, including a classification of embeddings with good asymptotic properties, see [3,6,10,11,27].

### The list of articles:

*Affine threefolds admitting*(with R.V. Gurjar, K.Masuda, M. Miyanishi and P. Russell). accepted to Math. Ann. (2017)**G**_{a}-actions-
*The Coolidge-Nagata conjecture*(with K. Palka), Duke Math. J. 166 (2017), no. 16, 3085-3145. *The geometry of sporadic*(with K. Palka and P. Russell), J. Algebra**C***-embeddings into**C**^{2}**456**(2016), 207-249.*A homology plane of general type can have at most a cyclic quotient singularity*(with R.V. Gurjar, M. Miyanishi and P. Russell). J. Algebraic Geom. 23 (2014), no. 1, 1-62.*Separable forms of*(with P. Russell). Transform. Groups 18 (2013), no. 4, 1155-1163.**G**_{m}-actions on**A**^{3}*Some properties of***C*** in**C**^{2}(with P. Russell). Affine algebraic geometry, 160-197, World Sci. Publ., Hackensack, NJ, 2013.*strong>C*-fibrations on affine threefolds*(with R.V. Gurjar, K.Masuda, M. Miyanishi and P. Russell). Affine algebraic geometry, 62-102, World Sci. Publ., Hackensack, NJ, 2013.*Singular*(with K. Palka). Osaka J. Math. 50 (2013), no. 1, 61-114.**Q**-homology planes of negative Kodaira dimension have smooth locus of non-general type*Affine normal surfaces with simply-connected smooth locus*(with R.V. Gurjar, M. Miyanishi and P. Russell). Math. Ann. 353 (2012), no. 1, 127-144.. Affine algebraic geometry, 165-191, CRM Proc. Lecture Notes, 54, Amer. Math. Soc., Providence, RI, 2011.**C*** in**C**^{2}is birationally equivalent to a line*Closed embeddings of*(with P. Cassou-Nogues and P. Russell). J. Algebra 322 (2009), no. 9, 2950-3002.**C*** in**C**^{2}I.*Two dimensional quotients of*(with R.V. Gurjar and P. Russell). Electron. Res. Announc. Math. Sci.**C**^{n}by a reductive group**15**(2008), 62-64.*On contractible plane curves.*Affine algebraic geometry, 275-288, Osaka Univ. Press, Osaka, 2007.*Contractible affine surfaces with quotient singularities*(with P. Russell). Transform. Groups 12 (2007), no. 2, 293-340.*Linearization problems. Algebraic group actions and quotients*(with P. Russell). 91-107, Hindawi Publ. Corp., Cairo, 2004.(with P. Russell). J. Algebraic Geom. 8 (1999), no. 4, 603-694.**C*** actions on**C**^{3}: the smooth locus of the quotient is not of hyperbolic type*Contractible threefolds and*(with P. Russell). J. Algebraic Geom. 6 (1997), no. 4, 671-695.**C*** actions on**C**^{3}(with S. Kaliman, L. Makar-Limanov and P. Russell). Electron. Res. Announc. Amer. Math. Soc.**C*** actions on**C**^{3}are linearizable**3**(1997), 63-71.*A characterization of*Compositio Math. 87 (1993), no. 3, 241-267.**A**^{2}/**Z**_{a}.*Codimension 2 torus actions on affine n-space*(with P. Russell). Group actions and invariant theory (Montreal, PQ, 1988), 103-110, CMS Conf. Proc., 10, Amer. Math. Soc., Providence, RI, 1989.*On linearizing "good"*(with P. Russell). Group actions and invariant theory (Montreal, PQ, 1988), 93-102, CMS Conf. Proc., 10, Amer. Math. Soc., Providence, RI, 1989.**C*** actions on**C**^{3}(with P. Russell). Proceedings of the 1984 Vancouver conference in algebraic geometry, 269-276, CMS Conf. Proc., 6, Amer. Math. Soc., Providence, RI, 1986.**G**_{m}-actions on**A**^{3}*Linearization of reductive group actions.*Group actions and vector fields (Vancouver, B.C., 1981), 92-98, Lecture Notes in Math., 956, Springer, Berlin, 1982.*Deformations of actions of linearly reductive groups.*Comment. Math. Prace Mat. 22 (1980), no. 1, 81-83.*On actions of an analytic torus.*Bull. Acad. Polon. Sci. Sér. Sci. Math. 27 (1979), no. 1, 21-26.

#### In preparation

*Planar rational cuspidal curves have at most 4 cusps*(with K. Palka)*Classification of sporadic*(with K. Palka)**C***-embeddings into**C**^{2}

**Climbing**

Mariusz Koras was an active member of the Warsaw group of climbers in years 1972-1982. His best known achievement is the Koras-Wolf route ("A Pole in the outer space") which was established on Kocioł Kazalnicy Mięguszowskiej (Tatra Mountains) on June 28-29^{th}, 1978 with Jan Wolf. This route, going mostly through steep or overhang rock, is described by a climbing guide as offbeat, audacious and one of the most beutiful routes in this region. He climbed many other difficult routes in Tatras and Alps, both in summer and winter. He ascended also many peaks in the Himalayas, including Tarapurn whose first ascent belongs to Mariusz and his team, as well as in the Canadian Rocky Mountains.

More about his climbing achievements (google-translation from Polish, may be inaccurate) can be found here.

#### Selected climbs:

- first one-day climb on the Heinrich-Chrobak route, Kazalnica (Tatra Mountains), Summer 1973
- first Polish and first free ascent on Łomnica (western wall, Tatra Mountains), Fall 1975
- first winter climb on Mały Kieżmarski Szczyt, 1974 first Winter climb on Cubryna 1975
- first Polish climbs in Alps: Mont Maudit (SE wall, 1973), Aiguille de Pouce (southern wall)
- expedition manager in Himalaya Kishtwar in India, where he did the first ascent to Tarparun (6013 m, Nanth Nullah Valley) in a 6-person team
- first Polish climbs to Bugaboo Spire (3185), Snowpatch Spire (3064) and Pigeon Spire (3125) in The Bugaboos (Purcell Mountains, British Columbia, Canada)