arXiv subjects
Publications and source records attributed to Sergey Finashin.
We give a complete deformation classification of real Zariski sextics, that is of generic apparent contours of nonsingular real cubic surfaces. As a by-product, we observe a certain "reversion" duality in the set of deformation classes of these sextics.
We present an infinite sequence of smooth embeddings of a connected sum of 6 projective planes in the 4-sphere, which are all ambient homeomorphic, but pairwise ambient non-diffeomorphic. The double covers of the 4-sphere ramified along these surfaces form a family of the exotic $\Bbb CP^2#5\bar{\Bbb CP^2}$ constructed recently by Park, Stipsicz and Szabó.
We study the invariants of surfaces in 4-manifolds extracted from the Seiberg-Witten and the Ozsvath-Szabo invariants of their fiber sums with auxiliary Lefschetz fibrations. Such invariants involve relative Spin_c structures and can be treated as refinements of the usual Seiberg-Witten and Ozsvath-Szabo invariants. We prove for instance the adjunction inequality for membranes, which estimates their genus.
A non-singular connected algebraic curve $A$ in a simply connected algebraic surface $X$ can be knotted so that its homology class and the fundamental group of its complement in $X$ is preserved, provided $A$ is sufficiently complex (not too ``rigid''). For example, it is true if $A$ admits a degeneration to an irreducible curve $A_0$ having a unique singularity of the type $X_9$ (a non-degenerate quadriple point), or more complicated one, and $A.A>16$. This generalizes the previous result of the author which concerns the curves in $CP^2$ of degree $d>4$ (the old preprint is included as a part of the current one).
I construct "fake algebraic curves" in $Cp^2$. More precisely, for any k>2, I construct infinitely many pairwise smoothly non-isotopic (and moreover not ambient diffeomorphic) smooth surfaces $F\subset Cp^2$ homeomorphic to a non-singular algebraic curve of degree 2k, realizing the same homology class as such a curve and having abelian fundamental group of the complement $Cp^2-F$. This gives an answer to Problem 4.110 of Kirby's list.
The topology of the orbit space, $Y$, for the action of the complex conjugation on a complex surface, $X$, defined over reals, is studied. I give a criterion for blow-up stable triviality of $Y$ (which implies vanishing of its Seiberg-Witten invariants). The main result concerns the double planes branched along the complexification of reducible real curves with 2 non-singular components. In connection with it, I analize the real singularities of $X$ which become smooth in $Y$ (after taking quotient).
The Arnold inequalities characterizing the topology of non-singular plane real algebraic curves and the generalization of these inequalities for nodal curves by Viro are extended in this paper for the curves whose singularities have non-degenerated Milnor form. This involves an invariant of such real singularities, the real linking form, which is calculated.
Quotients $Y=X/conj$ of complex surfaces by anti-holomorphic involutions $conj\: X\to X$ tend to be completely decomposable when they are simply connected, i.e., split into connected sums, $n CP^2\#m\barCP2$, if $w_2(Y)\ne0$, or into $n(S^2\times S^2)$ if $w_2(Y)=0$. If $X$ is a double branched covering over $CP^2$, this phenomenon is related to unknottedness of Arnold surfaces in $S^4=CP^2/conj$, which was conjectured by V.Rokhlin. The paper contains proof of Rokhlin Conjecture and of decomposability of quotients for plenty of double planes and in certain other cases. This results give, in particular, an elementary proof of Donaldson's result on decomposability of $Y$ for K3 surfaces.
It is proven that for any topological or analytical types of isolated singular points of plane curves, there exists a non-real irreducible plane algebraic curve of degree $d$ which goes through $d^2$ real distinct points and has imaginary singular points of the given types. This result is used to construct a series of examples of complex algebraic surfaces $X$ defined over $\R$ whose quotients $Y=X/\conj$ by the complex conjugation $\conj$ are $Spin$ simply connected 4-manifolds with signature $16k$, for arbitrary integer $k>0$.