Kiritchenko_Summary_2018_title (1136187), страница 2
Текст из файла (страница 2)
Inparticular, their rings of conditions are well-defined. Elements of the ring of conditions are classes of subvarieties of G/H under natural numerical equivalence relation.Namely, two subvarieties of the same dimension are equivalent if their intersectionindices with any subvariety of complementary dimension are equal. A transitiveaction of G is used to overcome the usual difficulty of intersecting non-transversesubvarieties. The ring product corresponds to the intersection of subvarieties.In particular, many problems of enumerative geometry (including Problems 2.1,2.2) reduce to computation of the self-intersection index of a hypersurface in G/H.In the toric case, the Kouchnirenko theorem yields an explicit formula for the selfintersection index of a hypersurface {f = 0} where f is a generic polynomial witha given Newton polytope.
In the spherical case, explicit formulas were obtained byBoris Kazarnovskii (case of (G × G)/Gdiag ) and Michel Brion (general case) [Kaz,Br89]. Though the Brion–Kazarnovskii formula was originally stated in differentterms, it can be reformulated using Newton–Okounkov polytopes [KaKh2].Example 2.3.
We now place Example 1.4 into the context of enumerative geometryand spherical homogeneous spaces. Let X = {(V 1 ⊂ V 2 ⊂ C3 ) | dim V i = i} bethe variety of complete flags in C3 . This is a homogeneous space under the actionof GL3 (C), namely, X = GL3 (C)/B where a Borel subgroup B is the subgroup ofupper-triangular matrices. It is easy to check that B acts on X with an open denseorbit U − B/B ' U − , in particular, X is spherical.We say that two flags V 1 ⊂ V 2 and W 1 ⊂ W 2 in C3 are not in general position ifeither V 1 ⊂ W 2 or W 1 ⊂ V 2 . How many flags in C3 are not in general position withthree given flags? On the one hand, it is easy to show that the answer is 6 usinghigh school geometry.
On the other hand, the same answer can be found using thesimplest projective embedding of X:Segrep : X ,→ P(C3 )×P(Λ2 C3 ) ,→ P(End(C3 ));1Dasp : (V 1 , V 2 ) 7→ V 1 ×V 2 7→ V 1 ⊗Λ2 V 2 ,Problem besteht darin, diejenigen geometrischen Anzahlen strenge und unter genauerFeststellung der Grenzen ihrer Gültigkeit zu beweisen, die insbesondere Schubert auf Grund dessogenannten Princips der speciellen Lage mittelst des von ihm ausgebildeten Abzählungskalkülsbestimmt hat (Hilbert).7and counting the number of intersection points of p(X) with 3 generic hyperplanesin CP8 (that is, the degree of p(X)).
Restricting the map p to the open dense B-orbitU − ⊂ X we get that the latter problem reduces to the problem from Example 1.4.In particular, we can show that the inclusion F F LV (1, 1) ⊂ ∆v (V ) is an equality.Indeed, by Theorem 1.6 the volume of ∆v (V ) times 3! is equal to the degree ofp(X), that is, to 6. Hence, the volume of ∆v (V ) is equal to 1. Since the volume ofF F LV (1, 1) is also equal to 1, the inclusion F F LV (1, 1) ⊂ ∆v (V ) implies the exactequality.3. Results and publicationsThis section contains a brief overview of the results of the habilitation thesis.
Themain purpose is to place these results into the general context (without going intoo much detail) and provide references to more recent developments. The precisestatements and all necessary definitions can be found in [1]–[10] (see the list ofpublications in the end of this section).3.1. Euler characteristic of complete intersections in reductive groups.
Inthe torus case, almost all invariants of a complete intersection Y = {f1 = 0} ∩. . . ∩ {fm = 0} ⊂ (C∗ )n can be computed in terms of Newton polytopes ∆f1 ,. . . ,∆fm . In the reductive case (for (G × G)/Gdiag ), the Brion–Kazarnovskii formula form = n (that is, for a zero-dimensional Y ) was the only explicit formula for quitesome time. Note that it can be interpreted as the formula for the (topological) Eulercharacteristic χ(Y ).
The main result of [1,2] is an explicit formula for χ(Y ) for allm ≤ n. The formula is obtained in two steps. First, (non-compact versions of)Chern classes of reductive groups are defined and studied as elements of the ringof conditions [1]. Second, an algorithm of De Concini–Procesi [CP83] is used tocompute intersection indices of these Chern classes with complete intersections inorder to use the adjunction formula [2].It is proved in [2] that the De Concini–Procesi algorithm works for the Chernclasses, which do not in general lie in the subring of conditions generated by completeintersections.
It is also shown how to convert this algorithm to an explicit formulausing the weight polytope of the representation associated with Y . In particular, thisyields another proof of the Brion–Kazarnovskii formula. While formulas of [1,2] donot use Newton–Okounkov polytopes directly they have the same convex geometricflavor (see Section 4.1 for a formula in the case m = 1). Recently, more invariants(in particular, arithmetic genus) of complete intersections in spherical homogenousspaces were found in terms of Newton–Okounkov polytopes (see [KaKh3], whichalso contains a historical account of this problem).3.2.
Convex geometric models for Schubert calculus. The ring of conditionsof a complex torus is generated by classes of hypersurfaces. The same is true forcomplete flag varieties G/B but not necessarily true for more general spherical homogeneous spaces. Thus complete flag varieties are first candidates for applying8convex geometric methods to computation of intersection products. Note that sinceG/B is compact its ring of conditions coincides with the Chow ring, and the latterhas a natural basis of Schubert cycles given by the closures of B-orbits. In [3,5,8],we built convex geometric models for Schubert calculus on G/B and found convexgeometric realizations of Schubert cycles.In [3], the Chevalley-Pieri formula for G/B in type A is interpreted in terms of theGZ polytopes.
In [5] (joint work with Evgeny Smirnov and Vladlen Timorin), wedevelop framework for realizing Schubert cycles by linear combinations of faces ofa polytope so that the intersection of faces corresponds to the intersection productof Schubert cycles. In type A, we get explicit realizations for every Schubert cyclethat allow us to represent the product of any two Schubert cycles by a nonnegativelinear combination of faces of a GZ polytope.
This result was inspired by the Ph.D.thesis of Mikhail Kogan who first associated faces of the GZ polytopes with Schubertvarieties [Ko]. We also get formulas for the Demazure characters of Schubert varietiesin terms of exponential sums over lattice points in Kogan faces of the GZ polytope.In [8], a geometric algorithm is developed for realizing Schubert cycles by facesof polytopes in arbitrary type. In type A, this algorithm reduces to the Knutson–Miller mitosis on pipe dreams.
In types B and C, it reduces to a new combinatorialalgorithm that might yield explicit realizations of Schubert cycles in symplectic andorthogonal flag varieties by faces of symplectic GZ polytopes (see Section 4.2 formore details).In [7] (joint work with Pavel Gusev and Vladlen Timorin), we study combinatoricsof the GZ polytopes corresponding to different partial flag varieties (or in combinatorial terms, to partitions 1i1 2i2 . .
. k ik ). Note that all GZ polytopes for a givenpartition have the same combinatorial type. We determine a recurrence relationfor the number of vertices V (1i1 2i2 . . . k ik ) and a PDE for the exponential generating function of the numbers V (1i1 2i2 . . . k ik ). Recently, the recurrence relation wasextended to f -vectors of the GZ polytopes in [ACK].3.3. Reincarnations of divided difference operators (DDO). DDO and Demazure operators (also known as push-pull operators) are important tools in Schubert calculus and representation theory. They were used in [BGG] and [D] to expressinductively Schubert cycles on complete flag varieties as polynomials in the Chernclasses of line bundles (both in cohomology and K-theory).
Another applicationis the Demazure formula for the Demazure characters of Schubert varieties [D]. In[4,6], we define and use analogs of DDO in (equivariant) Schubert calculus for algebraic cobordism. In [9], a convex geometric version of Demazure operators isdefined and used to construct polytopes that capture Demazure characters.
In [10],Newton–Okounkov polytopes of flag varieties for a geometric valuation are computedexplicitly using a simple convex geometric construction, which is also motivated byDDO.Note that the classical DDO and Demazure operators can be defined uniformlyusing the additive (Chow rings) and multiplicative (K-theory) formal group laws,9respectively. In [4] (joint work with Jens Hornbostel), we define DDO for the universal formal group law and apply them to study Schubert calculus in the algebraiccobordism ring (as defined by Levine–Morel and Levine–Pandharipande) of complete flag varieties.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
