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NATIONAL RESEARCH UNIVERSITYHIGHER SCHOOL OF ECONOMICSFACULTY OF MATHEMATICSValentina KiritchenkoGeometry of spherical varieties andNewton–Okounkov polytopesSummary of the thesisfor the purpose of obtaining academic degreeDoctor of Science in Mathematics HSEMoscow — 20182Toric geometry and theory of Newton polytopes exhibited fruitful connectionsbetween algebraic geometry and convex geometry. After the Kouchnirenko andBernstein–Khovanskii theorems were proved in the 1970-s (for a reminder see Section1), Askold Khovanskii asked how to extend these results to the setting where acomplex torus is replaced by an arbitrary connected reductive group.

In particular,he advertised widely the problem of finding the right analogs of Newton polytopesfor spherical varieties. The latter are natural generalizations of toric varieties andinclude classical examples such as Grassmannians, flag varieties and complete conics(see Section 2 for a reminder). Notion of Newton polytopes was extended to sphericalvarieties by Andrei Okounkov in the 1990-s [O97, O98]. Later, his constructionwas developed systematically in [KaKh, LM], and the resulting theory of Newton–Okounkov convex bodies is now an active field of algebraic geometry.While Newton–Okounkov convex bodies can be defined for line bundles on arbitrary varieties (without a group action), they are easier to deal with in thecase of spherical varieties because of connections with representation theory.

Forinstance, Gelfand–Zetlin (GZ) polytopes and Feigin–Fourier–Littelmann–Vinberg(FFLV) polytopes arise naturally as Newton–Okounkov polytopes of flag varieties.My research focuses on explicit description of geometric and topological invariantsof spherical varieties in terms of geometric and combinatorial invariants of theirNewton–Okounkov polytopes. The goal is to extend the toric picture to the moregeneral setting of varieties with a reductive group action. Section 3 is a survey ofmy results in this direction.

Section 4 contains precise formulations of main results.1. Newton–Okounkov convex bodiesIn this section, we recall construction of Newton–Okounkov convex bodies forthe general mathematical audience. Let us start from the definition of Newtonpolytopes.PDefinition 1.1. Let f = α∈Zn cα xα be a Laurent polynomial in n variables (herethe multiindex notation xα for x = (x1 , .

. . , xn ) and α = (α1 , . . . , αn ) ∈ Zn standsfor xα1 1 · · · xαnn ). The Newton polytope ∆f ⊂ Rn is the convex hull of all α ∈ Zn suchthat cα 6= 0.By definition, Newton polytope is a lattice polytope, that is, its vertices lie in Zn .Example 1.2. For n = 2 and f = 1 + 2x1 + x2 + 3x1 x2 , the Newton polytope ∆f isthe square with the vertices (0, 0), (1, 0), (0, 1) and (1, 1).Note that Laurent polynomials with complex coefficients are well-defined functionsat all points (x1 , . . . , xn ) ∈ Cn such thatSn x1 , . . . , xn 6= 0. They are regular functions∗ nnon the complex torus (C ) := C \ i=1 {xi = 0}.Theorem 1.3. [Kou] For a given lattice polytope ∆ ⊂ Rn , let f1 (x1 , .

. . , xn ),. . . ,fn (x1 , . . . , xn ) be a generic collection of Laurent polynomials with the Newton polytope ∆. Then the system f1 = . . . = fn = 0 has n!Volume(∆) solutions in thecomplex torus (C∗ )n .3The Kouchnirenko theorem can be viewed as a generalization of the classicalBezout theorem.

The Newton polytope serves as a refinement of the degree ofa polynomial. This makes the Kouchnirenko theorem applicable to collections ofpolynomials which are not generic among all polynomials of given degree but onlyamong polynomials with given Newton polytope. For instance, the Kouchnirenkotheorem applied to a pair of generic polynomials with Newton polytope as in Example 1.2 yields the correct answer 2 while Bezout theorem yields an incorrect answer4 (because of two extraneous solutions at infinity). A more geometric viewpoint onthe Bezout theorem and its extensions stems from enumerative geometry and willbe discussed in the next section.

The Koushnirenko theorem was extended to thesystems of Laurent polynomials with distinct Newton polytopes by David Bernsteinand Khovanskii using mixed volumes of polytopes [B75]. Further generalizationsinclude explicit formulas for the genus and Euler characteristic of complete intersections {f1 = 0} ∩ . . . ∩ {fm = 0} in (C∗ )n for m < n [Kh78].We now consider a bit more general situation. Fix a finite-dimensional vectorspace V ⊂ C(x1 , . . . , xn ) of rational functions on Cn .

Let f1 ,. . . , fn be a genericcollection of functions from V , and X0 ⊂ Cn an open dense subset obtained byremoving poles of these functions. How many solutions does a system f1 = . . . =fn = 0 have in X0 ? For instance, if V is the space spanned by all Laurent polynomialswith a given Newton polytope, and X0 = (C∗ )n , then the answer is given by theKouchnirenko theorem.

Here is a simple non-toric example from representationtheory.Example 1.4. Let n = 3. Consider the adjoint representation of GL3 (C) on the spaceEnd(C3 ) of all linear operators on C3 . That is, g ∈ GL3 (C) acts on an operatorX ∈ End(C3 ) as follows:Ad(g) : X 7→ gXg −1 .Let U − ⊂ GL3 (C) be the subgroup of lower triangular unipotent matrices: 1 0 0−3x1 1 0U =| (x1 , x2 , x3 ) ∈ C . x x 123To define a subspace V ⊂ C(x1 , x2 , x3 ) we restrict functions from the dual spaceEnd∗ (C3 ) to the U − -orbit Ad(U − )E13 of the operator E13 := e1 ⊗ e∗3 ∈ End(C3 )(here e1 , e2 , e3 is the standard basis in C3 ). More precisely, a linear functionf ∈ End∗ (C3 ) yields the polynomial fˆ(x1 , x2 , x3 ) as follows:−1 1 0 00 0 11 0 0fˆ(x1 , x2 , x3 ) := f x1 1 0  0 0 0  x1 1 0  x2 x3 10 0 0x2 x3 1It is easy to check that the space V is spanned by 8 polynomials: 1, x1 , x2 , x3 ,x1 x2 − x21 x3 , x1 x3 , x2 x3 , x22 − x1 x2 x3 .

It will be clear from the next section thatthe Kouchnirenko theorem does not apply to the space V , that is, the normalized4volume of the Newton polytope of a generic polynomial from V is bigger than thenumber of solutions of a generic system f1 = f2 = f3 = 0 with fi ∈ V .To assign the Newton–Okounkov convex body to V we need an extra ingredient.Choose a translation-invariant total order on the lattice Zn (e.g., we can take thelexicographic order).

Consider a mapv : C(x1 , . . . , xn ) \ {0} → Zn ,that behaves like the lowest order term of a polynomial, namely: v(f + g) ≥min{v(f ), v(g)} and v(f g) = v(f ) + v(g) for all nonzero f, g. Recall that mapswith such properties are called valuations. A straightforward construction of valuations is shown in Example 1.7 below.Definition 1.5. The Newton–Okounkov convex body ∆v (V ) is the closure of the convex hull of the set∞ [v(f )k⊂ Rn .|f ∈Vkk=1By V k we denote the subspace spanned by the k-th powers of the functions from V .Different valuations might yield different Newton–Okounkov convex bodies.

Animportant application of Newton–Okounkov bodies is the following analog of Kouchnirenko theorem. Recall that by X0 ⊂ Cn we denoted an open dense subset whereall functions from V are regular (that is, do not have poles).Theorem 1.6. [KaKh, LM] If V is sufficiently big, then a generic system f1 =. . . = fn = 0 with fi ∈ V has n!Volume(∆v (V )) solutions in X0 .In particular, it follows that all Newton–Okounkov convex bodies for V have thesame volume. For more details (in particular, for the precise meaning of “sufficientlybig”) we refer the reader to [KaKh, Theorem 4.9].Example 1.7. Let V be the space from Example 1.4.

Define a valuation v by assigningto a polynomial f ∈ C[x1 , x2 , x3 ] its lowest order term with respect to the lexicographic ordering of monomials. More precisely, we say that xk11 xk22 xk33  xl11 xl22 xl33 iffthere exists j ≤ 3 such that ki = li for i < j and kj > lj . It is easy to check thatv(V ) consists of 8 lattice points (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 0), (1, 0, 1),(0, 1, 1), (0, 2, 0). Their convex hull is depicted on Figure 1. This is the FFLV polytope F F LV (1, 1) for the adjoint representation of GL3 (in this case, it happens to beunimodularly equivalent to the GZ polytope). In particular, F F LV (1, 1) ⊂ ∆v (V ).2.

Spherical varietiesIn this section, we give a brief introduction to geometry of spherical varieties forthe general mathematical audience. Spherical varieties arise naturally in enumerative geometry. Recall two classical problems of enumerative geometry from the19-th century.5Figure 1Problem 2.1 (Schubert). How many lines in a 3-space intersect four given lines ingeneral position?We can identify lines in CP3 with vector planes in C4 , that is, a line can be viewedas a point on the Grassmannian G(2, 4).

The condition that a line l ∈ G(2, 4)intersects a fixed line l1 defines a hypersurface H1 ⊂ G(2, 4). Hence, the problemreduces to computing the number of intersection points of four hypersurfaces inG(2, 4). It is not hard to check that the hypersurface H1 is just a hyperplane sectionof the Grassmannian under the Plücker embedding G(2, 4) ,→ P(Λ2 C4 ) ' CP5 . Theimage of the Grassmannian is a quadric in CP5 . The number of intersection pointsof a quadric in CP5 with four hyperplanes in general position is equal to 2 by theBezout theorem.

Hence, the answer to the Schubert problem is 2.Problem 2.2 (Steiner). How many smooth conics are tangent to five given conics?Similarly to the Schubert problem, we can identify conics with points in CP5 ,namely, the conic given by an equation ax2 + bxy + cy 2 + dxz + eyz + f z 2 = 0corresponds to the point (a : b : c : d : e : f ) ∈ CP5 . Smooth conics form an opensubset C ⊂ CP5 (the complement CP5 \ C is the zero set of the discriminant). Thecondition that a conic is tangent to a given conic defines a hypersurface in CP5 ofdegree 6. Using Bezout theorem in CP5 one might guess (as Jacob Steiner himselfdid) that the answer to the Steiner problem is 65 . However, the correct answer ismuch smaller. This is similar to the difference between the Bezout and Kouchnirenkotheorems: the former yields extraneous solutions that have no enumerative meaning.The correct answer was found by Michel Chasles who used (in modern terms) awonderful compactification of C, namely, the space of complete conics.Hermann Schubert developed a powerful general method (calculus of conditions)for solving problems of enumerative geometry such as Problems 2.1, 2.2.

In a sense,his method was based on an informal version of intersection theory. The 15-th6Hilbert problem asked for a rigorous foundation of Schubert calculus1. In the firsthalf of the 20-th century, these foundations were developed both in the topological(cohomology rings) and algebraic (Chow rings) settings. However, Schubert’s versionof intersection theory was formalized only in the 1980-s by Corrado De Concini andClaudio Procesi [CP85].Let G be a connected reductive group, and H a spherical algebraic subgroup, thatis, a Borel subgroup B ⊂ G acts on G/H with an open dense orbit. For a sphericalhomogeneous space G/H (not necessarily compact), De Concini and Procesi constructed the ring of conditions of G/H that encodes simultaneously all enumerativeproblems on G/H. It is easy to check that complex torus (C∗ )n , GrassmannianG(2, 4) and the space C of smooth conics considered above are spherical homogenous spaces under the reductive groups (C∗ )n , GL4 (C) and GL3 (C), respectively.

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